Theorem grpidpropdg 13640

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 2243	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = 𝑦 ↔ (𝑥(+g‘𝐿)𝑦) = 𝑦))
3	1	oveqrspc2v 6085	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
4	3	oveqrspc2v 6085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(+g‘𝐾)𝑥) = (𝑦(+g‘𝐿)𝑥))
5	4	ancom2s 568	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(+g‘𝐾)𝑥) = (𝑦(+g‘𝐿)𝑥))
6	5	eqeq1d 2243	. . . . . . . 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 2540	. . . . 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 2304	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
13	11	raleqdv 2749	. . . . 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 2304	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐿)))
17	15	raleqdv 2749	. . . . 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 5340	. 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 2234	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
23		eqid 2234	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
24		eqid 2234	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
25	22, 23, 24	grpidvalg 13639	. . 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 2234	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
29		eqid 2234	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
30		eqid 2234	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
31	28, 29, 30	grpidvalg 13639	. . 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 2277	1 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ℩cio 5315  ‘cfv 5357  (class class class)co 6058  Basecbs 13299  +_gcplusg 13377  0_gc0g 13556

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9258  df-ndx 13302  df-slot 13303  df-base 13305  df-0g 13558

This theorem is referenced by:  gsumpropd  13658  gsumpropd2  13659  mhmpropd  13724  grppropd  13775  grpinvpropdg  13833  mulgpropdg  13920  rngidpropdg  14394  sralmod0g  14728