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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 2741 | . . . 4 ⊢ (𝜑 → 𝑥 = 𝑥) | |
4 | eqidd 2741 | . . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾)) | |
5 | nmpropd.2 | . . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) | |
6 | 5 | oveqdr 7476 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
7 | 4, 1, 6 | grpidpropd 18700 | . . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
8 | 2, 3, 7 | oveq123d 7469 | . . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿))) |
9 | 1, 8 | mpteq12dv 5257 | . 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))) |
10 | eqid 2740 | . . 3 ⊢ (norm‘𝐾) = (norm‘𝐾) | |
11 | eqid 2740 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | eqid 2740 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
13 | eqid 2740 | . . 3 ⊢ (dist‘𝐾) = (dist‘𝐾) | |
14 | 10, 11, 12, 13 | nmfval 24622 | . 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) |
15 | eqid 2740 | . . 3 ⊢ (norm‘𝐿) = (norm‘𝐿) | |
16 | eqid 2740 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
17 | eqid 2740 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
18 | eqid 2740 | . . 3 ⊢ (dist‘𝐿) = (dist‘𝐿) | |
19 | 15, 16, 17, 18 | nmfval 24622 | . 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))) |
20 | 9, 14, 19 | 3eqtr4g 2805 | 1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 +gcplusg 17311 distcds 17320 0gc0g 17499 normcnm 24610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-0g 17501 df-nm 24616 |
This theorem is referenced by: sranlm 24726 rlmnm 24731 zlmnm 33912 |
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