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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 2736 | . . . 4 ⊢ (𝜑 → 𝑥 = 𝑥) | |
4 | eqidd 2736 | . . . . 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 18688 | . . . 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 5239 | . 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))) |
10 | eqid 2735 | . . 3 ⊢ (norm‘𝐾) = (norm‘𝐾) | |
11 | eqid 2735 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | eqid 2735 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
13 | eqid 2735 | . . 3 ⊢ (dist‘𝐾) = (dist‘𝐾) | |
14 | 10, 11, 12, 13 | nmfval 24617 | . 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) |
15 | eqid 2735 | . . 3 ⊢ (norm‘𝐿) = (norm‘𝐿) | |
16 | eqid 2735 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
17 | eqid 2735 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
18 | eqid 2735 | . . 3 ⊢ (dist‘𝐿) = (dist‘𝐿) | |
19 | 15, 16, 17, 18 | nmfval 24617 | . 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))) |
20 | 9, 14, 19 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 distcds 17307 0gc0g 17486 normcnm 24605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-0g 17488 df-nm 24611 |
This theorem is referenced by: sranlm 24721 rlmnm 24726 zlmnm 33927 |
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