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Theorem frmdup1 14801
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 14796 . . . 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 11792 . . . . . . 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 14778 . . . . . 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 5885 . . . 4  |-  ( ph  ->  E :Word  I --> B )
18 eqid 2435 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
192, 18frmdbas 14789 . . . . . 6  |-  ( I  e.  X  ->  ( Base `  M )  = Word 
I )
201, 19syl 16 . . . . 5  |-  ( ph  ->  ( Base `  M
)  = Word  I )
2120feq2d 5573 . . . 4  |-  ( ph  ->  ( E : (
Base `  M ) --> B 
<->  E :Word  I --> B ) )
2217, 21mpbird 224 . . 3  |-  ( ph  ->  E : ( Base `  M ) --> B )
232, 18frmdelbas 14790 . . . . . . . . 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 14790 . . . . . . . . 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 11796 . . . . . . . 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 6089 . . . . . 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 11792 . . . . . . . 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 11792 . . . . . . . 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 2435 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
3713, 36gsumccat 14779 . . . . . . 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 2467 . . . . 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 2435 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
412, 18, 40frmdadd 14792 . . . . . . . 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 5724 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( E `  (
y concat  z ) ) )
44 ccatcl 11735 . . . . . . . 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 5023 . . . . . . . . 9  |-  ( x  =  ( y concat  z
)  ->  ( A  o.  x )  =  ( A  o.  ( y concat 
z ) ) )
4746oveq2d 6089 . . . . . . . 8  |-  ( x  =  ( y concat  z
)  ->  ( G  gsumg  ( A  o.  x ) )  =  ( G 
gsumg  ( A  o.  (
y concat  z ) ) ) )
48 ovex 6098 . . . . . . . 8  |-  ( G 
gsumg  ( A  o.  x
) )  e.  _V
4947, 16, 48fvmpt3i 5801 . . . . . . 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 2467 . . . . 5  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( G  gsumg  ( A  o.  (
y concat  z ) ) ) )
52 coeq2 5023 . . . . . . . . 9  |-  ( x  =  y  ->  ( A  o.  x )  =  ( A  o.  y ) )
5352oveq2d 6089 . . . . . . . 8  |-  ( x  =  y  ->  ( G  gsumg  ( A  o.  x
) )  =  ( G  gsumg  ( A  o.  y
) ) )
5453, 16, 48fvmpt3i 5801 . . . . . . 7  |-  ( y  e. Word  I  ->  ( E `  y )  =  ( G  gsumg  ( A  o.  y ) ) )
55 coeq2 5023 . . . . . . . . 9  |-  ( x  =  z  ->  ( A  o.  x )  =  ( A  o.  z ) )
5655oveq2d 6089 . . . . . . . 8  |-  ( x  =  z  ->  ( G  gsumg  ( A  o.  x
) )  =  ( G  gsumg  ( A  o.  z
) ) )
5756, 16, 48fvmpt3i 5801 . . . . . . 7  |-  ( z  e. Word  I  ->  ( E `  z )  =  ( G  gsumg  ( A  o.  z ) ) )
5854, 57oveqan12d 6092 . . . . . 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 2477 . . . 4  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( ( E `  y ) ( +g  `  G ) ( E `
 z ) ) )
6160ralrimivva 2790 . . 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 11724 . . . 4  |-  (/)  e. Word  I
63 coeq2 5023 . . . . . . . 8  |-  ( x  =  (/)  ->  ( A  o.  x )  =  ( A  o.  (/) ) )
64 co02 5375 . . . . . . . 8  |-  ( A  o.  (/) )  =  (/)
6563, 64syl6eq 2483 . . . . . . 7  |-  ( x  =  (/)  ->  ( A  o.  x )  =  (/) )
6665oveq2d 6089 . . . . . 6  |-  ( x  =  (/)  ->  ( G 
gsumg  ( A  o.  x
) )  =  ( G  gsumg  (/) ) )
67 eqid 2435 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
6867gsum0 14772 . . . . . 6  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
6966, 68syl6eq 2483 . . . . 5  |-  ( x  =  (/)  ->  ( G 
gsumg  ( A  o.  x
) )  =  ( 0g `  G ) )
7069, 16, 48fvmpt3i 5801 . . . 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 14797 . . 3  |-  (/)  =  ( 0g `  M )
7418, 13, 40, 36, 73, 67ismhm 14732 . 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 1652    e. wcel 1725   A.wral 2697   (/)c0 3620    e. cmpt 4258    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073  Word cword 11709   concat cconcat 11710   Basecbs 13461   +g cplusg 13521   0gc0g 13715    gsumg cgsu 13716   Mndcmnd 14676   MndHom cmhm 14728  freeMndcfrmd 14784
This theorem is referenced by:  frmdup3  14803
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-word 11715  df-concat 11716  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-gsum 13720  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-frmd 14786
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