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

Theorem grpodivval 30297
Description: Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdiv.1 𝑋 = ran 𝐺
grpdiv.2 𝑁 = (invβ€˜πΊ)
grpdiv.3 𝐷 = ( /𝑔 β€˜πΊ)
Assertion
Ref Expression
grpodivval ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐴𝐺(π‘β€˜π΅)))

Proof of Theorem grpodivval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpdiv.1 . . . . 5 𝑋 = ran 𝐺
2 grpdiv.2 . . . . 5 𝑁 = (invβ€˜πΊ)
3 grpdiv.3 . . . . 5 𝐷 = ( /𝑔 β€˜πΊ)
41, 2, 3grpodivfval 30296 . . . 4 (𝐺 ∈ GrpOp β†’ 𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))))
54oveqd 7422 . . 3 (𝐺 ∈ GrpOp β†’ (𝐴𝐷𝐡) = (𝐴(π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦)))𝐡))
6 oveq1 7412 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯𝐺(π‘β€˜π‘¦)) = (𝐴𝐺(π‘β€˜π‘¦)))
7 fveq2 6885 . . . . 5 (𝑦 = 𝐡 β†’ (π‘β€˜π‘¦) = (π‘β€˜π΅))
87oveq2d 7421 . . . 4 (𝑦 = 𝐡 β†’ (𝐴𝐺(π‘β€˜π‘¦)) = (𝐴𝐺(π‘β€˜π΅)))
9 eqid 2726 . . . 4 (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦)))
10 ovex 7438 . . . 4 (𝐴𝐺(π‘β€˜π΅)) ∈ V
116, 8, 9, 10ovmpo 7564 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴(π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦)))𝐡) = (𝐴𝐺(π‘β€˜π΅)))
125, 11sylan9eq 2786 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺(π‘β€˜π΅)))
13123impb 1112 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐴𝐺(π‘β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5670  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  GrpOpcgr 30251  invcgn 30253   /𝑔 cgs 30254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-gdiv 30258
This theorem is referenced by:  grpodivinv  30298  grpoinvdiv  30299  grpodivdiv  30302  grpomuldivass  30303  grpodivid  30304  grponpcan  30305  ablodivdiv4  30316  nvmval  30404  rngosub  37311
  Copyright terms: Public domain W3C validator