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Theorem frmdup1 14729
Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypotheses
Ref Expression
frmdup.m  |-  M  =  (freeMnd `  I )
frmdup.b  |-  B  =  ( Base `  G
)
frmdup.e  |-  E  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )
frmdup.g  |-  ( ph  ->  G  e.  Mnd )
frmdup.i  |-  ( ph  ->  I  e.  X )
frmdup.a  |-  ( ph  ->  A : I --> B )
Assertion
Ref Expression
frmdup1  |-  ( ph  ->  E  e.  ( M MndHom  G ) )
Distinct variable groups:    x, A    x, B    x, G    ph, x    x, I
Allowed substitution hints:    E( x)    M( x)    X( x)

Proof of Theorem frmdup1
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frmdup.i . . . 4  |-  ( ph  ->  I  e.  X )
2 frmdup.m . . . . 5  |-  M  =  (freeMnd `  I )
32frmdmnd 14724 . . . 4  |-  ( I  e.  X  ->  M  e.  Mnd )
41, 3syl 16 . . 3  |-  ( ph  ->  M  e.  Mnd )
5 frmdup.g . . 3  |-  ( ph  ->  G  e.  Mnd )
64, 5jca 519 . 2  |-  ( ph  ->  ( M  e.  Mnd  /\  G  e.  Mnd )
)
75adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e. Word  I )  ->  G  e.  Mnd )
8 simpr 448 . . . . . . 7  |-  ( (
ph  /\  x  e. Word  I )  ->  x  e. Word  I )
9 frmdup.a . . . . . . . 8  |-  ( ph  ->  A : I --> B )
109adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e. Word  I )  ->  A :
I --> B )
11 wrdco 11720 . . . . . . 7  |-  ( ( x  e. Word  I  /\  A : I --> B )  ->  ( A  o.  x )  e. Word  B
)
128, 10, 11syl2anc 643 . . . . . 6  |-  ( (
ph  /\  x  e. Word  I )  ->  ( A  o.  x )  e. Word  B
)
13 frmdup.b . . . . . . 7  |-  B  =  ( Base `  G
)
1413gsumwcl 14706 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( A  o.  x
)  e. Word  B )  ->  ( G  gsumg  ( A  o.  x
) )  e.  B
)
157, 12, 14syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e. Word  I )  ->  ( G  gsumg  ( A  o.  x ) )  e.  B )
16 frmdup.e . . . . 5  |-  E  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x
) ) )
1715, 16fmptd 5825 . . . 4  |-  ( ph  ->  E :Word  I --> B )
18 eqid 2380 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
192, 18frmdbas 14717 . . . . . 6  |-  ( I  e.  X  ->  ( Base `  M )  = Word 
I )
201, 19syl 16 . . . . 5  |-  ( ph  ->  ( Base `  M
)  = Word  I )
2120feq2d 5514 . . . 4  |-  ( ph  ->  ( E : (
Base `  M ) --> B 
<->  E :Word  I --> B ) )
2217, 21mpbird 224 . . 3  |-  ( ph  ->  E : ( Base `  M ) --> B )
232, 18frmdelbas 14718 . . . . . . . . 9  |-  ( y  e.  ( Base `  M
)  ->  y  e. Word  I )
2423ad2antrl 709 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  y  e. Word  I )
252, 18frmdelbas 14718 . . . . . . . . 9  |-  ( z  e.  ( Base `  M
)  ->  z  e. Word  I )
2625ad2antll 710 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  z  e. Word  I )
279adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  A : I --> B )
28 ccatco 11724 . . . . . . . 8  |-  ( ( y  e. Word  I  /\  z  e. Word  I  /\  A : I --> B )  ->  ( A  o.  ( y concat  z )
)  =  ( ( A  o.  y ) concat 
( A  o.  z
) ) )
2924, 26, 27, 28syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( A  o.  ( y concat  z ) )  =  ( ( A  o.  y
) concat  ( A  o.  z
) ) )
3029oveq2d 6029 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( G  gsumg  ( A  o.  (
y concat  z ) ) )  =  ( G  gsumg  ( ( A  o.  y ) concat 
( A  o.  z
) ) ) )
315adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  G  e.  Mnd )
32 wrdco 11720 . . . . . . . 8  |-  ( ( y  e. Word  I  /\  A : I --> B )  ->  ( A  o.  y )  e. Word  B
)
3324, 27, 32syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( A  o.  y )  e. Word  B )
34 wrdco 11720 . . . . . . . 8  |-  ( ( z  e. Word  I  /\  A : I --> B )  ->  ( A  o.  z )  e. Word  B
)
3526, 27, 34syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( A  o.  z )  e. Word  B )
36 eqid 2380 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
3713, 36gsumccat 14707 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( A  o.  y
)  e. Word  B  /\  ( A  o.  z
)  e. Word  B )  ->  ( G  gsumg  ( ( A  o.  y ) concat  ( A  o.  z ) ) )  =  ( ( G 
gsumg  ( A  o.  y
) ) ( +g  `  G ) ( G 
gsumg  ( A  o.  z
) ) ) )
3831, 33, 35, 37syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( G  gsumg  ( ( A  o.  y ) concat  ( A  o.  z ) ) )  =  ( ( G 
gsumg  ( A  o.  y
) ) ( +g  `  G ) ( G 
gsumg  ( A  o.  z
) ) ) )
3930, 38eqtrd 2412 . . . . 5  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( G  gsumg  ( A  o.  (
y concat  z ) ) )  =  ( ( G 
gsumg  ( A  o.  y
) ) ( +g  `  G ) ( G 
gsumg  ( A  o.  z
) ) ) )
40 eqid 2380 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
412, 18, 40frmdadd 14720 . . . . . . . 8  |-  ( ( y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) )  ->  (
y ( +g  `  M
) z )  =  ( y concat  z ) )
4241adantl 453 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  (
y ( +g  `  M
) z )  =  ( y concat  z ) )
4342fveq2d 5665 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( E `  (
y concat  z ) ) )
44 ccatcl 11663 . . . . . . . 8  |-  ( ( y  e. Word  I  /\  z  e. Word  I )  ->  ( y concat  z )  e. Word  I )
4524, 26, 44syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  (
y concat  z )  e. Word  I
)
46 coeq2 4964 . . . . . . . . 9  |-  ( x  =  ( y concat  z
)  ->  ( A  o.  x )  =  ( A  o.  ( y concat 
z ) ) )
4746oveq2d 6029 . . . . . . . 8  |-  ( x  =  ( y concat  z
)  ->  ( G  gsumg  ( A  o.  x ) )  =  ( G 
gsumg  ( A  o.  (
y concat  z ) ) ) )
48 ovex 6038 . . . . . . . 8  |-  ( G 
gsumg  ( A  o.  x
) )  e.  _V
4947, 16, 48fvmpt3i 5741 . . . . . . 7  |-  ( ( y concat  z )  e. Word 
I  ->  ( E `  ( y concat  z ) )  =  ( G 
gsumg  ( A  o.  (
y concat  z ) ) ) )
5045, 49syl 16 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y concat  z ) )  =  ( G  gsumg  ( A  o.  (
y concat  z ) ) ) )
5143, 50eqtrd 2412 . . . . 5  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( G  gsumg  ( A  o.  (
y concat  z ) ) ) )
52 coeq2 4964 . . . . . . . . 9  |-  ( x  =  y  ->  ( A  o.  x )  =  ( A  o.  y ) )
5352oveq2d 6029 . . . . . . . 8  |-  ( x  =  y  ->  ( G  gsumg  ( A  o.  x
) )  =  ( G  gsumg  ( A  o.  y
) ) )
5453, 16, 48fvmpt3i 5741 . . . . . . 7  |-  ( y  e. Word  I  ->  ( E `  y )  =  ( G  gsumg  ( A  o.  y ) ) )
55 coeq2 4964 . . . . . . . . 9  |-  ( x  =  z  ->  ( A  o.  x )  =  ( A  o.  z ) )
5655oveq2d 6029 . . . . . . . 8  |-  ( x  =  z  ->  ( G  gsumg  ( A  o.  x
) )  =  ( G  gsumg  ( A  o.  z
) ) )
5756, 16, 48fvmpt3i 5741 . . . . . . 7  |-  ( z  e. Word  I  ->  ( E `  z )  =  ( G  gsumg  ( A  o.  z ) ) )
5854, 57oveqan12d 6032 . . . . . 6  |-  ( ( y  e. Word  I  /\  z  e. Word  I )  ->  ( ( E `  y ) ( +g  `  G ) ( E `
 z ) )  =  ( ( G 
gsumg  ( A  o.  y
) ) ( +g  `  G ) ( G 
gsumg  ( A  o.  z
) ) ) )
5924, 26, 58syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  (
( E `  y
) ( +g  `  G
) ( E `  z ) )  =  ( ( G  gsumg  ( A  o.  y ) ) ( +g  `  G
) ( G  gsumg  ( A  o.  z ) ) ) )
6039, 51, 593eqtr4d 2422 . . . 4  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( ( E `  y ) ( +g  `  G ) ( E `
 z ) ) )
6160ralrimivva 2734 . . 3  |-  ( ph  ->  A. y  e.  (
Base `  M ) A. z  e.  ( Base `  M ) ( E `  ( y ( +g  `  M
) z ) )  =  ( ( E `
 y ) ( +g  `  G ) ( E `  z
) ) )
62 wrd0 11652 . . . 4  |-  (/)  e. Word  I
63 coeq2 4964 . . . . . . . 8  |-  ( x  =  (/)  ->  ( A  o.  x )  =  ( A  o.  (/) ) )
64 co02 5316 . . . . . . . 8  |-  ( A  o.  (/) )  =  (/)
6563, 64syl6eq 2428 . . . . . . 7  |-  ( x  =  (/)  ->  ( A  o.  x )  =  (/) )
6665oveq2d 6029 . . . . . 6  |-  ( x  =  (/)  ->  ( G 
gsumg  ( A  o.  x
) )  =  ( G  gsumg  (/) ) )
67 eqid 2380 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
6867gsum0 14700 . . . . . 6  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
6966, 68syl6eq 2428 . . . . 5  |-  ( x  =  (/)  ->  ( G 
gsumg  ( A  o.  x
) )  =  ( 0g `  G ) )
7069, 16, 48fvmpt3i 5741 . . . 4  |-  ( (/)  e. Word  I  ->  ( E `  (/) )  =  ( 0g `  G ) )
7162, 70mp1i 12 . . 3  |-  ( ph  ->  ( E `  (/) )  =  ( 0g `  G
) )
7222, 61, 713jca 1134 . 2  |-  ( ph  ->  ( E : (
Base `  M ) --> B  /\  A. y  e.  ( Base `  M
) A. z  e.  ( Base `  M
) ( E `  ( y ( +g  `  M ) z ) )  =  ( ( E `  y ) ( +g  `  G
) ( E `  z ) )  /\  ( E `  (/) )  =  ( 0g `  G
) ) )
732frmd0 14725 . . 3  |-  (/)  =  ( 0g `  M )
7418, 13, 40, 36, 73, 67ismhm 14660 . 2  |-  ( E  e.  ( M MndHom  G
)  <->  ( ( M  e.  Mnd  /\  G  e.  Mnd )  /\  ( E : ( Base `  M
) --> B  /\  A. y  e.  ( Base `  M ) A. z  e.  ( Base `  M
) ( E `  ( y ( +g  `  M ) z ) )  =  ( ( E `  y ) ( +g  `  G
) ( E `  z ) )  /\  ( E `  (/) )  =  ( 0g `  G
) ) ) )
756, 72, 74sylanbrc 646 1  |-  ( ph  ->  E  e.  ( M MndHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   (/)c0 3564    e. cmpt 4200    o. ccom 4815   -->wf 5383   ` cfv 5387  (class class class)co 6013  Word cword 11637   concat cconcat 11638   Basecbs 13389   +g cplusg 13449   0gc0g 13643    gsumg cgsu 13644   Mndcmnd 14604   MndHom cmhm 14656  freeMndcfrmd 14712
This theorem is referenced by:  frmdup3  14731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969  df-fzo 11059  df-seq 11244  df-hash 11539  df-word 11643  df-concat 11644  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-0g 13647  df-gsum 13648  df-mnd 14610  df-mhm 14658  df-submnd 14659  df-frmd 14714
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