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Theorem grpodivval 29519
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 29518 . . . 4 (𝐺 ∈ GrpOp β†’ 𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))))
54oveqd 7375 . . 3 (𝐺 ∈ GrpOp β†’ (𝐴𝐷𝐡) = (𝐴(π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦)))𝐡))
6 oveq1 7365 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯𝐺(π‘β€˜π‘¦)) = (𝐴𝐺(π‘β€˜π‘¦)))
7 fveq2 6843 . . . . 5 (𝑦 = 𝐡 β†’ (π‘β€˜π‘¦) = (π‘β€˜π΅))
87oveq2d 7374 . . . 4 (𝑦 = 𝐡 β†’ (𝐴𝐺(π‘β€˜π‘¦)) = (𝐴𝐺(π‘β€˜π΅)))
9 eqid 2733 . . . 4 (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦)))
10 ovex 7391 . . . 4 (𝐴𝐺(π‘β€˜π΅)) ∈ V
116, 8, 9, 10ovmpo 7516 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴(π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦)))𝐡) = (𝐴𝐺(π‘β€˜π΅)))
125, 11sylan9eq 2793 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺(π‘β€˜π΅)))
13123impb 1116 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ran crn 5635  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  GrpOpcgr 29473  invcgn 29475   /𝑔 cgs 29476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-gdiv 29480
This theorem is referenced by:  grpodivinv  29520  grpoinvdiv  29521  grpodivdiv  29524  grpomuldivass  29525  grpodivid  29526  grponpcan  29527  ablodivdiv4  29538  nvmval  29626  rngosub  36435
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