Theorem grpinvpropdg 12804

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 12658	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
7	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
8	1, 7	eqeq12d 2190	. . . . . 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 5837	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
11	10	mpteq2dva 4088	. . 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 5822	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
13	2, 12	mpteq12dv 4080	. . 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 5822	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)) = (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
15	3, 14	mpteq12dv 4080	. . 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 2216	. 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 2175	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2175	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
19		eqid 2175	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2175	. . . 4 ⊢ (invg‘𝐾) = (invg‘𝐾)
21	17, 18, 19, 20	grpinvfvalg 12775	. . 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 2175	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
24		eqid 2175	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
25		eqid 2175	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
26		eqid 2175	. . . 4 ⊢ (invg‘𝐿) = (invg‘𝐿)
27	23, 24, 25, 26	grpinvfvalg 12775	. . 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 2218	1 ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2146   ↦ cmpt 4059  ‘cfv 5208  ℩crio 5820  (class class class)co 5865  Basecbs 12428  +_gcplusg 12492  0_gc0g 12626  inv_gcminusg 12739

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-inn 8891  df-ndx 12431  df-slot 12432  df-base 12434  df-0g 12628  df-minusg 12742

This theorem is referenced by:  grpsubpropdg  12833  grpsubpropd2  12834