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Theorem mhmid 12868
Description: A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.)
Hypotheses
Ref Expression
ghmgrp.f  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  .+  y
) )  =  ( ( F `  x
)  .+^  ( F `  y ) ) )
ghmgrp.x  |-  X  =  ( Base `  G
)
ghmgrp.y  |-  Y  =  ( Base `  H
)
ghmgrp.p  |-  .+  =  ( +g  `  G )
ghmgrp.q  |-  .+^  =  ( +g  `  H )
ghmgrp.1  |-  ( ph  ->  F : X -onto-> Y
)
mhmmnd.3  |-  ( ph  ->  G  e.  Mnd )
mhmid.0  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
mhmid  |-  ( ph  ->  ( F `  .0.  )  =  ( 0g `  H ) )
Distinct variable groups:    x, F, y   
x, G, y    x,  .+ , y    x, H, y   
x, X, y    x, Y, y    x,  .+^ , y    ph, x, y    x,  .0. , y

Proof of Theorem mhmid
Dummy variables  a  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmgrp.y . 2  |-  Y  =  ( Base `  H
)
2 eqid 2177 . 2  |-  ( 0g
`  H )  =  ( 0g `  H
)
3 ghmgrp.q . 2  |-  .+^  =  ( +g  `  H )
4 ghmgrp.1 . . . 4  |-  ( ph  ->  F : X -onto-> Y
)
5 fof 5434 . . . 4  |-  ( F : X -onto-> Y  ->  F : X --> Y )
64, 5syl 14 . . 3  |-  ( ph  ->  F : X --> Y )
7 mhmmnd.3 . . . 4  |-  ( ph  ->  G  e.  Mnd )
8 ghmgrp.x . . . . 5  |-  X  =  ( Base `  G
)
9 mhmid.0 . . . . 5  |-  .0.  =  ( 0g `  G )
108, 9mndidcl 12723 . . . 4  |-  ( G  e.  Mnd  ->  .0.  e.  X )
117, 10syl 14 . . 3  |-  ( ph  ->  .0.  e.  X )
126, 11ffvelcdmd 5648 . 2  |-  ( ph  ->  ( F `  .0.  )  e.  Y )
13 simplll 533 . . . . . . 7  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ph )
14 ghmgrp.f . . . . . . 7  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  .+  y
) )  =  ( ( F `  x
)  .+^  ( F `  y ) ) )
1513, 14syl3an1 1271 . . . . . 6  |-  ( ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  /\  x  e.  X  /\  y  e.  X )  ->  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
 y ) ) )
167ad3antrrr 492 . . . . . . 7  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  G  e.  Mnd )
1716, 10syl 14 . . . . . 6  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  .0.  e.  X )
18 simplr 528 . . . . . 6  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  i  e.  X )
1915, 17, 18mhmlem 12867 . . . . 5  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( F `  (  .0.  .+  i
) )  =  ( ( F `  .0.  )  .+^  ( F `  i ) ) )
20 ghmgrp.p . . . . . . . 8  |-  .+  =  ( +g  `  G )
218, 20, 9mndlid 12728 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  i  e.  X )  ->  (  .0.  .+  i
)  =  i )
2216, 18, 21syl2anc 411 . . . . . 6  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  (  .0.  .+  i )  =  i )
2322fveq2d 5515 . . . . 5  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( F `  (  .0.  .+  i
) )  =  ( F `  i ) )
2419, 23eqtr3d 2212 . . . 4  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( ( F `  .0.  )  .+^  ( F `  i ) )  =  ( F `
 i ) )
25 simpr 110 . . . . 5  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( F `  i )  =  a )
2625oveq2d 5885 . . . 4  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( ( F `  .0.  )  .+^  ( F `  i ) )  =  ( ( F `  .0.  )  .+^  a ) )
2724, 26, 253eqtr3d 2218 . . 3  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( ( F `  .0.  )  .+^  a )  =  a )
28 foelcdmi 5564 . . . 4  |-  ( ( F : X -onto-> Y  /\  a  e.  Y
)  ->  E. i  e.  X  ( F `  i )  =  a )
294, 28sylan 283 . . 3  |-  ( (
ph  /\  a  e.  Y )  ->  E. i  e.  X  ( F `  i )  =  a )
3027, 29r19.29a 2620 . 2  |-  ( (
ph  /\  a  e.  Y )  ->  (
( F `  .0.  )  .+^  a )  =  a )
3115, 18, 17mhmlem 12867 . . . . 5  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( F `  ( i  .+  .0.  ) )  =  ( ( F `  i
)  .+^  ( F `  .0.  ) ) )
328, 20, 9mndrid 12729 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  i  e.  X )  ->  ( i  .+  .0.  )  =  i )
3316, 18, 32syl2anc 411 . . . . . 6  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( i  .+  .0.  )  =  i )
3433fveq2d 5515 . . . . 5  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( F `  ( i  .+  .0.  ) )  =  ( F `  i ) )
3531, 34eqtr3d 2212 . . . 4  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( ( F `  i )  .+^  ( F `  .0.  ) )  =  ( F `  i ) )
3625oveq1d 5884 . . . 4  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( ( F `  i )  .+^  ( F `  .0.  ) )  =  ( a  .+^  ( F `  .0.  ) ) )
3735, 36, 253eqtr3d 2218 . . 3  |-  ( ( ( ( ph  /\  a  e.  Y )  /\  i  e.  X
)  /\  ( F `  i )  =  a )  ->  ( a  .+^  ( F `  .0.  ) )  =  a )
3837, 29r19.29a 2620 . 2  |-  ( (
ph  /\  a  e.  Y )  ->  (
a  .+^  ( F `  .0.  ) )  =  a )
391, 2, 3, 12, 30, 38ismgmid2 12691 1  |-  ( ph  ->  ( F `  .0.  )  =  ( 0g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   -->wf 5208   -onto->wfo 5210   ` cfv 5212  (class class class)co 5869   Basecbs 12445   +g cplusg 12518   0gc0g 12653   Mndcmnd 12709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fo 5218  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-0g 12655  df-mgm 12667  df-sgrp 12700  df-mnd 12710
This theorem is referenced by:  mhmfmhm  12870  ghmgrp  12871
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