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Theorem frmdup1 14502
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 14497 . . . 4  |-  ( I  e.  X  ->  M  e.  Mnd )
41, 3syl 15 . . 3  |-  ( ph  ->  M  e.  Mnd )
5 frmdup.g . . 3  |-  ( ph  ->  G  e.  Mnd )
64, 5jca 518 . 2  |-  ( ph  ->  ( M  e.  Mnd  /\  G  e.  Mnd )
)
75adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e. Word  I )  ->  G  e.  Mnd )
8 simpr 447 . . . . . . 7  |-  ( (
ph  /\  x  e. Word  I )  ->  x  e. Word  I )
9 frmdup.a . . . . . . . 8  |-  ( ph  ->  A : I --> B )
109adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e. Word  I )  ->  A :
I --> B )
11 wrdco 11502 . . . . . . 7  |-  ( ( x  e. Word  I  /\  A : I --> B )  ->  ( A  o.  x )  e. Word  B
)
128, 10, 11syl2anc 642 . . . . . 6  |-  ( (
ph  /\  x  e. Word  I )  ->  ( A  o.  x )  e. Word  B
)
13 frmdup.b . . . . . . 7  |-  B  =  ( Base `  G
)
1413gsumwcl 14479 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( A  o.  x
)  e. Word  B )  ->  ( G  gsumg  ( A  o.  x
) )  e.  B
)
157, 12, 14syl2anc 642 . . . . 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 5700 . . . 4  |-  ( ph  ->  E :Word  I --> B )
18 eqid 2296 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
192, 18frmdbas 14490 . . . . . 6  |-  ( I  e.  X  ->  ( Base `  M )  = Word 
I )
201, 19syl 15 . . . . 5  |-  ( ph  ->  ( Base `  M
)  = Word  I )
2120feq2d 5396 . . . 4  |-  ( ph  ->  ( E : (
Base `  M ) --> B 
<->  E :Word  I --> B ) )
2217, 21mpbird 223 . . 3  |-  ( ph  ->  E : ( Base `  M ) --> B )
232, 18frmdelbas 14491 . . . . . . . . 9  |-  ( y  e.  ( Base `  M
)  ->  y  e. Word  I )
2423ad2antrl 708 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  y  e. Word  I )
252, 18frmdelbas 14491 . . . . . . . . 9  |-  ( z  e.  ( Base `  M
)  ->  z  e. Word  I )
2625ad2antll 709 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  z  e. Word  I )
279adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  A : I --> B )
28 ccatco 11506 . . . . . . . 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 1182 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( A  o.  ( y concat  z ) )  =  ( ( A  o.  y
) concat  ( A  o.  z
) ) )
3029oveq2d 5890 . . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  G  e.  Mnd )
32 wrdco 11502 . . . . . . . 8  |-  ( ( y  e. Word  I  /\  A : I --> B )  ->  ( A  o.  y )  e. Word  B
)
3324, 27, 32syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( A  o.  y )  e. Word  B )
34 wrdco 11502 . . . . . . . 8  |-  ( ( z  e. Word  I  /\  A : I --> B )  ->  ( A  o.  z )  e. Word  B
)
3526, 27, 34syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( A  o.  z )  e. Word  B )
36 eqid 2296 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
3713, 36gsumccat 14480 . . . . . . 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 1182 . . . . . 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 2328 . . . . 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 2296 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
412, 18, 40frmdadd 14493 . . . . . . . 8  |-  ( ( y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) )  ->  (
y ( +g  `  M
) z )  =  ( y concat  z ) )
4241adantl 452 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  (
y ( +g  `  M
) z )  =  ( y concat  z ) )
4342fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( E `  (
y concat  z ) ) )
44 ccatcl 11445 . . . . . . . 8  |-  ( ( y  e. Word  I  /\  z  e. Word  I )  ->  ( y concat  z )  e. Word  I )
4524, 26, 44syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  (
y concat  z )  e. Word  I
)
46 coeq2 4858 . . . . . . . . 9  |-  ( x  =  ( y concat  z
)  ->  ( A  o.  x )  =  ( A  o.  ( y concat 
z ) ) )
4746oveq2d 5890 . . . . . . . 8  |-  ( x  =  ( y concat  z
)  ->  ( G  gsumg  ( A  o.  x ) )  =  ( G 
gsumg  ( A  o.  (
y concat  z ) ) ) )
48 ovex 5899 . . . . . . . 8  |-  ( G 
gsumg  ( A  o.  x
) )  e.  _V
4947, 16, 48fvmpt3i 5621 . . . . . . 7  |-  ( ( y concat  z )  e. Word 
I  ->  ( E `  ( y concat  z ) )  =  ( G 
gsumg  ( A  o.  (
y concat  z ) ) ) )
5045, 49syl 15 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y concat  z ) )  =  ( G  gsumg  ( A  o.  (
y concat  z ) ) ) )
5143, 50eqtrd 2328 . . . . 5  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( G  gsumg  ( A  o.  (
y concat  z ) ) ) )
52 coeq2 4858 . . . . . . . . 9  |-  ( x  =  y  ->  ( A  o.  x )  =  ( A  o.  y ) )
5352oveq2d 5890 . . . . . . . 8  |-  ( x  =  y  ->  ( G  gsumg  ( A  o.  x
) )  =  ( G  gsumg  ( A  o.  y
) ) )
5453, 16, 48fvmpt3i 5621 . . . . . . 7  |-  ( y  e. Word  I  ->  ( E `  y )  =  ( G  gsumg  ( A  o.  y ) ) )
55 coeq2 4858 . . . . . . . . 9  |-  ( x  =  z  ->  ( A  o.  x )  =  ( A  o.  z ) )
5655oveq2d 5890 . . . . . . . 8  |-  ( x  =  z  ->  ( G  gsumg  ( A  o.  x
) )  =  ( G  gsumg  ( A  o.  z
) ) )
5756, 16, 48fvmpt3i 5621 . . . . . . 7  |-  ( z  e. Word  I  ->  ( E `  z )  =  ( G  gsumg  ( A  o.  z ) ) )
5854, 57oveqan12d 5893 . . . . . 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 642 . . . . 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 2338 . . . 4  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( ( E `  y ) ( +g  `  G ) ( E `
 z ) ) )
6160ralrimivva 2648 . . 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 11434 . . . 4  |-  (/)  e. Word  I
63 coeq2 4858 . . . . . . . 8  |-  ( x  =  (/)  ->  ( A  o.  x )  =  ( A  o.  (/) ) )
64 co02 5202 . . . . . . . 8  |-  ( A  o.  (/) )  =  (/)
6563, 64syl6eq 2344 . . . . . . 7  |-  ( x  =  (/)  ->  ( A  o.  x )  =  (/) )
6665oveq2d 5890 . . . . . 6  |-  ( x  =  (/)  ->  ( G 
gsumg  ( A  o.  x
) )  =  ( G  gsumg  (/) ) )
67 eqid 2296 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
6867gsum0 14473 . . . . . 6  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
6966, 68syl6eq 2344 . . . . 5  |-  ( x  =  (/)  ->  ( G 
gsumg  ( A  o.  x
) )  =  ( 0g `  G ) )
7069, 16, 48fvmpt3i 5621 . . . 4  |-  ( (/)  e. Word  I  ->  ( E `  (/) )  =  ( 0g `  G ) )
7162, 70mp1i 11 . . 3  |-  ( ph  ->  ( E `  (/) )  =  ( 0g `  G
) )
7222, 61, 713jca 1132 . 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 14498 . . 3  |-  (/)  =  ( 0g `  M )
7418, 13, 40, 36, 73, 67ismhm 14433 . 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 645 1  |-  ( ph  ->  E  e.  ( M MndHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   (/)c0 3468    e. cmpt 4093    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874  Word cword 11419   concat cconcat 11420   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   MndHom cmhm 14429  freeMndcfrmd 14485
This theorem is referenced by:  frmdup3  14504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-word 11425  df-concat 11426  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-frmd 14487
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