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 17084 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
5 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿)) |
6 | 1, 5 | eqeq12d 2625 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
7 | 6 | anass1rs 845 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
8 | 7 | riotabidva 6527 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
9 | 8 | mpteq2dva 4672 | . . 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 6512 | . . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) |
11 | 2, 10 | mpteq12dv 4663 | . . 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 6512 | . . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)) = (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
13 | 3, 12 | mpteq12dv 4663 | . . 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 2652 | . 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 2610 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | eqid 2610 | . . 3 ⊢ (+g‘𝐾) = (+g‘𝐾) | |
17 | eqid 2610 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
18 | eqid 2610 | . . 3 ⊢ (invg‘𝐾) = (invg‘𝐾) | |
19 | 15, 16, 17, 18 | grpinvfval 17283 | . 2 ⊢ (invg‘𝐾) = (𝑦 ∈ (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) |
20 | eqid 2610 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
21 | eqid 2610 | . . 3 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
22 | eqid 2610 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
23 | eqid 2610 | . . 3 ⊢ (invg‘𝐿) = (invg‘𝐿) | |
24 | 20, 21, 22, 23 | grpinvfval 17283 | . 2 ⊢ (invg‘𝐿) = (𝑦 ∈ (Base‘𝐿) ↦ (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) |
25 | 14, 19, 24 | 3eqtr4g 2669 | 1 ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 invgcminusg 17246 |
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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
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-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 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-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-minusg 17249 |
This theorem is referenced by: grpsubpropd 17343 grpsubpropd2 17344 mulgpropd 17407 invrpropd 18521 rlmvneg 19028 matinvg 20043 tngngp3 22270 |
Copyright terms: Public domain | W3C validator |