Theorem grpidpropdg 13576

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 2241	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = 𝑦 ↔ (𝑥(+g‘𝐿)𝑦) = 𝑦))
3	1	oveqrspc2v 6076	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
4	3	oveqrspc2v 6076	. . . . . . . . . 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 2241	. . . . . . . 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 2538	. . . . 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 2302	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
13	11	raleqdv 2746	. . . . 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 2302	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐿)))
17	15	raleqdv 2746	. . . . 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 5334	. 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 2232	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
23		eqid 2232	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
24		eqid 2232	. . . 4 ⊢ (0g‘𝐾) = (0g‘𝐾)
25	22, 23, 24	grpidvalg 13575	. . 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 2232	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
29		eqid 2232	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
30		eqid 2232	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
31	28, 29, 30	grpidvalg 13575	. . 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 2275	1 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ℩cio 5309  ‘cfv 5351  (class class class)co 6049  Basecbs 13201  +_gcplusg 13279  0_gc0g 13458

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-riota 6002  df-ov 6052  df-inn 9234  df-ndx 13204  df-slot 13205  df-base 13207  df-0g 13460

This theorem is referenced by:  gsumpropd  13594  gsumpropd2  13595  mhmpropd  13668  grppropd  13719  grpinvpropdg  13777  mulgpropdg  13870  rngidpropdg  14280  sralmod0g  14586