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Theorem grpodivinv 28323
Description: Group division by an inverse. (Contributed by NM, 15-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
grpodivinv ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝑁𝐵)) = (𝐴𝐺𝐵))

Proof of Theorem grpodivinv
StepHypRef Expression
1 grpdiv.1 . . . . 5 𝑋 = ran 𝐺
2 grpdiv.2 . . . . 5 𝑁 = (inv‘𝐺)
31, 2grpoinvcl 28311 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
433adant2 1128 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
5 grpdiv.3 . . . 4 𝐷 = ( /𝑔𝐺)
61, 2, 5grpodivval 28322 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ (𝑁𝐵) ∈ 𝑋) → (𝐴𝐷(𝑁𝐵)) = (𝐴𝐺(𝑁‘(𝑁𝐵))))
74, 6syld3an3 1406 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝑁𝐵)) = (𝐴𝐺(𝑁‘(𝑁𝐵))))
81, 2grpo2inv 28318 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁‘(𝑁𝐵)) = 𝐵)
983adant2 1128 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝑁𝐵)) = 𝐵)
109oveq2d 7155 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁‘(𝑁𝐵))) = (𝐴𝐺𝐵))
117, 10eqtrd 2836 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝑁𝐵)) = (𝐴𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  ran crn 5524  cfv 6328  (class class class)co 7139  GrpOpcgr 28276  invcgn 28278   /𝑔 cgs 28279
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-grpo 28280  df-gid 28281  df-ginv 28282  df-gdiv 28283
This theorem is referenced by:  ablodivdiv4  28341
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