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Theorem grpodivval 30389
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 30388 . . . 4 (𝐺 ∈ GrpOp β†’ 𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))))
54oveqd 7433 . . 3 (𝐺 ∈ GrpOp β†’ (𝐴𝐷𝐡) = (𝐴(π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦)))𝐡))
6 oveq1 7423 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯𝐺(π‘β€˜π‘¦)) = (𝐴𝐺(π‘β€˜π‘¦)))
7 fveq2 6892 . . . . 5 (𝑦 = 𝐡 β†’ (π‘β€˜π‘¦) = (π‘β€˜π΅))
87oveq2d 7432 . . . 4 (𝑦 = 𝐡 β†’ (𝐴𝐺(π‘β€˜π‘¦)) = (𝐴𝐺(π‘β€˜π΅)))
9 eqid 2725 . . . 4 (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦)))
10 ovex 7449 . . . 4 (𝐴𝐺(π‘β€˜π΅)) ∈ V
116, 8, 9, 10ovmpo 7578 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴(π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦)))𝐡) = (𝐴𝐺(π‘β€˜π΅)))
125, 11sylan9eq 2785 . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5673  β€˜cfv 6543  (class class class)co 7416   ∈ cmpo 7418  GrpOpcgr 30343  invcgn 30345   /𝑔 cgs 30346
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-gdiv 30350
This theorem is referenced by:  grpodivinv  30390  grpoinvdiv  30391  grpodivdiv  30394  grpomuldivass  30395  grpodivid  30396  grponpcan  30397  ablodivdiv4  30408  nvmval  30496  rngosub  37460
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