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Theorem cmpidelt 25007
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 25005 . . . 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 2475 . . 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 25006 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X
)
61exidu1 25004 . . . 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 6289 . . . . . . . 8  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
87eqeq1d 2469 . . . . . . 7  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
9 oveq2 6290 . . . . . . . 8  |-  ( u  =  U  ->  (
x G u )  =  ( x G U ) )
109eqeq1d 2469 . . . . . . 7  |-  ( u  =  U  ->  (
( x G u )  =  x  <->  ( x G U )  =  x ) )
118, 10anbi12d 710 . . . . . 6  |-  ( u  =  U  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
1211ralbidv 2903 . . . . 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 6266 . . . 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 661 . . 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 232 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  A. x  e.  X  ( ( U G x )  =  x  /\  ( x G U )  =  x ) )
16 oveq2 6290 . . . . 5  |-  ( x  =  A  ->  ( U G x )  =  ( U G A ) )
17 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1816, 17eqeq12d 2489 . . . 4  |-  ( x  =  A  ->  (
( U G x )  =  x  <->  ( U G A )  =  A ) )
19 oveq1 6289 . . . . 5  |-  ( x  =  A  ->  (
x G U )  =  ( A G U ) )
2019, 17eqeq12d 2489 . . . 4  |-  ( x  =  A  ->  (
( x G U )  =  x  <->  ( A G U )  =  A ) )
2118, 20anbi12d 710 . . 3  |-  ( x  =  A  ->  (
( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( ( U G A )  =  A  /\  ( A G U )  =  A ) ) )
2221rspccva 3213 . 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 471 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E!wreu 2816    i^i cin 3475   ran crn 5000   ` cfv 5586   iota_crio 6242  (class class class)co 6282  GIdcgi 24865    ExId cexid 24992   Magmacmagm 24996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-riota 6243  df-ov 6285  df-gid 24870  df-exid 24993  df-mgm 24997
This theorem is referenced by:  rngoidmlem  25101  exidreslem  29942
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