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| Mirrors > Home > MPE Home > Th. List > nmpropd | Structured version Visualization version GIF version | ||
| Description: Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| Ref | Expression |
|---|---|
| nmpropd.1 | ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
| nmpropd.2 | ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) |
| nmpropd.3 | ⊢ (𝜑 → (dist‘𝐾) = (dist‘𝐿)) |
| Ref | Expression |
|---|---|
| nmpropd | ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nmpropd.1 | . . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) | |
| 2 | nmpropd.3 | . . . 4 ⊢ (𝜑 → (dist‘𝐾) = (dist‘𝐿)) | |
| 3 | eqidd 2738 | . . . 4 ⊢ (𝜑 → 𝑥 = 𝑥) | |
| 4 | eqidd 2738 | . . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾)) | |
| 5 | nmpropd.2 | . . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) | |
| 6 | 5 | oveqdr 7459 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
| 7 | 4, 1, 6 | grpidpropd 18675 | . . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
| 8 | 2, 3, 7 | oveq123d 7452 | . . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿))) |
| 9 | 1, 8 | mpteq12dv 5233 | . 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))) |
| 10 | eqid 2737 | . . 3 ⊢ (norm‘𝐾) = (norm‘𝐾) | |
| 11 | eqid 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 12 | eqid 2737 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
| 13 | eqid 2737 | . . 3 ⊢ (dist‘𝐾) = (dist‘𝐾) | |
| 14 | 10, 11, 12, 13 | nmfval 24601 | . 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) |
| 15 | eqid 2737 | . . 3 ⊢ (norm‘𝐿) = (norm‘𝐿) | |
| 16 | eqid 2737 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
| 17 | eqid 2737 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
| 18 | eqid 2737 | . . 3 ⊢ (dist‘𝐿) = (dist‘𝐿) | |
| 19 | 15, 16, 17, 18 | nmfval 24601 | . 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))) |
| 20 | 9, 14, 19 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 distcds 17306 0gc0g 17484 normcnm 24589 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-0g 17486 df-nm 24595 |
| This theorem is referenced by: sranlm 24705 rlmnm 24710 zlmnm 33965 |
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