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Theorem grpoinvdiv 28091
Description: Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdiv.1 𝑋 = ran 𝐺
grpdiv.2 𝑁 = (inv‘𝐺)
grpdiv.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
grpoinvdiv ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴))

Proof of Theorem grpoinvdiv
StepHypRef Expression
1 grpdiv.1 . . . 4 𝑋 = ran 𝐺
2 grpdiv.2 . . . 4 𝑁 = (inv‘𝐺)
3 grpdiv.3 . . . 4 𝐷 = ( /𝑔𝐺)
41, 2, 3grpodivval 28089 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
54fveq2d 6503 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝑁‘(𝐴𝐺(𝑁𝐵))))
61, 2grpoinvcl 28078 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
763adant2 1111 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
81, 2grpoinvop 28087 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ (𝑁𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐺(𝑁𝐵))) = ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)))
97, 8syld3an3 1389 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺(𝑁𝐵))) = ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)))
101, 2grpo2inv 28085 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁‘(𝑁𝐵)) = 𝐵)
11103adant2 1111 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝑁𝐵)) = 𝐵)
1211oveq1d 6991 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺(𝑁𝐴)))
131, 2, 3grpodivval 28089 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐴𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁𝐴)))
14133com23 1106 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁𝐴)))
1512, 14eqtr4d 2817 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐷𝐴))
165, 9, 153eqtrd 2818 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1068   = wceq 1507  wcel 2050  ran crn 5408  cfv 6188  (class class class)co 6976  GrpOpcgr 28043  invcgn 28045   /𝑔 cgs 28046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-grpo 28047  df-gid 28048  df-ginv 28049  df-gdiv 28050
This theorem is referenced by:  grpodivdiv  28094
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