Theorem grpidpropd 18745

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 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = 𝑦 ↔ (𝑥(+g‘𝐿)𝑦) = 𝑦))
3	1	oveqrspc2v 7450	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
4	3	oveqrspc2v 7450	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(+g‘𝐾)𝑥) = (𝑦(+g‘𝐿)𝑥))
5	4	ancom2s 663	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(+g‘𝐾)𝑥) = (𝑦(+g‘𝐿)𝑥))
6	5	eqeq1d 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦(+g‘𝐾)𝑥) = 𝑦 ↔ (𝑦(+g‘𝐿)𝑥) = 𝑦))
7	2, 6	anbi12d 644	. . . . . . 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 473	. . . . . 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 3189	. . . . 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 590	. . . 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 2852	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
13	11	raleqdv 3326	. . . . 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 644	. . . 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 2852	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐿)))
17	15	raleqdv 3326	. . . . 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 644	. . . 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 312	. . 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 6527	. 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 2766	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
22		eqid 2766	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
23		eqid 2766	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
24	21, 22, 23	grpidval 18744	. 2 ⊢ (0g‘𝐾) = (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g‘𝐾)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐾)𝑥) = 𝑦)))
25		eqid 2766	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
26		eqid 2766	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
27		eqid 2766	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
28	25, 26, 27	grpidval 18744	. 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 2826	1 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ℩cio 6497  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  0_gc0g 17517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-0g 17519

This theorem is used by:  gsumpropd  18761  gsumpropd2lem  18762  mhmpropd  18875  grppropd  19043  grpinvpropd  19106  mulgpropd  19207  prds1  20430  rngidpropd  20523  nzrpropd  20648  drngprop  20874  drngpropd  20903  abvpropd  20968  lbspropd  21250  sralmod0  21339  phlpropd  21835  opsr0  22408  mplbaspropd  22426  ply1mpl0  22446  mat0  22604  nmpropd  24781  nmpropd2  24782  tng0  24830  mdegpropd  26271  ply1divalg2  26326  domnpropd  33624  resv0g  33682  zlm0  34374  hlhils0  42752  hlhil0  42762  mnring0gd  44978