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Theorem grpodivf 30300
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 2726 . . . . . . . 8 (invβ€˜πΊ) = (invβ€˜πΊ)
31, 2grpoinvcl 30286 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π‘¦) ∈ 𝑋)
433adant2 1128 . . . . . 6 ((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π‘¦) ∈ 𝑋)
51grpocl 30262 . . . . . 6 ((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π‘¦) ∈ 𝑋) β†’ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦)) ∈ 𝑋)
64, 5syld3an3 1406 . . . . 5 ((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦)) ∈ 𝑋)
763expib 1119 . . . 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 2726 . . . 4 (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦))) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦)))
109fmpo 8053 . . 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 30296 . . 3 (𝐺 ∈ GrpOp β†’ 𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (π‘₯𝐺((invβ€˜πΊ)β€˜π‘¦))))
1413feq1d 6696 . 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 1533   ∈ wcel 2098  βˆ€wral 3055   Γ— cxp 5667  ran crn 5670  βŸΆwf 6533  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-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-grpo 30255  df-gid 30256  df-ginv 30257  df-gdiv 30258
This theorem is referenced by:  grpodivcl  30301
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