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

Theorem grpodivf 28328
 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 2798 . . . . . . . 8 (inv‘𝐺) = (inv‘𝐺)
31, 2grpoinvcl 28314 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
433adant2 1128 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋)
51grpocl 28290 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋 ∧ ((inv‘𝐺)‘𝑦) ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
64, 5syld3an3 1406 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
763expib 1119 . . . 4 (𝐺 ∈ GrpOp → ((𝑥𝑋𝑦𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋))
87ralrimivv 3155 . . 3 (𝐺 ∈ GrpOp → ∀𝑥𝑋𝑦𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)
9 eqid 2798 . . . 4 (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))
109fmpo 7750 . . 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 28324 . . 3 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))))
1413feq1d 6472 . 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 1538   ∈ wcel 2111  ∀wral 3106   × cxp 5517  ran crn 5520  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  GrpOpcgr 28279  invcgn 28281   /𝑔 cgs 28282 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 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 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 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7673  df-2nd 7674  df-grpo 28283  df-gid 28284  df-ginv 28285  df-gdiv 28286 This theorem is referenced by:  grpodivcl  28329
 Copyright terms: Public domain W3C validator