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Theorem grpodivf 28092
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 2778 . . . . . . . 8 (inv‘𝐺) = (inv‘𝐺)
31, 2grpoinvcl 28078 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
433adant2 1111 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
51grpocl 28054 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋 ∧ ((inv‘𝐺)‘𝑦) ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
64, 5syld3an3 1389 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
763expib 1102 . . . 4 (𝐺 ∈ GrpOp → ((𝑥𝑋𝑦𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋))
87ralrimivv 3140 . . 3 (𝐺 ∈ GrpOp → ∀𝑥𝑋𝑦𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
9 eqid 2778 . . . 4 (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))
109fmpo 7574 . . 3 (∀𝑥𝑋𝑦𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋 ↔ (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋)
118, 10sylib 210 . 2 (𝐺 ∈ GrpOp → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋)
12 grpdivf.3 . . . 4 𝐷 = ( /𝑔𝐺)
131, 2, 12grpodivfval 28088 . . 3 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))))
1413feq1d 6329 . 2 (𝐺 ∈ GrpOp → (𝐷:(𝑋 × 𝑋)⟶𝑋 ↔ (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋))
1511, 14mpbird 249 1 (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  wral 3088   × cxp 5405  ran crn 5408  wf 6184  cfv 6188  (class class class)co 6976  cmpo 6978  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:  grpodivcl  28093
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