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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  |-  X  =  ran  G
cmpidelt.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
cmpidelt  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X )  ->  (
( U G A )  =  A  /\  ( A G U )  =  A ) )

Proof of Theorem cmpidelt
Dummy variables  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmpidelt.1 . . . . 5  |-  X  =  ran  G
2 cmpidelt.2 . . . . 5  |-  U  =  (GId `  G )
31, 2idrval 21010 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  =  (
iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
43eqcomd 2301 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  =  U )
51, 2iorlid 21011 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X
)
61exidu1 21009 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E! u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
7 oveq1 5881 . . . . . . . 8  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
87eqeq1d 2304 . . . . . . 7  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
9 oveq2 5882 . . . . . . . 8  |-  ( u  =  U  ->  (
x G u )  =  ( x G U ) )
109eqeq1d 2304 . . . . . . 7  |-  ( u  =  U  ->  (
( x G u )  =  x  <->  ( x G U )  =  x ) )
118, 10anbi12d 691 . . . . . 6  |-  ( u  =  U  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
1211ralbidv 2576 . . . . 5  |-  ( u  =  U  ->  ( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  A. x  e.  X  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
1312riota2 6343 . . . 4  |-  ( ( U  e.  X  /\  E! u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  ->  ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  =  U ) )
145, 6, 13syl2anc 642 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( iota_ u  e.  X A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) )  =  U ) )
154, 14mpbird 223 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  A. x  e.  X  ( ( U G x )  =  x  /\  ( x G U )  =  x ) )
16 oveq2 5882 . . . . 5  |-  ( x  =  A  ->  ( U G x )  =  ( U G A ) )
17 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1816, 17eqeq12d 2310 . . . 4  |-  ( x  =  A  ->  (
( U G x )  =  x  <->  ( U G A )  =  A ) )
19 oveq1 5881 . . . . 5  |-  ( x  =  A  ->  (
x G U )  =  ( A G U ) )
2019, 17eqeq12d 2310 . . . 4  |-  ( x  =  A  ->  (
( x G U )  =  x  <->  ( A G U )  =  A ) )
2118, 20anbi12d 691 . . 3  |-  ( x  =  A  ->  (
( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( ( U G A )  =  A  /\  ( A G U )  =  A ) ) )
2221rspccva 2896 . 2  |-  ( ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G U )  =  x )  /\  A  e.  X )  ->  (
( U G A )  =  A  /\  ( A G U )  =  A ) )
2315, 22sylan 457 1  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X )  ->  (
( U G A )  =  A  /\  ( A G U )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E!wreu 2558    i^i cin 3164   ran crn 4706   ` cfv 5271  (class class class)co 5874   iota_crio 6313  GIdcgi 20870    ExId cexid 20997   Magmacmagm 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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