MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpodivf Structured version   Visualization version   GIF version

Theorem grpodivf 28619
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 2737 . . . . . . . 8 (inv‘𝐺) = (inv‘𝐺)
31, 2grpoinvcl 28605 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
433adant2 1133 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
51grpocl 28581 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋 ∧ ((inv‘𝐺)‘𝑦) ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
64, 5syld3an3 1411 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
763expib 1124 . . . 4 (𝐺 ∈ GrpOp → ((𝑥𝑋𝑦𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋))
87ralrimivv 3111 . . 3 (𝐺 ∈ GrpOp → ∀𝑥𝑋𝑦𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
9 eqid 2737 . . . 4 (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))
109fmpo 7838 . . 3 (∀𝑥𝑋𝑦𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋 ↔ (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋)
118, 10sylib 221 . 2 (𝐺 ∈ GrpOp → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋)
12 grpdivf.3 . . . 4 𝐷 = ( /𝑔𝐺)
131, 2, 12grpodivfval 28615 . . 3 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))))
1413feq1d 6530 . 2 (𝐺 ∈ GrpOp → (𝐷:(𝑋 × 𝑋)⟶𝑋 ↔ (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋))
1511, 14mpbird 260 1 (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  wral 3061   × cxp 5549  ran crn 5552  wf 6376  cfv 6380  (class class class)co 7213  cmpo 7215  GrpOpcgr 28570  invcgn 28572   /𝑔 cgs 28573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-grpo 28574  df-gid 28575  df-ginv 28576  df-gdiv 28577
This theorem is referenced by:  grpodivcl  28620
  Copyright terms: Public domain W3C validator