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Theorem cmpidelt 21012
 Description: A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmpidelt.1
cmpidelt.2 GId
Assertion
Ref Expression
cmpidelt

Proof of Theorem cmpidelt
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmpidelt.1 . . . . 5
2 cmpidelt.2 . . . . 5 GId
31, 2idrval 21010 . . . 4
43eqcomd 2301 . . 3
51, 2iorlid 21011 . . . 4
61exidu1 21009 . . . 4
7 oveq1 5881 . . . . . . . 8
87eqeq1d 2304 . . . . . . 7
9 oveq2 5882 . . . . . . . 8
109eqeq1d 2304 . . . . . . 7
118, 10anbi12d 691 . . . . . 6
1211ralbidv 2576 . . . . 5
1312riota2 6343 . . . 4
145, 6, 13syl2anc 642 . . 3
154, 14mpbird 223 . 2
16 oveq2 5882 . . . . 5
17 id 19 . . . . 5
1816, 17eqeq12d 2310 . . . 4
19 oveq1 5881 . . . . 5
2019, 17eqeq12d 2310 . . . 4
2118, 20anbi12d 691 . . 3
2221rspccva 2896 . 2
2315, 22sylan 457 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1632   wcel 1696  wral 2556  wreu 2558   cin 3164   crn 4706  cfv 5271  (class class class)co 5874  crio 6313  GIdcgi 20870   cexid 20997  cmagm 21001 This theorem is referenced by:  rngoidmlem  21106  fincmpzer  25453  glmrngo  25585  exidreslem  26670 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-riota 6320  df-gid 20875  df-exid 20998  df-mgm 21002
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