Theorem grpinvpropdg 13721

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 13520	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
7	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
8	1, 7	eqeq12d 2246	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
9	8	anass1rs 573	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
10	9	riotabidva 5999	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
11	10	mpteq2dva 4184	. . 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 5982	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) = (℩𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
13	2, 12	mpteq12dv 4176	. . 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 5982	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)) = (℩𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
15	3, 14	mpteq12dv 4176	. . 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 2272	. 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 2231	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2231	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
19		eqid 2231	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
20		eqid 2231	. . . 4 ⊢ (invg‘𝐾) = (invg‘𝐾)
21	17, 18, 19, 20	grpinvfvalg 13688	. . 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 2231	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
24		eqid 2231	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
25		eqid 2231	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
26		eqid 2231	. . . 4 ⊢ (invg‘𝐿) = (invg‘𝐿)
27	23, 24, 25, 26	grpinvfvalg 13688	. . 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 2274	1 ⊢ (𝜑 → (invg‘𝐾) = (invg‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202   ↦ cmpt 4155  ‘cfv 5333  ℩crio 5980  (class class class)co 6028  Basecbs 13145  +_gcplusg 13223  0_gc0g 13402  inv_gcminusg 13647

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9187  df-ndx 13148  df-slot 13149  df-base 13151  df-0g 13404  df-minusg 13650

This theorem is referenced by:  grpsubpropdg  13750  grpsubpropd2  13751  mulgpropdg  13814  invrpropdg  14227  rlmvnegg  14544