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Theorem cmpidelt 21917
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 21915 . . . 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 2441 . . 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 21916 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X
)
61exidu1 21914 . . . 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 6088 . . . . . . . 8  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
87eqeq1d 2444 . . . . . . 7  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
9 oveq2 6089 . . . . . . . 8  |-  ( u  =  U  ->  (
x G u )  =  ( x G U ) )
109eqeq1d 2444 . . . . . . 7  |-  ( u  =  U  ->  (
( x G u )  =  x  <->  ( x G U )  =  x ) )
118, 10anbi12d 692 . . . . . 6  |-  ( u  =  U  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
1211ralbidv 2725 . . . . 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 6572 . . . 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 643 . . 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 224 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  A. x  e.  X  ( ( U G x )  =  x  /\  ( x G U )  =  x ) )
16 oveq2 6089 . . . . 5  |-  ( x  =  A  ->  ( U G x )  =  ( U G A ) )
17 id 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1816, 17eqeq12d 2450 . . . 4  |-  ( x  =  A  ->  (
( U G x )  =  x  <->  ( U G A )  =  A ) )
19 oveq1 6088 . . . . 5  |-  ( x  =  A  ->  (
x G U )  =  ( A G U ) )
2019, 17eqeq12d 2450 . . . 4  |-  ( x  =  A  ->  (
( x G U )  =  x  <->  ( A G U )  =  A ) )
2118, 20anbi12d 692 . . 3  |-  ( x  =  A  ->  (
( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( ( U G A )  =  A  /\  ( A G U )  =  A ) ) )
2221rspccva 3051 . 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 458 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E!wreu 2707    i^i cin 3319   ran crn 4879   ` cfv 5454  (class class class)co 6081   iota_crio 6542  GIdcgi 21775    ExId cexid 21902   Magmacmagm 21906
This theorem is referenced by:  rngoidmlem  22011  exidreslem  26552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084  df-riota 6549  df-gid 21780  df-exid 21903  df-mgm 21907
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