Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2824 | . . . 4 ⊢ (𝜑 → 𝑥 = 𝑥) | |
4 | eqidd 2824 | . . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾)) | |
5 | nmpropd.2 | . . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) | |
6 | 5 | oveqdr 7186 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
7 | 4, 1, 6 | grpidpropd 17874 | . . . 4 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
8 | 2, 3, 7 | oveq123d 7179 | . . 3 ⊢ (𝜑 → (𝑥(dist‘𝐾)(0g‘𝐾)) = (𝑥(dist‘𝐿)(0g‘𝐿))) |
9 | 1, 8 | mpteq12dv 5153 | . 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿)))) |
10 | eqid 2823 | . . 3 ⊢ (norm‘𝐾) = (norm‘𝐾) | |
11 | eqid 2823 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | eqid 2823 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
13 | eqid 2823 | . . 3 ⊢ (dist‘𝐾) = (dist‘𝐾) | |
14 | 10, 11, 12, 13 | nmfval 23200 | . 2 ⊢ (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g‘𝐾))) |
15 | eqid 2823 | . . 3 ⊢ (norm‘𝐿) = (norm‘𝐿) | |
16 | eqid 2823 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
17 | eqid 2823 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
18 | eqid 2823 | . . 3 ⊢ (dist‘𝐿) = (dist‘𝐿) | |
19 | 15, 16, 17, 18 | nmfval 23200 | . 2 ⊢ (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g‘𝐿))) |
20 | 9, 14, 19 | 3eqtr4g 2883 | 1 ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 distcds 16576 0gc0g 16715 normcnm 23188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-0g 16717 df-nm 23194 |
This theorem is referenced by: sranlm 23295 rlmnm 23300 zlmnm 31209 |
Copyright terms: Public domain | W3C validator |