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Theorem gsumwmhm 14483
Description: Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Hypothesis
Ref Expression
gsumwmhm.b  |-  B  =  ( Base `  M
)
Assertion
Ref Expression
gsumwmhm  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )

Proof of Theorem gsumwmhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5882 . . . . 5  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( M  gsumg  (/) ) )
2 eqid 2296 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
32gsum0 14473 . . . . 5  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
41, 3syl6eq 2344 . . . 4  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( 0g `  M ) )
54fveq2d 5545 . . 3  |-  ( W  =  (/)  ->  ( H `
 ( M  gsumg  W ) )  =  ( H `
 ( 0g `  M ) ) )
6 coeq2 4858 . . . . . 6  |-  ( W  =  (/)  ->  ( H  o.  W )  =  ( H  o.  (/) ) )
7 co02 5202 . . . . . 6  |-  ( H  o.  (/) )  =  (/)
86, 7syl6eq 2344 . . . . 5  |-  ( W  =  (/)  ->  ( H  o.  W )  =  (/) )
98oveq2d 5890 . . . 4  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( N  gsumg  (/) ) )
10 eqid 2296 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
1110gsum0 14473 . . . 4  |-  ( N 
gsumg  (/) )  =  ( 0g
`  N )
129, 11syl6eq 2344 . . 3  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( 0g `  N ) )
135, 12eqeq12d 2310 . 2  |-  ( W  =  (/)  ->  ( ( H `  ( M 
gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) )  <->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) ) )
14 mhmrcl1 14434 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  M  e.  Mnd )
1514ad2antrr 706 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  M  e.  Mnd )
16 gsumwmhm.b . . . . . . 7  |-  B  =  ( Base `  M
)
17 eqid 2296 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
1816, 17mndcl 14388 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
19183expb 1152 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
2015, 19sylan 457 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  M ) y )  e.  B )
21 wrdf 11435 . . . . . . 7  |-  ( W  e. Word  B  ->  W : ( 0..^ (
# `  W )
) --> B )
2221ad2antlr 707 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W : ( 0..^ (
# `  W )
) --> B )
23 wrdfin 11436 . . . . . . . . . . . 12  |-  ( W  e. Word  B  ->  W  e.  Fin )
2423adantl 452 . . . . . . . . . . 11  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  W  e.  Fin )
25 hashnncl 11370 . . . . . . . . . . 11  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
2624, 25syl 15 . . . . . . . . . 10  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
2726biimpar 471 . . . . . . . . 9  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  NN )
2827nnzd 10132 . . . . . . . 8  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  ZZ )
29 fzoval 10892 . . . . . . . 8  |-  ( (
# `  W )  e.  ZZ  ->  ( 0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
3028, 29syl 15 . . . . . . 7  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( 0..^ ( # `  W ) )  =  ( 0 ... (
( # `  W )  -  1 ) ) )
3130feq2d 5396 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( W : ( 0..^ ( # `  W
) ) --> B  <->  W :
( 0 ... (
( # `  W )  -  1 ) ) --> B ) )
3222, 31mpbid 201 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B )
33 ffvelrn 5679 . . . . 5  |-  ( ( W : ( 0 ... ( ( # `  W )  -  1 ) ) --> B  /\  x  e.  ( 0 ... ( ( # `  W )  -  1 ) ) )  -> 
( W `  x
)  e.  B )
3432, 33sylan 457 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( W `  x )  e.  B
)
35 nnm1nn0 10021 . . . . . 6  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  NN0 )
3627, 35syl 15 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( # `  W
)  -  1 )  e.  NN0 )
37 nn0uz 10278 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
3836, 37syl6eleq 2386 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( # `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
39 simpll 730 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  H  e.  ( M MndHom  N ) )
40 eqid 2296 . . . . . . 7  |-  ( +g  `  N )  =  ( +g  `  N )
4116, 17, 40mhmlin 14438 . . . . . 6  |-  ( ( H  e.  ( M MndHom  N )  /\  x  e.  B  /\  y  e.  B )  ->  ( H `  ( x
( +g  `  M ) y ) )  =  ( ( H `  x ) ( +g  `  N ) ( H `
 y ) ) )
42413expb 1152 . . . . 5  |-  ( ( H  e.  ( M MndHom  N )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( H `  ( x ( +g  `  M ) y ) )  =  ( ( H `  x ) ( +g  `  N
) ( H `  y ) ) )
4339, 42sylan 457 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( H `  (
x ( +g  `  M
) y ) )  =  ( ( H `
 x ) ( +g  `  N ) ( H `  y
) ) )
44 ffn 5405 . . . . . . 7  |-  ( W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B  ->  W  Fn  ( 0 ... (
( # `  W )  -  1 ) ) )
4532, 44syl 15 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W  Fn  ( 0 ... ( ( # `  W )  -  1 ) ) )
46 fvco2 5610 . . . . . 6  |-  ( ( W  Fn  ( 0 ... ( ( # `  W )  -  1 ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
4745, 46sylan 457 . . . . 5  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
4847eqcomd 2301 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( H `  ( W `  x
) )  =  ( ( H  o.  W
) `  x )
)
4920, 34, 38, 43, 48seqhomo 11109 . . 3  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  (  seq  0 ( ( +g  `  M ) ,  W
) `  ( ( # `
 W )  - 
1 ) ) )  =  (  seq  0
( ( +g  `  N
) ,  ( H  o.  W ) ) `
 ( ( # `  W )  -  1 ) ) )
5016, 17, 15, 38, 32gsumval2 14476 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( M  gsumg  W )  =  (  seq  0 ( ( +g  `  M ) ,  W ) `  ( ( # `  W
)  -  1 ) ) )
5150fveq2d 5545 . . 3  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( H `  (  seq  0 ( ( +g  `  M ) ,  W
) `  ( ( # `
 W )  - 
1 ) ) ) )
52 eqid 2296 . . . 4  |-  ( Base `  N )  =  (
Base `  N )
53 mhmrcl2 14435 . . . . 5  |-  ( H  e.  ( M MndHom  N
)  ->  N  e.  Mnd )
5453ad2antrr 706 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  N  e.  Mnd )
5516, 52mhmf 14436 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  H : B
--> ( Base `  N
) )
5655ad2antrr 706 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  H : B --> ( Base `  N ) )
57 fco 5414 . . . . 5  |-  ( ( H : B --> ( Base `  N )  /\  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B )  -> 
( H  o.  W
) : ( 0 ... ( ( # `  W )  -  1 ) ) --> ( Base `  N ) )
5856, 32, 57syl2anc 642 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H  o.  W
) : ( 0 ... ( ( # `  W )  -  1 ) ) --> ( Base `  N ) )
5952, 40, 54, 38, 58gsumval2 14476 . . 3  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( N  gsumg  ( H  o.  W
) )  =  (  seq  0 ( ( +g  `  N ) ,  ( H  o.  W ) ) `  ( ( # `  W
)  -  1 ) ) )
6049, 51, 593eqtr4d 2338 . 2  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )
612, 10mhm0 14439 . . 3  |-  ( H  e.  ( M MndHom  N
)  ->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) )
6261adantr 451 . 2  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H `  ( 0g `  M ) )  =  ( 0g `  N
) )
6313, 60, 62pm2.61ne 2534 1  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   0cc0 8753   1c1 8754    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798  ..^cfzo 10886    seq cseq 11062   #chash 11353  Word cword 11419   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   MndHom cmhm 14429
This theorem is referenced by:  frmdup3  14504  frgpup3lem  15102  symgtrinv  27516
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-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-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-word 11425  df-0g 13420  df-gsum 13421  df-mnd 14383  df-mhm 14431
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