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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 17576 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
5 | 4 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿)) |
6 | 1, 5 | eqeq12d 2814 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
7 | 6 | anass1rs 646 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
8 | 7 | riotabidva 6855 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
9 | 8 | mpteq2dva 4937 | . . 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 6840 | . . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) |
11 | 2, 10 | mpteq12dv 4926 | . . 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 6840 | . . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)) = (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
13 | 3, 12 | mpteq12dv 4926 | . . 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 2841 | . 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 2799 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | eqid 2799 | . . 3 ⊢ (+g‘𝐾) = (+g‘𝐾) | |
17 | eqid 2799 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
18 | eqid 2799 | . . 3 ⊢ (invg‘𝐾) = (invg‘𝐾) | |
19 | 15, 16, 17, 18 | grpinvfval 17776 | . 2 ⊢ (invg‘𝐾) = (𝑦 ∈ (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) |
20 | eqid 2799 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
21 | eqid 2799 | . . 3 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
22 | eqid 2799 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
23 | eqid 2799 | . . 3 ⊢ (invg‘𝐿) = (invg‘𝐿) | |
24 | 20, 21, 22, 23 | grpinvfval 17776 | . 2 ⊢ (invg‘𝐿) = (𝑦 ∈ (Base‘𝐿) ↦ (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
25 | 14, 19, 24 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ↦ cmpt 4922 ‘cfv 6101 ℩crio 6838 (class class class)co 6878 Basecbs 16184 +gcplusg 16267 0gc0g 16415 invgcminusg 17739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-0g 16417 df-minusg 17742 |
This theorem is referenced by: grpsubpropd 17836 grpsubpropd2 17837 mulgpropd 17897 invrpropd 19014 rlmvneg 19530 matinvg 20549 tngngp3 22788 |
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