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Mirrors > Home > MPE Home > Th. List > grpinvpropd | Structured version Visualization version GIF version |
Description: If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
grpinvpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
grpinvpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
grpinvpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
Ref | Expression |
---|---|
grpinvpropd | ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvpropd.3 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
2 | grpinvpropd.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
3 | grpinvpropd.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
4 | 2, 3, 1 | grpidpropd 17468 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
5 | 4 | adantr 466 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿)) |
6 | 1, 5 | eqeq12d 2786 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
7 | 6 | anass1rs 626 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
8 | 7 | riotabidva 6769 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
9 | 8 | mpteq2dva 4878 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) = (𝑦 ∈ 𝐵 ↦ (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))) |
10 | 2 | riotaeqdv 6754 | . . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) |
11 | 2, 10 | mpteq12dv 4867 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) = (𝑦 ∈ (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))) |
12 | 3 | riotaeqdv 6754 | . . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)) = (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
13 | 3, 12 | mpteq12dv 4867 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) = (𝑦 ∈ (Base‘𝐿) ↦ (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))) |
14 | 9, 11, 13 | 3eqtr3d 2813 | . 2 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) = (𝑦 ∈ (Base‘𝐿) ↦ (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))) |
15 | eqid 2771 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | eqid 2771 | . . 3 ⊢ (+g‘𝐾) = (+g‘𝐾) | |
17 | eqid 2771 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
18 | eqid 2771 | . . 3 ⊢ (invg‘𝐾) = (invg‘𝐾) | |
19 | 15, 16, 17, 18 | grpinvfval 17667 | . 2 ⊢ (invg‘𝐾) = (𝑦 ∈ (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) |
20 | eqid 2771 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
21 | eqid 2771 | . . 3 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
22 | eqid 2771 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
23 | eqid 2771 | . . 3 ⊢ (invg‘𝐿) = (invg‘𝐿) | |
24 | 20, 21, 22, 23 | grpinvfval 17667 | . 2 ⊢ (invg‘𝐿) = (𝑦 ∈ (Base‘𝐿) ↦ (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
25 | 14, 19, 24 | 3eqtr4g 2830 | 1 ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ↦ cmpt 4863 ‘cfv 6031 ℩crio 6752 (class class class)co 6792 Basecbs 16063 +gcplusg 16148 0gc0g 16307 invgcminusg 17630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-0g 16309 df-minusg 17633 |
This theorem is referenced by: grpsubpropd 17727 grpsubpropd2 17728 mulgpropd 17791 invrpropd 18905 rlmvneg 19421 matinvg 20440 tngngp3 22679 |
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