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Theorem grpidd 14710
Description: Deduce the identity element of a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grpidd.b  |-  ( ph  ->  B  =  ( Base `  G ) )
grpidd.p  |-  ( ph  ->  .+  =  ( +g  `  G ) )
grpidd.z  |-  ( ph  ->  .0.  e.  B )
grpidd.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
grpidd.j  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  x )
Assertion
Ref Expression
grpidd  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Distinct variable groups:    x, G    ph, x    x,  .0.
Allowed substitution hints:    B( x)    .+ ( x)

Proof of Theorem grpidd
StepHypRef Expression
1 eqid 2435 . 2  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2435 . 2  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2435 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 grpidd.z . . 3  |-  ( ph  ->  .0.  e.  B )
5 grpidd.b . . 3  |-  ( ph  ->  B  =  ( Base `  G ) )
64, 5eleqtrd 2511 . 2  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
75eleq2d 2502 . . . 4  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  G
) ) )
87biimpar 472 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  x  e.  B )
9 grpidd.p . . . . . 6  |-  ( ph  ->  .+  =  ( +g  `  G ) )
109adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  .+  =  ( +g  `  G ) )
1110oveqd 6090 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  G ) x ) )
12 grpidd.i . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
1311, 12eqtr3d 2469 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  ( +g  `  G
) x )  =  x )
148, 13syldan 457 . 2  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  (  .0.  ( +g  `  G ) x )  =  x )
1510oveqd 6090 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  ( x ( +g  `  G )  .0.  ) )
16 grpidd.j . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  x )
1715, 16eqtr3d 2469 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x ( +g  `  G
)  .0.  )  =  x )
188, 17syldan 457 . 2  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  ( x
( +g  `  G )  .0.  )  =  x )
191, 2, 3, 6, 14, 18ismgmid2 14705 1  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   0gc0g 13715
This theorem is referenced by:  imasmnd2  14724  isgrpde  14821  ress0g  24174  xrs0  24189
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-riota 6541  df-0g 13719
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