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

Theorem grpodivcl 28320
Description: Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1 𝑋 = ran 𝐺
grpdivf.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
grpodivcl ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)

Proof of Theorem grpodivcl
StepHypRef Expression
1 grpdivf.1 . . 3 𝑋 = ran 𝐺
2 grpdivf.3 . . 3 𝐷 = ( /𝑔𝐺)
31, 2grpodivf 28319 . 2 (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)
4 fovrn 7309 . 2 ((𝐷:(𝑋 × 𝑋)⟶𝑋𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
53, 4syl3an1 1160 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115   × cxp 5541  ran crn 5544  wf 6340  cfv 6344  (class class class)co 7146  GrpOpcgr 28270   /𝑔 cgs 28273
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-grpo 28274  df-gid 28275  df-ginv 28276  df-gdiv 28277
This theorem is referenced by:  grpodivdiv  28321  ablomuldiv  28333  ablodivdiv4  28335  ablonnncan1  28338  ablo4pnp  35230  ghomdiv  35242  grpokerinj  35243  dmncan1  35426
  Copyright terms: Public domain W3C validator