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Theorem grpodivfval 20909
 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
grpdiv.2
grpdiv.3
Assertion
Ref Expression
grpodivfval
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem grpodivfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 grpdiv.3 . 2
2 grpdiv.1 . . . . 5
3 rnexg 4940 . . . . 5
42, 3syl5eqel 2367 . . . 4
5 mpt2exga 6197 . . . 4
64, 4, 5syl2anc 642 . . 3
7 rneq 4904 . . . . . 6
87, 2syl6eqr 2333 . . . . 5
9 id 19 . . . . . 6
10 eqidd 2284 . . . . . 6
11 fveq2 5525 . . . . . . . 8
12 grpdiv.2 . . . . . . . 8
1311, 12syl6eqr 2333 . . . . . . 7
1413fveq1d 5527 . . . . . 6
159, 10, 14oveq123d 5879 . . . . 5
168, 8, 15mpt2eq123dv 5910 . . . 4
17 df-gdiv 20861 . . . 4
1816, 17fvmptg 5600 . . 3
196, 18mpdan 649 . 2
201, 19syl5eq 2327 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1623   wcel 1684  cvv 2788   crn 4690  cfv 5255  (class class class)co 5858   cmpt2 5860  cgr 20853  cgn 20855   cgs 20856 This theorem is referenced by:  grpodivval  20910  grpodivf  20913  nvmfval  21202 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-gdiv 20861
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