Theorem grpinvpropdg 12944

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.)

Hypotheses

Ref	Expression
grpinvpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
grpinvpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
grpinvpropdg.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
grpinvpropdg.l	⊢ (𝜑 → 𝐿 ∈ 𝑊)
grpinvpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))

Assertion

Ref	Expression
grpinvpropdg	⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem grpinvpropdg

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		grpinvpropdg.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		grpinvpropdg.l	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ 𝑊)
6	2, 3, 4, 5, 1	grpidpropdg 12792	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
7	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
8	1, 7	eqeq12d 2192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
9	8	anass1rs 571	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
10	9	riotabidva 5846	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
11	10	mpteq2dva 4093	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) = (𝑦 ∈ 𝐵 ↦ (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))))
12	2	riotaeqdv 5831	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
13	2, 12	mpteq12dv 4085	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))) = (𝑦 ∈ (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))))
14	3	riotaeqdv 5831	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)) = (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
15	3, 14	mpteq12dv 4085	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))) = (𝑦 ∈ (Base‘𝐿) ↦ (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))))
16	11, 13, 15	3eqtr3d 2218	. 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))))
17		eqid 2177	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2177	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
19		eqid 2177	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2177	. . . 4 ⊢ (invg‘𝐾) = (invg‘𝐾)
21	17, 18, 19, 20	grpinvfvalg 12914	. . 3 ⊢ (𝐾 ∈ 𝑉 → (invg‘𝐾) = (𝑦 ∈ (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))))
22	4, 21	syl 14	. 2 ⊢ (𝜑 → (invg‘𝐾) = (𝑦 ∈ (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾))))
23		eqid 2177	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
24		eqid 2177	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
25		eqid 2177	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
26		eqid 2177	. . . 4 ⊢ (invg‘𝐿) = (invg‘𝐿)
27	23, 24, 25, 26	grpinvfvalg 12914	. . 3 ⊢ (𝐿 ∈ 𝑊 → (invg‘𝐿) = (𝑦 ∈ (Base‘𝐿) ↦ (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))))
28	5, 27	syl 14	. 2 ⊢ (𝜑 → (invg‘𝐿) = (𝑦 ∈ (Base‘𝐿) ↦ (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))))
29	16, 22, 28	3eqtr4d 2220	1 ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   ↦ cmpt 4064  ‘cfv 5216  ℩crio 5829  (class class class)co 5874  Basecbs 12461  +_gcplusg 12535  0_gc0g 12704  inv_gcminusg 12877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-inn 8919  df-ndx 12464  df-slot 12465  df-base 12467  df-0g 12706  df-minusg 12880

This theorem is referenced by:  grpsubpropdg  12973  grpsubpropd2  12974  mulgpropdg  13023  invrpropdg  13316