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Theorem grpodivf 29522
Description: Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1 𝑋 = ran 𝐺
grpdivf.3 𝐷 = ( /𝑔 β€˜πΊ)
Assertion
Ref Expression
grpodivf (𝐺 ∈ GrpOp β†’ 𝐷:(𝑋 Γ— 𝑋)βŸΆπ‘‹)

Proof of Theorem grpodivf
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpdivf.1 . . . . . . . 8 𝑋 = ran 𝐺
2 eqid 2733 . . . . . . . 8 (invβ€˜πΊ) = (invβ€˜πΊ)
31, 2grpoinvcl 29508 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π‘¦) ∈ 𝑋)
433adant2 1132 . . . . . 6 ((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π‘¦) ∈ 𝑋)
51grpocl 29484 . . . . . 6 ((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π‘¦) ∈ 𝑋) β†’ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦)) ∈ 𝑋)
64, 5syld3an3 1410 . . . . 5 ((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦)) ∈ 𝑋)
763expib 1123 . . . 4 (𝐺 ∈ GrpOp β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦)) ∈ 𝑋))
87ralrimivv 3192 . . 3 (𝐺 ∈ GrpOp β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦)) ∈ 𝑋)
9 eqid 2733 . . . 4 (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦))) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦)))
109fmpo 8001 . . 3 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦)) ∈ 𝑋 ↔ (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦))):(𝑋 Γ— 𝑋)βŸΆπ‘‹)
118, 10sylib 217 . 2 (𝐺 ∈ GrpOp β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦))):(𝑋 Γ— 𝑋)βŸΆπ‘‹)
12 grpdivf.3 . . . 4 𝐷 = ( /𝑔 β€˜πΊ)
131, 2, 12grpodivfval 29518 . . 3 (𝐺 ∈ GrpOp β†’ 𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦))))
1413feq1d 6654 . 2 (𝐺 ∈ GrpOp β†’ (𝐷:(𝑋 Γ— 𝑋)βŸΆπ‘‹ ↔ (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦))):(𝑋 Γ— 𝑋)βŸΆπ‘‹))
1511, 14mpbird 257 1 (𝐺 ∈ GrpOp β†’ 𝐷:(𝑋 Γ— 𝑋)βŸΆπ‘‹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   Γ— cxp 5632  ran crn 5635  βŸΆwf 6493  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-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-grpo 29477  df-gid 29478  df-ginv 29479  df-gdiv 29480
This theorem is referenced by:  grpodivcl  29523
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