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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 2611 | . . . 4 ⊢ (𝜑 → 𝑥 = 𝑥) | |
4 | eqidd 2611 | . . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾)) | |
5 | nmpropd.2 | . . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) | |
6 | 5 | oveqdr 6573 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
7 | 4, 1, 6 | grpidpropd 17084 | . . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
8 | 2, 3, 7 | oveq123d 6570 | . . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿))) |
9 | 1, 8 | mpteq12dv 4663 | . 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))) |
10 | eqid 2610 | . . 3 ⊢ (norm‘𝐾) = (norm‘𝐾) | |
11 | eqid 2610 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | eqid 2610 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
13 | eqid 2610 | . . 3 ⊢ (dist‘𝐾) = (dist‘𝐾) | |
14 | 10, 11, 12, 13 | nmfval 22203 | . 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) |
15 | eqid 2610 | . . 3 ⊢ (norm‘𝐿) = (norm‘𝐿) | |
16 | eqid 2610 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
17 | eqid 2610 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
18 | eqid 2610 | . . 3 ⊢ (dist‘𝐿) = (dist‘𝐿) | |
19 | 15, 16, 17, 18 | nmfval 22203 | . 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))) |
20 | 9, 14, 19 | 3eqtr4g 2669 | 1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 distcds 15777 0gc0g 15923 normcnm 22191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-0g 15925 df-nm 22197 |
This theorem is referenced by: sranlm 22298 rlmnm 22303 zlmnm 29338 |
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