Theorem grpinvpropdg 13933

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 13747	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
7	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
8	1, 7	eqeq12d 2253	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
9	8	anass1rs 577	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
10	9	riotabidva 6056	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
11	10	mpteq2dva 4221	. . 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 6039	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
13	2, 12	mpteq12dv 4213	. . 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 6039	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)) = (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
15	3, 14	mpteq12dv 4213	. . 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 2279	. 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 2238	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2238	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
19		eqid 2238	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2238	. . . 4 ⊢ (invg‘𝐾) = (invg‘𝐾)
21	17, 18, 19, 20	grpinvfvalg 13900	. . 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 2238	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
24		eqid 2238	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
25		eqid 2238	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
26		eqid 2238	. . . 4 ⊢ (invg‘𝐿) = (invg‘𝐿)
27	23, 24, 25, 26	grpinvfvalg 13900	. . 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 2281	1 ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ↦ cmpt 4192  ‘cfv 5377  ℩crio 6037  (class class class)co 6085  Basecbs 13404  +_gcplusg 13484  0_gc0g 13663  inv_gcminusg 13859

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9308  df-ndx 13407  df-slot 13408  df-base 13410  df-0g 13665  df-minusg 13862

This theorem is used by:  grpsubpropdg  13962  grpsubpropd2  13963  mulgpropdg  14020  invrpropdg  14540  rlmvnegg  14886