Theorem grponpcan 30562

Description: Cancellation law for group division. (npcan 11517 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpdivf.1	⊢ 𝑋 = ran 𝐺
grpdivf.3	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
grponpcan	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)

Proof of Theorem grponpcan

Step	Hyp	Ref	Expression
1		grpdivf.1	. . . 4 ⊢ 𝑋 = ran 𝐺
2		eqid 2737	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
3		grpdivf.3	. . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
4	1, 2, 3	grpodivval 30554	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
5	4	oveq1d 7446	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵))
6		simp1 1137	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ GrpOp)
7		simp2 1138	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8	1, 2	grpoinvcl 30543	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
9	8	3adant2 1132	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
10		simp3 1139	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
11	1	grpoass 30522	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)))
12	6, 7, 9, 10, 11	syl13anc 1374	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)))
13		eqid 2737	. . . . . . 7 ⊢ (GId‘𝐺) = (GId‘𝐺)
14	1, 13, 2	grpolinv 30545	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (((inv‘𝐺)‘𝐵)𝐺𝐵) = (GId‘𝐺))
15	14	oveq2d 7447	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = (𝐴𝐺(GId‘𝐺)))
16	15	3adant2 1132	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = (𝐴𝐺(GId‘𝐺)))
17	1, 13	grporid 30536	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
18	17	3adant3 1133	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
19	16, 18	eqtrd 2777	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = 𝐴)
20	12, 19	eqtrd 2777	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = 𝐴)
21	5, 20	eqtrd 2777	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ran crn 5686  ‘cfv 6561  (class class class)co 7431  GrpOpcgr 30508  GIdcgi 30509  invcgn 30510   /_𝑔 cgs 30511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ginv 30514  df-gdiv 30515

This theorem is referenced by:  grpoeqdivid  37888  ghomdiv  37899