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Theorem grpodivf 27259
Description: Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1 𝑋 = ran 𝐺
grpdivf.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
grpodivf (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)

Proof of Theorem grpodivf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpdivf.1 . . . . . . . 8 𝑋 = ran 𝐺
2 eqid 2621 . . . . . . . 8 (inv‘𝐺) = (inv‘𝐺)
31, 2grpoinvcl 27245 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
433adant2 1078 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
51grpocl 27221 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋 ∧ ((inv‘𝐺)‘𝑦) ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
64, 5syld3an3 1368 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
763expib 1265 . . . 4 (𝐺 ∈ GrpOp → ((𝑥𝑋𝑦𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋))
87ralrimivv 2965 . . 3 (𝐺 ∈ GrpOp → ∀𝑥𝑋𝑦𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
9 eqid 2621 . . . 4 (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))
109fmpt2 7189 . . 3 (∀𝑥𝑋𝑦𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋 ↔ (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋)
118, 10sylib 208 . 2 (𝐺 ∈ GrpOp → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋)
12 grpdivf.3 . . . 4 𝐷 = ( /𝑔𝐺)
131, 2, 12grpodivfval 27255 . . 3 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))))
1413feq1d 5992 . 2 (𝐺 ∈ GrpOp → (𝐷:(𝑋 × 𝑋)⟶𝑋 ↔ (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋))
1511, 14mpbird 247 1 (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907   × cxp 5077  ran crn 5080  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  GrpOpcgr 27210  invcgn 27212   /𝑔 cgs 27213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-grpo 27214  df-gid 27215  df-ginv 27216  df-gdiv 27217
This theorem is referenced by:  grpodivcl  27260
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