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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 2737 | . . . 4 ⊢ (𝜑 → 𝑥 = 𝑥) | |
4 | eqidd 2737 | . . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾)) | |
5 | nmpropd.2 | . . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) | |
6 | 5 | oveqdr 7381 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
7 | 4, 1, 6 | grpidpropd 18509 | . . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
8 | 2, 3, 7 | oveq123d 7374 | . . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿))) |
9 | 1, 8 | mpteq12dv 5194 | . 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))) |
10 | eqid 2736 | . . 3 ⊢ (norm‘𝐾) = (norm‘𝐾) | |
11 | eqid 2736 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | eqid 2736 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
13 | eqid 2736 | . . 3 ⊢ (dist‘𝐾) = (dist‘𝐾) | |
14 | 10, 11, 12, 13 | nmfval 23928 | . 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) |
15 | eqid 2736 | . . 3 ⊢ (norm‘𝐿) = (norm‘𝐿) | |
16 | eqid 2736 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
17 | eqid 2736 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
18 | eqid 2736 | . . 3 ⊢ (dist‘𝐿) = (dist‘𝐿) | |
19 | 15, 16, 17, 18 | nmfval 23928 | . 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))) |
20 | 9, 14, 19 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 +gcplusg 17125 distcds 17134 0gc0g 17313 normcnm 23916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7356 df-0g 17315 df-nm 23922 |
This theorem is referenced by: sranlm 24032 rlmnm 24037 zlmnm 32416 |
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