![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 18522 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
5 | 4 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿)) |
6 | 1, 5 | eqeq12d 2749 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
7 | 6 | anass1rs 654 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
8 | 7 | riotabidva 7334 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
9 | 8 | mpteq2dva 5206 | . . 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 7315 | . . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) |
11 | 2, 10 | mpteq12dv 5197 | . . 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 7315 | . . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)) = (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
13 | 3, 12 | mpteq12dv 5197 | . . 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 2781 | . 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 2733 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | eqid 2733 | . . 3 ⊢ (+g‘𝐾) = (+g‘𝐾) | |
17 | eqid 2733 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
18 | eqid 2733 | . . 3 ⊢ (invg‘𝐾) = (invg‘𝐾) | |
19 | 15, 16, 17, 18 | grpinvfval 18794 | . 2 ⊢ (invg‘𝐾) = (𝑦 ∈ (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) |
20 | eqid 2733 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
21 | eqid 2733 | . . 3 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
22 | eqid 2733 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
23 | eqid 2733 | . . 3 ⊢ (invg‘𝐿) = (invg‘𝐿) | |
24 | 20, 21, 22, 23 | grpinvfval 18794 | . 2 ⊢ (invg‘𝐿) = (𝑦 ∈ (Base‘𝐿) ↦ (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
25 | 14, 19, 24 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5189 ‘cfv 6497 ℩crio 7313 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 0gc0g 17326 invgcminusg 18754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-riota 7314 df-ov 7361 df-0g 17328 df-minusg 18757 |
This theorem is referenced by: grpsubpropd 18857 grpsubpropd2 18858 mulgpropd 18923 invrpropd 20134 rlmvneg 20693 matinvg 21783 tngngp3 24036 |
Copyright terms: Public domain | W3C validator |