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Theorem exidresid 26545
 Description: The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
exidres.1
exidres.2 GId
exidres.3
Assertion
Ref Expression
exidresid GId

Proof of Theorem exidresid
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exidres.3 . . . . . 6
2 resexg 5177 . . . . . 6
31, 2syl5eqel 2519 . . . . 5
4 eqid 2435 . . . . . 6
54gidval 21793 . . . . 5 GId
63, 5syl 16 . . . 4 GId
763ad2ant1 978 . . 3 GId
9 exidres.1 . . . . . . 7
10 exidres.2 . . . . . . 7 GId
119, 10, 1exidreslem 26543 . . . . . 6
1211simprd 450 . . . . 5
1312adantr 452 . . . 4
149, 10, 1exidres 26544 . . . . . 6
15 elin 3522 . . . . . . . 8
16 rngopid 21903 . . . . . . . 8
1715, 16sylbir 205 . . . . . . 7
1817ancoms 440 . . . . . 6
1914, 18sylan 458 . . . . 5
2019raleqdv 2902 . . . 4
2113, 20mpbird 224 . . 3
2211simpld 446 . . . . . 6
2322adantr 452 . . . . 5
2423, 19eleqtrrd 2512 . . . 4
254exidu1 21906 . . . . . . 7
2615, 25sylbir 205 . . . . . 6
2726ancoms 440 . . . . 5
2814, 27sylan 458 . . . 4
29 oveq1 6080 . . . . . . . 8
3029eqeq1d 2443 . . . . . . 7
31 oveq2 6081 . . . . . . . 8
3231eqeq1d 2443 . . . . . . 7
3330, 32anbi12d 692 . . . . . 6
3433ralbidv 2717 . . . . 5
3534riota2 6564 . . . 4
3624, 28, 35syl2anc 643 . . 3
3721, 36mpbid 202 . 2
388, 37eqtrd 2467 1 GId
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  wreu 2699  cvv 2948   cin 3311   wss 3312   cxp 4868   cdm 4870   crn 4871   cres 4872  cfv 5446  (class class class)co 6073  crio 6534  GIdcgi 21767   cexid 21894  cmagm 21898 This theorem is referenced by:  isdrngo2  26565 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-reu 2704  df-rmo 2705  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-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-riota 6541  df-gid 21772  df-exid 21895  df-mgm 21899
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