MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cmpidelt Structured version   Unicode version

Theorem cmpidelt 25745
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 25743 . . . 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 2410 . . 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 25744 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X
)
61exidu1 25742 . . . 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 6285 . . . . . . . 8  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
87eqeq1d 2404 . . . . . . 7  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
9 oveq2 6286 . . . . . . . 8  |-  ( u  =  U  ->  (
x G u )  =  ( x G U ) )
109eqeq1d 2404 . . . . . . 7  |-  ( u  =  U  ->  (
( x G u )  =  x  <->  ( x G U )  =  x ) )
118, 10anbi12d 709 . . . . . 6  |-  ( u  =  U  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
1211ralbidv 2843 . . . . 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 6262 . . . 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 659 . . 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 6286 . . . . 5  |-  ( x  =  A  ->  ( U G x )  =  ( U G A ) )
17 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1816, 17eqeq12d 2424 . . . 4  |-  ( x  =  A  ->  (
( U G x )  =  x  <->  ( U G A )  =  A ) )
19 oveq1 6285 . . . . 5  |-  ( x  =  A  ->  (
x G U )  =  ( A G U ) )
2019, 17eqeq12d 2424 . . . 4  |-  ( x  =  A  ->  (
( x G U )  =  x  <->  ( A G U )  =  A ) )
2118, 20anbi12d 709 . . 3  |-  ( x  =  A  ->  (
( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( ( U G A )  =  A  /\  ( A G U )  =  A ) ) )
2221rspccva 3159 . 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 469 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 367    = wceq 1405    e. wcel 1842   A.wral 2754   E!wreu 2756    i^i cin 3413   ran crn 4824   ` cfv 5569   iota_crio 6239  (class class class)co 6278  GIdcgi 25603    ExId cexid 25730   Magmacmagm 25734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fo 5575  df-fv 5577  df-riota 6240  df-ov 6281  df-gid 25608  df-exid 25731  df-mgmOLD 25735
This theorem is referenced by:  rngoidmlem  25839  exidreslem  31621
  Copyright terms: Public domain W3C validator