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Theorem cmpidelt 26050
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 26048 . . . 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 2456 . . 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 26049 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X
)
61exidu1 26047 . . . 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 6295 . . . . . . . 8  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
87eqeq1d 2452 . . . . . . 7  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
9 oveq2 6296 . . . . . . . 8  |-  ( u  =  U  ->  (
x G u )  =  ( x G U ) )
109eqeq1d 2452 . . . . . . 7  |-  ( u  =  U  ->  (
( x G u )  =  x  <->  ( x G U )  =  x ) )
118, 10anbi12d 716 . . . . . 6  |-  ( u  =  U  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
1211ralbidv 2826 . . . . 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 6272 . . . 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 666 . . 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 236 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  A. x  e.  X  ( ( U G x )  =  x  /\  ( x G U )  =  x ) )
16 oveq2 6296 . . . . 5  |-  ( x  =  A  ->  ( U G x )  =  ( U G A ) )
17 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1816, 17eqeq12d 2465 . . . 4  |-  ( x  =  A  ->  (
( U G x )  =  x  <->  ( U G A )  =  A ) )
19 oveq1 6295 . . . . 5  |-  ( x  =  A  ->  (
x G U )  =  ( A G U ) )
2019, 17eqeq12d 2465 . . . 4  |-  ( x  =  A  ->  (
( x G U )  =  x  <->  ( A G U )  =  A ) )
2118, 20anbi12d 716 . . 3  |-  ( x  =  A  ->  (
( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( ( U G A )  =  A  /\  ( A G U )  =  A ) ) )
2221rspccva 3148 . 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 474 1  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  A  e.  X )  ->  (
( U G A )  =  A  /\  ( A G U )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736   E!wreu 2738    i^i cin 3402   ran crn 4834   ` cfv 5581   iota_crio 6249  (class class class)co 6288  GIdcgi 25908    ExId cexid 26035   Magmacmagm 26039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-riota 6250  df-ov 6291  df-gid 25913  df-exid 26036  df-mgmOLD 26040
This theorem is referenced by:  rngoidmlem  26144  exidreslem  32168
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