Theorem grpidpropd 18619

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))
grpidpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))

Assertion

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

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

Proof of Theorem grpidpropd

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 2739	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = 𝑦 ↔ (𝑥(+g‘𝐿)𝑦) = 𝑦))
3	1	oveqrspc2v 7385	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
4	3	oveqrspc2v 7385	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(+g‘𝐾)𝑥) = (𝑦(+g‘𝐿)𝑥))
5	4	ancom2s 651	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(+g‘𝐾)𝑥) = (𝑦(+g‘𝐿)𝑥))
6	5	eqeq1d 2739	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦(+g‘𝐾)𝑥) = 𝑦 ↔ (𝑦(+g‘𝐿)𝑥) = 𝑦))
7	2, 6	anbi12d 633	. . . . . . 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 467	. . . . . 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 3159	. . . . 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 579	. . . 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 2823	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
13	11	raleqdv 3296	. . . . 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 633	. . . 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 2823	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐿)))
17	15	raleqdv 3296	. . . . 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 633	. . . 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 309	. . 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 6474	. 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		eqid 2737	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
22		eqid 2737	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
23		eqid 2737	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
24	21, 22, 23	grpidval 18618	. 2 ⊢ (0g‘𝐾) = (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦)))
25		eqid 2737	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
26		eqid 2737	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
27		eqid 2737	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
28	25, 26, 27	grpidval 18618	. 2 ⊢ (0g‘𝐿) = (℩𝑥(𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g‘𝐿)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐿)𝑥) = 𝑦)))
29	20, 24, 28	3eqtr4g 2797	1 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ℩cio 6444  ‘cfv 6490  (class class class)co 7358  Basecbs 17168  +_gcplusg 17209  0_gc0g 17391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7361  df-0g 17393

This theorem is referenced by:  gsumpropd  18635  gsumpropd2lem  18636  mhmpropd  18749  grppropd  18916  grpinvpropd  18980  mulgpropd  19081  prds1  20291  rngidpropd  20384  nzrpropd  20486  drngprop  20710  drngpropd  20735  abvpropd  20801  lbspropd  21084  sralmod0  21173  phlpropd  21643  opsr0  22191  mplbaspropd  22209  ply1mpl0  22229  mat0  22391  nmpropd  24568  nmpropd2  24569  tng0  24617  mdegpropd  26061  ply1divalg2  26116  domnpropd  33358  resv0g  33418  zlm0  34125  hlhils0  42402  hlhil0  42412  mnring0gd  44663