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Theorem frmdup3lem 16728
Description: Lemma for frmdup3 16729. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
frmdup3.m  |-  M  =  (freeMnd `  I )
frmdup3.b  |-  B  =  ( Base `  G
)
frmdup3.u  |-  U  =  (varFMnd `  I )
Assertion
Ref Expression
frmdup3lem  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  ->  F  =  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) ) )
Distinct variable groups:    x, A    x, B    x, G    x, I    x, M    x, F    x, U    x, V

Proof of Theorem frmdup3lem
StepHypRef Expression
1 eqid 2471 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
2 frmdup3.b . . . . . 6  |-  B  =  ( Base `  G
)
31, 2mhmf 16665 . . . . 5  |-  ( F  e.  ( M MndHom  G
)  ->  F :
( Base `  M ) --> B )
43ad2antrl 742 . . . 4  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  ->  F : ( Base `  M
) --> B )
5 frmdup3.m . . . . . . . 8  |-  M  =  (freeMnd `  I )
65, 1frmdbas 16714 . . . . . . 7  |-  ( I  e.  V  ->  ( Base `  M )  = Word 
I )
763ad2ant2 1052 . . . . . 6  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( Base `  M
)  = Word  I )
87adantr 472 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  ->  ( Base `  M )  = Word 
I )
98feq2d 5725 . . . 4  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  ->  ( F : ( Base `  M
) --> B  <->  F :Word  I
--> B ) )
104, 9mpbid 215 . . 3  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  ->  F :Word  I --> B )
1110feqmptd 5932 . 2  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  ->  F  =  ( x  e. Word 
I  |->  ( F `  x ) ) )
12 simplrl 778 . . . . 5  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  F  e.  ( M MndHom  G ) )
13 simpr 468 . . . . . 6  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  x  e. Word  I )
14 frmdup3.u . . . . . . . . . 10  |-  U  =  (varFMnd `  I )
1514vrmdf 16720 . . . . . . . . 9  |-  ( I  e.  V  ->  U : I -->Word  I )
16153ad2ant2 1052 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I -->Word  I )
177feq3d 5726 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  ( U :
I --> ( Base `  M
)  <->  U : I -->Word  I )
)
1816, 17mpbird 240 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  ->  U : I --> ( Base `  M
) )
1918ad2antrr 740 . . . . . 6  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  U : I --> ( Base `  M ) )
20 wrdco 12987 . . . . . 6  |-  ( ( x  e. Word  I  /\  U : I --> ( Base `  M ) )  -> 
( U  o.  x
)  e. Word  ( Base `  M ) )
2113, 19, 20syl2anc 673 . . . . 5  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  ( U  o.  x
)  e. Word  ( Base `  M ) )
221gsumwmhm 16707 . . . . 5  |-  ( ( F  e.  ( M MndHom  G )  /\  ( U  o.  x )  e. Word  ( Base `  M
) )  ->  ( F `  ( M  gsumg  ( U  o.  x ) ) )  =  ( G  gsumg  ( F  o.  ( U  o.  x )
) ) )
2312, 21, 22syl2anc 673 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  ( F `  ( M  gsumg  ( U  o.  x
) ) )  =  ( G  gsumg  ( F  o.  ( U  o.  x )
) ) )
24 simpll2 1070 . . . . . 6  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  I  e.  V )
255, 14frmdgsum 16724 . . . . . 6  |-  ( ( I  e.  V  /\  x  e. Word  I )  ->  ( M  gsumg  ( U  o.  x
) )  =  x )
2624, 13, 25syl2anc 673 . . . . 5  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  ( M  gsumg  ( U  o.  x
) )  =  x )
2726fveq2d 5883 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  ( F `  ( M  gsumg  ( U  o.  x
) ) )  =  ( F `  x
) )
28 coass 5361 . . . . . 6  |-  ( ( F  o.  U )  o.  x )  =  ( F  o.  ( U  o.  x )
)
29 simplrr 779 . . . . . . 7  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  ( F  o.  U
)  =  A )
3029coeq1d 5001 . . . . . 6  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  ( ( F  o.  U )  o.  x
)  =  ( A  o.  x ) )
3128, 30syl5eqr 2519 . . . . 5  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  ( F  o.  ( U  o.  x )
)  =  ( A  o.  x ) )
3231oveq2d 6324 . . . 4  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  ( G  gsumg  ( F  o.  ( U  o.  x )
) )  =  ( G  gsumg  ( A  o.  x
) ) )
3323, 27, 323eqtr3d 2513 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  /\  x  e. Word  I )  ->  ( F `  x
)  =  ( G 
gsumg  ( A  o.  x
) ) )
3433mpteq2dva 4482 . 2  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  ->  (
x  e. Word  I  |->  ( F `  x ) )  =  ( x  e. Word  I  |->  ( G 
gsumg  ( A  o.  x
) ) ) )
3511, 34eqtrd 2505 1  |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  ->  F  =  ( x  e. Word 
I  |->  ( G  gsumg  ( A  o.  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    |-> cmpt 4454    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308  Word cword 12703   Basecbs 15199    gsumg cgsu 15417   Mndcmnd 16613   MndHom cmhm 16658  freeMndcfrmd 16709  varFMndcvrmd 16710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-frmd 16711  df-vrmd 16712
This theorem is referenced by:  frmdup3  16729  elmrsubrn  30230
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