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Theorem grpoinvdiv 28361
 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 28359 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
54fveq2d 6656 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝑁‘(𝐴𝐺(𝑁𝐵))))
61, 2grpoinvcl 28348 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
763adant2 1128 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
81, 2grpoinvop 28357 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ (𝑁𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐺(𝑁𝐵))) = ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)))
97, 8syld3an3 1406 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺(𝑁𝐵))) = ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)))
101, 2grpo2inv 28355 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁‘(𝑁𝐵)) = 𝐵)
11103adant2 1128 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝑁𝐵)) = 𝐵)
1211oveq1d 7157 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺(𝑁𝐴)))
131, 2, 3grpodivval 28359 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐴𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁𝐴)))
14133com23 1123 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁𝐴)))
1512, 14eqtr4d 2836 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐷𝐴))
165, 9, 153eqtrd 2837 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ran crn 5523  ‘cfv 6329  (class class class)co 7142  GrpOpcgr 28313  invcgn 28315   /𝑔 cgs 28316 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-1st 7681  df-2nd 7682  df-grpo 28317  df-gid 28318  df-ginv 28319  df-gdiv 28320 This theorem is referenced by:  grpodivdiv  28364
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