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Theorem grponpcan 30060
Description: Cancellation law for group division. (npcan 11474 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1 𝑋 = ran 𝐺
grpdivf.3 𝐷 = ( /𝑔 β€˜πΊ)
Assertion
Ref Expression
grponpcan ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐷𝐡)𝐺𝐡) = 𝐴)
Proof of Theorem grponpcan
StepHypRef Expression
1 grpdivf.1 . . . 4 𝑋 = ran 𝐺
2 eqid 2731 . . . 4 (invβ€˜πΊ) = (invβ€˜πΊ)
3 grpdivf.3 . . . 4 𝐷 = ( /𝑔 β€˜πΊ)
41, 2, 3grpodivval 30052 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
54oveq1d 7427 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐷𝐡)𝐺𝐡) = ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐺𝐡))
6 simp1 1135 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐺 ∈ GrpOp)
7 simp2 1136 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
81, 2grpoinvcl 30041 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋)
983adant2 1130 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋)
10 simp3 1137 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐡 ∈ 𝑋)
111grpoass 30020 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐺𝐡) = (𝐴𝐺(((invβ€˜πΊ)β€˜π΅)𝐺𝐡)))
126, 7, 9, 10, 11syl13anc 1371 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐺𝐡) = (𝐴𝐺(((invβ€˜πΊ)β€˜π΅)𝐺𝐡)))
13 eqid 2731 . . . . . . 7 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
141, 13, 2grpolinv 30043 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜π΅)𝐺𝐡) = (GIdβ€˜πΊ))
1514oveq2d 7428 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺(((invβ€˜πΊ)β€˜π΅)𝐺𝐡)) = (𝐴𝐺(GIdβ€˜πΊ)))
16153adant2 1130 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺(((invβ€˜πΊ)β€˜π΅)𝐺𝐡)) = (𝐴𝐺(GIdβ€˜πΊ)))
171, 13grporid 30034 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(GIdβ€˜πΊ)) = 𝐴)
18173adant3 1131 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺(GIdβ€˜πΊ)) = 𝐴)
1916, 18eqtrd 2771 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺(((invβ€˜πΊ)β€˜π΅)𝐺𝐡)) = 𝐴)
2012, 19eqtrd 2771 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐺𝐡) = 𝐴)
215, 20eqtrd 2771 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐷𝐡)𝐺𝐡) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ran crn 5678  β€˜cfv 6544  (class class class)co 7412  GrpOpcgr 30006  GIdcgi 30007  invcgn 30008   /𝑔 cgs 30009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-grpo 30010  df-gid 30011  df-ginv 30012  df-gdiv 30013
This theorem is referenced by:  grpoeqdivid  37053  ghomdiv  37064
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