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Theorem grponpcan 30471
Description: Cancellation law for group division. (npcan 11508 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1 𝑋 = ran 𝐺
grpdivf.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
grponpcan ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)

Proof of Theorem grponpcan
StepHypRef Expression
1 grpdivf.1 . . . 4 𝑋 = ran 𝐺
2 eqid 2726 . . . 4 (inv‘𝐺) = (inv‘𝐺)
3 grpdivf.3 . . . 4 𝐷 = ( /𝑔𝐺)
41, 2, 3grpodivval 30463 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
54oveq1d 7429 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵))
6 simp1 1133 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐺 ∈ GrpOp)
7 simp2 1134 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐴𝑋)
81, 2grpoinvcl 30452 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
983adant2 1128 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
10 simp3 1135 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐵𝑋)
111grpoass 30431 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋𝐵𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)))
126, 7, 9, 10, 11syl13anc 1369 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)))
13 eqid 2726 . . . . . . 7 (GId‘𝐺) = (GId‘𝐺)
141, 13, 2grpolinv 30454 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (((inv‘𝐺)‘𝐵)𝐺𝐵) = (GId‘𝐺))
1514oveq2d 7430 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = (𝐴𝐺(GId‘𝐺)))
16153adant2 1128 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = (𝐴𝐺(GId‘𝐺)))
171, 13grporid 30445 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
18173adant3 1129 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
1916, 18eqtrd 2766 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(((inv‘𝐺)‘𝐵)𝐺𝐵)) = 𝐴)
2012, 19eqtrd 2766 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐺𝐵) = 𝐴)
215, 20eqtrd 2766 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  ran crn 5674  cfv 6544  (class class class)co 7414  GrpOpcgr 30417  GIdcgi 30418  invcgn 30419   /𝑔 cgs 30420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7993  df-2nd 7994  df-grpo 30421  df-gid 30422  df-ginv 30423  df-gdiv 30424
This theorem is referenced by:  grpoeqdivid  37593  ghomdiv  37604
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