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Theorem grpoinvdiv 27731
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 27729 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
54fveq2d 6336 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝑁‘(𝐴𝐺(𝑁𝐵))))
61, 2grpoinvcl 27718 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
763adant2 1125 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
81, 2grpoinvop 27727 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ (𝑁𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐺(𝑁𝐵))) = ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)))
97, 8syld3an3 1515 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺(𝑁𝐵))) = ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)))
101, 2grpo2inv 27725 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁‘(𝑁𝐵)) = 𝐵)
11103adant2 1125 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝑁𝐵)) = 𝐵)
1211oveq1d 6808 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺(𝑁𝐴)))
131, 2, 3grpodivval 27729 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐴𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁𝐴)))
14133com23 1120 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁𝐴)))
1512, 14eqtr4d 2808 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐷𝐴))
165, 9, 153eqtrd 2809 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  ran crn 5250  cfv 6031  (class class class)co 6793  GrpOpcgr 27683  invcgn 27685   /𝑔 cgs 27686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-grpo 27687  df-gid 27688  df-ginv 27689  df-gdiv 27690
This theorem is referenced by:  grpodivdiv  27734
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