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Theorem grpidd 16462
Description: Deduce the identity element of a magma 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 2429 . 2  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2429 . 2  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2429 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 grpidd.z . . 3  |-  ( ph  ->  .0.  e.  B )
5 grpidd.b . . 3  |-  ( ph  ->  B  =  ( Base `  G ) )
64, 5eleqtrd 2519 . 2  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
75eleq2d 2499 . . . 4  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  G
) ) )
87biimpar 487 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  x  e.  B )
9 grpidd.p . . . . . 6  |-  ( ph  ->  .+  =  ( +g  `  G ) )
109adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  .+  =  ( +g  `  G ) )
1110oveqd 6322 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  G ) x ) )
12 grpidd.i . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
1311, 12eqtr3d 2472 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  ( +g  `  G
) x )  =  x )
148, 13syldan 472 . 2  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  (  .0.  ( +g  `  G ) x )  =  x )
1510oveqd 6322 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  ( x ( +g  `  G )  .0.  ) )
16 grpidd.j . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  x )
1715, 16eqtr3d 2472 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x ( +g  `  G
)  .0.  )  =  x )
188, 17syldan 472 . 2  |-  ( (
ph  /\  x  e.  ( Base `  G )
)  ->  ( x
( +g  `  G )  .0.  )  =  x )
191, 2, 3, 6, 14, 18ismgmid2 16461 1  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   0gc0g 15297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-riota 6267  df-ov 6308  df-0g 15299
This theorem is referenced by:  ress0g  16516  imasmnd2  16524  isgrpde  16641  xrs0  28274
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