Theorem grponpcan 30748

Description: Cancellation law for group division. (npcan 11441 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 2764	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
3		grpdivf.3	. . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
4	1, 2, 3	grpodivval 30740	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
5	4	oveq1d 7413	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵))
6		simp1 1150	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ GrpOp)
7		simp2 1151	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8	1, 2	grpoinvcl 30729	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
9	8	3adant2 1145	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
10		simp3 1152	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
11	1	grpoass 30708	. . . 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 1393	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)))
13		eqid 2764	. . . . . . 7 ⊢ (GId‘𝐺) = (GId‘𝐺)
14	1, 13, 2	grpolinv 30731	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (((inv‘𝐺)‘𝐵)𝐺𝐵) = (GId‘𝐺))
15	14	oveq2d 7414	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = (𝐴𝐺(GId‘𝐺)))
16	15	3adant2 1145	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = (𝐴𝐺(GId‘𝐺)))
17	1, 13	grporid 30722	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
18	17	3adant3 1146	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
19	16, 18	eqtrd 2799	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = 𝐴)
20	12, 19	eqtrd 2799	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = 𝐴)
21	5, 20	eqtrd 2799	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ran crn 5650  ‘cfv 6523  (class class class)co 7398  GrpOpcgr 30694  GIdcgi 30695  invcgn 30696   /_𝑔 cgs 30697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-grpo 30698  df-gid 30699  df-ginv 30700  df-gdiv 30701

This theorem is referenced by:  grpoeqdivid  38385  ghomdiv  38396