Theorem grpinvpropdg 13277

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 13076	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
7	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
8	1, 7	eqeq12d 2211	. . . . . 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 5897	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
11	10	mpteq2dva 4124	. . 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 5881	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
13	2, 12	mpteq12dv 4116	. . 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 5881	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)) = (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
15	3, 14	mpteq12dv 4116	. . 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 2237	. 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 2196	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2196	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
19		eqid 2196	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2196	. . . 4 ⊢ (invg‘𝐾) = (invg‘𝐾)
21	17, 18, 19, 20	grpinvfvalg 13244	. . 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 2196	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
24		eqid 2196	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
25		eqid 2196	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
26		eqid 2196	. . . 4 ⊢ (invg‘𝐿) = (invg‘𝐿)
27	23, 24, 25, 26	grpinvfvalg 13244	. . 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 2239	1 ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167   ↦ cmpt 4095  ‘cfv 5259  ℩crio 5879  (class class class)co 5925  Basecbs 12703  +_gcplusg 12780  0_gc0g 12958  inv_gcminusg 13203

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-0g 12960  df-minusg 13206

This theorem is referenced by:  grpsubpropdg  13306  grpsubpropd2  13307  mulgpropdg  13370  invrpropdg  13781  rlmvnegg  14097