Theorem grpidpropdg 12685

Description: If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)

Hypotheses

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

Assertion

Ref	Expression
grpidpropdg	⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))

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

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

Proof of Theorem grpidpropdg

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpidpropd.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
2	1	eqeq1d 2186	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = 𝑦 ↔ (𝑥(+g‘𝐿)𝑦) = 𝑦))
3	1	oveqrspc2v 5896	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
4	3	oveqrspc2v 5896	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(+g‘𝐾)𝑥) = (𝑦(+g‘𝐿)𝑥))
5	4	ancom2s 566	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(+g‘𝐾)𝑥) = (𝑦(+g‘𝐿)𝑥))
6	5	eqeq1d 2186	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦(+g‘𝐾)𝑥) = 𝑦 ↔ (𝑦(+g‘𝐿)𝑥) = 𝑦))
7	2, 6	anbi12d 473	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦) ↔ ((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦)))
8	7	anassrs 400	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦) ↔ ((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦)))
9	8	ralbidva 2473	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦)))
10	9	pm5.32da 452	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦)) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦))))
11		grpidpropd.1	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
12	11	eleq2d 2247	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
13	11	raleqdv 2678	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦)))
14	12, 13	anbi12d 473	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦))))
15		grpidpropd.2	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
16	15	eleq2d 2247	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐿)))
17	15	raleqdv 2678	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦)))
18	16, 17	anbi12d 473	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦))))
19	10, 14, 18	3bitr3d 218	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦))))
20	19	iotabidv 5195	. 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦))))
21		grpidproddg.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
22		eqid 2177	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
23		eqid 2177	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
24		eqid 2177	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
25	22, 23, 24	grpidvalg 12684	. . 3 ⊢ (𝐾 ∈ 𝑉 → (0g‘𝐾) = (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦))))
26	21, 25	syl 14	. 2 ⊢ (𝜑 → (0g‘𝐾) = (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦))))
27		grpidproddg.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ 𝑊)
28		eqid 2177	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
29		eqid 2177	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
30		eqid 2177	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
31	28, 29, 30	grpidvalg 12684	. . 3 ⊢ (𝐿 ∈ 𝑊 → (0g‘𝐿) = (℩𝑥(𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦))))
32	27, 31	syl 14	. 2 ⊢ (𝜑 → (0g‘𝐿) = (℩𝑥(𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦))))
33	20, 26, 32	3eqtr4d 2220	1 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ℩cio 5172  ‘cfv 5212  (class class class)co 5869  Basecbs 12445  +_gcplusg 12518  0_gc0g 12653

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-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899

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-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8909  df-ndx 12448  df-slot 12449  df-base 12451  df-0g 12655

This theorem is referenced by:  mhmpropd  12747  grppropd  12783  grpinvpropdg  12834  mulgpropdg  12913  rngidpropdg  13138