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Theorem grpoidinv 21788
 Description: A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1
Assertion
Ref Expression
grpoidinv
Distinct variable groups:   ,,,   ,,,

Proof of Theorem grpoidinv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpfo.1 . . 3
21grpolidinv 21781 . 2
3 simpl 444 . . . . . . . . . 10
43ralimi 2773 . . . . . . . . 9
5 oveq2 6081 . . . . . . . . . . 11
6 id 20 . . . . . . . . . . 11
75, 6eqeq12d 2449 . . . . . . . . . 10
87rspccva 3043 . . . . . . . . 9
94, 8sylan 458 . . . . . . . 8
109adantll 695 . . . . . . 7
1110adantll 695 . . . . . 6
12 simpl 444 . . . . . . . . 9
1312anim1i 552 . . . . . . . 8
14 id 20 . . . . . . . . . . . 12
1514adantrr 698 . . . . . . . . . . 11
1615adantr 452 . . . . . . . . . 10
174adantl 453 . . . . . . . . . . 11
1817ad2antlr 708 . . . . . . . . . 10
19 simpr 448 . . . . . . . . . . . . 13
2019ralimi 2773 . . . . . . . . . . . 12
2120adantl 453 . . . . . . . . . . 11
2221ad2antlr 708 . . . . . . . . . 10
2316, 18, 22jca32 522 . . . . . . . . 9
24 biid 228 . . . . . . . . . 10
25 biid 228 . . . . . . . . . 10
261, 24, 25grpoidinvlem3 21786 . . . . . . . . 9
2723, 26sylancom 649 . . . . . . . 8
281grpoidinvlem4 21787 . . . . . . . 8
2913, 27, 28syl2anc 643 . . . . . . 7
3029, 11eqtrd 2467 . . . . . 6
3111, 30, 27jca31 521 . . . . 5
3231ralrimiva 2781 . . . 4
3332exp32 589 . . 3
3433reximdvai 2808 . 2
352, 34mpd 15 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698   crn 4871  (class class class)co 6073  cgr 21766 This theorem is referenced by:  grpoideu  21789  grpoidval  21796  grpoidinv2  21798  grpomndo  21926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-grpo 21771
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