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Theorem grpodivf 21787
 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
grpdivf.3
Assertion
Ref Expression
grpodivf

Proof of Theorem grpodivf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpdivf.1 . . . . . . . 8
2 eqid 2404 . . . . . . . 8
31, 2grpoinvcl 21767 . . . . . . 7
433adant2 976 . . . . . 6
51grpocl 21741 . . . . . 6
64, 5syld3an3 1229 . . . . 5
763expib 1156 . . . 4
87ralrimivv 2757 . . 3
9 eqid 2404 . . . 4
109fmpt2 6377 . . 3
118, 10sylib 189 . 2
12 grpdivf.3 . . . 4
131, 2, 12grpodivfval 21783 . . 3
1413feq1d 5539 . 2
1511, 14mpbird 224 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1649   wcel 1721  wral 2666   cxp 4835   crn 4838  wf 5409  cfv 5413  (class class class)co 6040   cmpt2 6042  cgr 21727  cgn 21729   cgs 21730 This theorem is referenced by:  grpodivcl  21788 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735
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