Normed real vector spaces #1704
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…shake-out-signed-rational
…shake-out-signed-rational
Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
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| ≤ zero-ℝ | ||
| by leq-sim-ℝ (right-zero-law-mul-ℝ _) | ||
| ≤ raise-ℝ l1 zero-ℝ | ||
| by leq-sim-ℝ (sim-raise-ℝ l1 zero-ℝ) | ||
| ≤ dist-Seminormed-ℝ-Vector-Space V v v | ||
| by | ||
| leq-eq-ℝ | ||
| ( inv (is-zero-diagonal-dist-Seminormed-ℝ-Vector-Space V v)) | ||
| ≤ ( map-seminorm-Seminormed-ℝ-Vector-Space V v) +ℝ | ||
| ( map-seminorm-Seminormed-ℝ-Vector-Space | ||
| ( V) | ||
| ( neg-Seminormed-ℝ-Vector-Space V v)) | ||
| by triangular-Seminormed-ℝ-Vector-Space V _ _ | ||
| ≤ ( map-seminorm-Seminormed-ℝ-Vector-Space V v) +ℝ | ||
| ( map-seminorm-Seminormed-ℝ-Vector-Space V v) | ||
| by | ||
| leq-eq-ℝ | ||
| ( ap-add-ℝ | ||
| ( refl) | ||
| ( seminorm-neg-Seminormed-ℝ-Vector-Space V v)) | ||
| ≤ real-ℕ 2 *ℝ map-seminorm-Seminormed-ℝ-Vector-Space V v | ||
| by leq-eq-ℝ (inv (left-mul-real-ℕ 2 _))) |
Collaborator
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It should be a lemma that zero-ℝ is less than or equal to any nonnegative real, and real-ℕ 2 *ℝ map-seminorm-Seminormed-ℝ-Vector-Space V v is nonnegative.
Collaborator
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Oh, this whole chain of inequalities should be simplified to the argument that multiplying by two preserves inequality, and zero is less than map-seminorm-Seminormed-ℝ-Vector-Space V v.
Collaborator
Author
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I think your reasoning is circular. We don't know beforehand that "real-ℕ 2 *ℝ map-seminorm-Seminormed-ℝ-Vector-Space V v is nonnegative" or that "zero is less than map-seminorm-Seminormed-ℝ-Vector-Space V v", that's what we're here to prove.
Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
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Builds on #1689. Completes #1702 .