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Theorem frmdup1 15547
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 15542 . . . 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 532 . 2  |-  ( ph  ->  ( M  e.  Mnd  /\  G  e.  Mnd )
)
75adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e. Word  I )  ->  G  e.  Mnd )
8 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e. Word  I )  ->  x  e. Word  I )
9 frmdup.a . . . . . . . 8  |-  ( ph  ->  A : I --> B )
109adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e. Word  I )  ->  A :
I --> B )
11 wrdco 12464 . . . . . . 7  |-  ( ( x  e. Word  I  /\  A : I --> B )  ->  ( A  o.  x )  e. Word  B
)
128, 10, 11syl2anc 661 . . . . . 6  |-  ( (
ph  /\  x  e. Word  I )  ->  ( A  o.  x )  e. Word  B
)
13 frmdup.b . . . . . . 7  |-  B  =  ( Base `  G
)
1413gsumwcl 15523 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( A  o.  x
)  e. Word  B )  ->  ( G  gsumg  ( A  o.  x
) )  e.  B
)
157, 12, 14syl2anc 661 . . . . 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 5872 . . . 4  |-  ( ph  ->  E :Word  I --> B )
18 eqid 2443 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
192, 18frmdbas 15535 . . . . . 6  |-  ( I  e.  X  ->  ( Base `  M )  = Word 
I )
201, 19syl 16 . . . . 5  |-  ( ph  ->  ( Base `  M
)  = Word  I )
2120feq2d 5552 . . . 4  |-  ( ph  ->  ( E : (
Base `  M ) --> B 
<->  E :Word  I --> B ) )
2217, 21mpbird 232 . . 3  |-  ( ph  ->  E : ( Base `  M ) --> B )
232, 18frmdelbas 15536 . . . . . . . . 9  |-  ( y  e.  ( Base `  M
)  ->  y  e. Word  I )
2423ad2antrl 727 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  y  e. Word  I )
252, 18frmdelbas 15536 . . . . . . . . 9  |-  ( z  e.  ( Base `  M
)  ->  z  e. Word  I )
2625ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  z  e. Word  I )
279adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  A : I --> B )
28 ccatco 12468 . . . . . . . 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 1218 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( A  o.  ( y concat  z ) )  =  ( ( A  o.  y
) concat  ( A  o.  z
) ) )
3029oveq2d 6112 . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  G  e.  Mnd )
32 wrdco 12464 . . . . . . . 8  |-  ( ( y  e. Word  I  /\  A : I --> B )  ->  ( A  o.  y )  e. Word  B
)
3324, 27, 32syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( A  o.  y )  e. Word  B )
34 wrdco 12464 . . . . . . . 8  |-  ( ( z  e. Word  I  /\  A : I --> B )  ->  ( A  o.  z )  e. Word  B
)
3526, 27, 34syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( A  o.  z )  e. Word  B )
36 eqid 2443 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
3713, 36gsumccat 15524 . . . . . . 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 1218 . . . . . 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 2475 . . . . 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 2443 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
412, 18, 40frmdadd 15538 . . . . . . . 8  |-  ( ( y  e.  ( Base `  M )  /\  z  e.  ( Base `  M
) )  ->  (
y ( +g  `  M
) z )  =  ( y concat  z ) )
4241adantl 466 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  (
y ( +g  `  M
) z )  =  ( y concat  z ) )
4342fveq2d 5700 . . . . . 6  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( E `  (
y concat  z ) ) )
44 ccatcl 12279 . . . . . . . 8  |-  ( ( y  e. Word  I  /\  z  e. Word  I )  ->  ( y concat  z )  e. Word  I )
4524, 26, 44syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  (
y concat  z )  e. Word  I
)
46 coeq2 5003 . . . . . . . . 9  |-  ( x  =  ( y concat  z
)  ->  ( A  o.  x )  =  ( A  o.  ( y concat 
z ) ) )
4746oveq2d 6112 . . . . . . . 8  |-  ( x  =  ( y concat  z
)  ->  ( G  gsumg  ( A  o.  x ) )  =  ( G 
gsumg  ( A  o.  (
y concat  z ) ) ) )
48 ovex 6121 . . . . . . . 8  |-  ( G 
gsumg  ( A  o.  x
) )  e.  _V
4947, 16, 48fvmpt3i 5783 . . . . . . 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 2475 . . . . 5  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( G  gsumg  ( A  o.  (
y concat  z ) ) ) )
52 coeq2 5003 . . . . . . . . 9  |-  ( x  =  y  ->  ( A  o.  x )  =  ( A  o.  y ) )
5352oveq2d 6112 . . . . . . . 8  |-  ( x  =  y  ->  ( G  gsumg  ( A  o.  x
) )  =  ( G  gsumg  ( A  o.  y
) ) )
5453, 16, 48fvmpt3i 5783 . . . . . . 7  |-  ( y  e. Word  I  ->  ( E `  y )  =  ( G  gsumg  ( A  o.  y ) ) )
55 coeq2 5003 . . . . . . . . 9  |-  ( x  =  z  ->  ( A  o.  x )  =  ( A  o.  z ) )
5655oveq2d 6112 . . . . . . . 8  |-  ( x  =  z  ->  ( G  gsumg  ( A  o.  x
) )  =  ( G  gsumg  ( A  o.  z
) ) )
5756, 16, 48fvmpt3i 5783 . . . . . . 7  |-  ( z  e. Word  I  ->  ( E `  z )  =  ( G  gsumg  ( A  o.  z ) ) )
5854, 57oveqan12d 6115 . . . . . 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 661 . . . . 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 2485 . . . 4  |-  ( (
ph  /\  ( y  e.  ( Base `  M
)  /\  z  e.  ( Base `  M )
) )  ->  ( E `  ( y
( +g  `  M ) z ) )  =  ( ( E `  y ) ( +g  `  G ) ( E `
 z ) ) )
6160ralrimivva 2813 . . 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 12257 . . . 4  |-  (/)  e. Word  I
63 coeq2 5003 . . . . . . . 8  |-  ( x  =  (/)  ->  ( A  o.  x )  =  ( A  o.  (/) ) )
64 co02 5356 . . . . . . . 8  |-  ( A  o.  (/) )  =  (/)
6563, 64syl6eq 2491 . . . . . . 7  |-  ( x  =  (/)  ->  ( A  o.  x )  =  (/) )
6665oveq2d 6112 . . . . . 6  |-  ( x  =  (/)  ->  ( G 
gsumg  ( A  o.  x
) )  =  ( G  gsumg  (/) ) )
67 eqid 2443 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
6867gsum0 15515 . . . . . 6  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
6966, 68syl6eq 2491 . . . . 5  |-  ( x  =  (/)  ->  ( G 
gsumg  ( A  o.  x
) )  =  ( 0g `  G ) )
7069, 16, 48fvmpt3i 5783 . . . 4  |-  ( (/)  e. Word  I  ->  ( E `  (/) )  =  ( 0g `  G ) )
7162, 70mp1i 12 . . 3  |-  ( ph  ->  ( E `  (/) )  =  ( 0g `  G
) )
7222, 61, 713jca 1168 . 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 15543 . . 3  |-  (/)  =  ( 0g `  M )
7418, 13, 40, 36, 73, 67ismhm 15471 . 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 664 1  |-  ( ph  ->  E  e.  ( M MndHom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   (/)c0 3642    e. cmpt 4355    o. ccom 4849   -->wf 5419   ` cfv 5423  (class class class)co 6096  Word cword 12226   concat cconcat 12228   Basecbs 14179   +g cplusg 14243   0gc0g 14383    gsumg cgsu 14384   Mndcmnd 15414   MndHom cmhm 15467  freeMndcfrmd 15530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-word 12234  df-concat 12236  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-gsum 14386  df-mnd 15420  df-mhm 15469  df-submnd 15470  df-frmd 15532
This theorem is referenced by:  frmdup3  15549
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