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Theorem grpodivfval 29305
Description: Group division (or subtraction) operation. (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
grpodivfval (𝐺 ∈ GrpOp β†’ 𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))))
Distinct variable groups:   π‘₯,𝑦,𝐺   π‘₯,𝑁,𝑦   π‘₯,𝑋,𝑦
Allowed substitution hints:   𝐷(π‘₯,𝑦)

Proof of Theorem grpodivfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpdiv.3 . 2 𝐷 = ( /𝑔 β€˜πΊ)
2 grpdiv.1 . . . . 5 𝑋 = ran 𝐺
3 rnexg 7833 . . . . 5 (𝐺 ∈ GrpOp β†’ ran 𝐺 ∈ V)
42, 3eqeltrid 2842 . . . 4 (𝐺 ∈ GrpOp β†’ 𝑋 ∈ V)
5 mpoexga 8002 . . . 4 ((𝑋 ∈ V ∧ 𝑋 ∈ V) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))) ∈ V)
64, 4, 5syl2anc 584 . . 3 (𝐺 ∈ GrpOp β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))) ∈ V)
7 rneq 5889 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
87, 2eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 β†’ ran 𝑔 = 𝑋)
9 id 22 . . . . . 6 (𝑔 = 𝐺 β†’ 𝑔 = 𝐺)
10 eqidd 2738 . . . . . 6 (𝑔 = 𝐺 β†’ π‘₯ = π‘₯)
11 fveq2 6839 . . . . . . . 8 (𝑔 = 𝐺 β†’ (invβ€˜π‘”) = (invβ€˜πΊ))
12 grpdiv.2 . . . . . . . 8 𝑁 = (invβ€˜πΊ)
1311, 12eqtr4di 2795 . . . . . . 7 (𝑔 = 𝐺 β†’ (invβ€˜π‘”) = 𝑁)
1413fveq1d 6841 . . . . . 6 (𝑔 = 𝐺 β†’ ((invβ€˜π‘”)β€˜π‘¦) = (π‘β€˜π‘¦))
159, 10, 14oveq123d 7372 . . . . 5 (𝑔 = 𝐺 β†’ (π‘₯𝑔((invβ€˜π‘”)β€˜π‘¦)) = (π‘₯𝐺(π‘β€˜π‘¦)))
168, 8, 15mpoeq123dv 7426 . . . 4 (𝑔 = 𝐺 β†’ (π‘₯ ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (π‘₯𝑔((invβ€˜π‘”)β€˜π‘¦))) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))))
17 df-gdiv 29267 . . . 4 /𝑔 = (𝑔 ∈ GrpOp ↦ (π‘₯ ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (π‘₯𝑔((invβ€˜π‘”)β€˜π‘¦))))
1816, 17fvmptg 6943 . . 3 ((𝐺 ∈ GrpOp ∧ (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))) ∈ V) β†’ ( /𝑔 β€˜πΊ) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))))
196, 18mpdan 685 . 2 (𝐺 ∈ GrpOp β†’ ( /𝑔 β€˜πΊ) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))))
201, 19eqtrid 2789 1 (𝐺 ∈ GrpOp β†’ 𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺(π‘β€˜π‘¦))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3443  ran crn 5632  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  GrpOpcgr 29260  invcgn 29262   /𝑔 cgs 29263
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-gdiv 29267
This theorem is referenced by:  grpodivval  29306  grpodivf  29309  nvmfval  29415
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