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Theorem gsumwmhm 14782
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 6081 . . . . 5  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( M  gsumg  (/) ) )
2 eqid 2435 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
32gsum0 14772 . . . . 5  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
41, 3syl6eq 2483 . . . 4  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( 0g `  M ) )
54fveq2d 5724 . . 3  |-  ( W  =  (/)  ->  ( H `
 ( M  gsumg  W ) )  =  ( H `
 ( 0g `  M ) ) )
6 coeq2 5023 . . . . . 6  |-  ( W  =  (/)  ->  ( H  o.  W )  =  ( H  o.  (/) ) )
7 co02 5375 . . . . . 6  |-  ( H  o.  (/) )  =  (/)
86, 7syl6eq 2483 . . . . 5  |-  ( W  =  (/)  ->  ( H  o.  W )  =  (/) )
98oveq2d 6089 . . . 4  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( N  gsumg  (/) ) )
10 eqid 2435 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
1110gsum0 14772 . . . 4  |-  ( N 
gsumg  (/) )  =  ( 0g
`  N )
129, 11syl6eq 2483 . . 3  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( 0g `  N ) )
135, 12eqeq12d 2449 . 2  |-  ( W  =  (/)  ->  ( ( H `  ( M 
gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) )  <->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) ) )
14 mhmrcl1 14733 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  M  e.  Mnd )
1514ad2antrr 707 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  M  e.  Mnd )
16 gsumwmhm.b . . . . . . 7  |-  B  =  ( Base `  M
)
17 eqid 2435 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
1816, 17mndcl 14687 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
19183expb 1154 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
2015, 19sylan 458 . . . 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 11725 . . . . . . 7  |-  ( W  e. Word  B  ->  W : ( 0..^ (
# `  W )
) --> B )
2221ad2antlr 708 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W : ( 0..^ (
# `  W )
) --> B )
23 wrdfin 11726 . . . . . . . . . . . 12  |-  ( W  e. Word  B  ->  W  e.  Fin )
2423adantl 453 . . . . . . . . . . 11  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  W  e.  Fin )
25 hashnncl 11637 . . . . . . . . . . 11  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
2726biimpar 472 . . . . . . . . 9  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  NN )
2827nnzd 10366 . . . . . . . 8  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  ZZ )
29 fzoval 11133 . . . . . . . 8  |-  ( (
# `  W )  e.  ZZ  ->  ( 0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
3028, 29syl 16 . . . . . . 7  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( 0..^ ( # `  W ) )  =  ( 0 ... (
( # `  W )  -  1 ) ) )
3130feq2d 5573 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( W : ( 0..^ ( # `  W
) ) --> B  <->  W :
( 0 ... (
( # `  W )  -  1 ) ) --> B ) )
3222, 31mpbid 202 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B )
3332ffvelrnda 5862 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( W `  x )  e.  B
)
34 nnm1nn0 10253 . . . . . 6  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  NN0 )
3527, 34syl 16 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( # `  W
)  -  1 )  e.  NN0 )
36 nn0uz 10512 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
3735, 36syl6eleq 2525 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( # `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
38 simpll 731 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  H  e.  ( M MndHom  N ) )
39 eqid 2435 . . . . . . 7  |-  ( +g  `  N )  =  ( +g  `  N )
4016, 17, 39mhmlin 14737 . . . . . 6  |-  ( ( H  e.  ( M MndHom  N )  /\  x  e.  B  /\  y  e.  B )  ->  ( H `  ( x
( +g  `  M ) y ) )  =  ( ( H `  x ) ( +g  `  N ) ( H `
 y ) ) )
41403expb 1154 . . . . 5  |-  ( ( H  e.  ( M MndHom  N )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( H `  ( x ( +g  `  M ) y ) )  =  ( ( H `  x ) ( +g  `  N
) ( H `  y ) ) )
4238, 41sylan 458 . . . 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
) ) )
43 ffn 5583 . . . . . . 7  |-  ( W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B  ->  W  Fn  ( 0 ... (
( # `  W )  -  1 ) ) )
4432, 43syl 16 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W  Fn  ( 0 ... ( ( # `  W )  -  1 ) ) )
45 fvco2 5790 . . . . . 6  |-  ( ( W  Fn  ( 0 ... ( ( # `  W )  -  1 ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
4644, 45sylan 458 . . . . 5  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
4746eqcomd 2440 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( H `  ( W `  x
) )  =  ( ( H  o.  W
) `  x )
)
4820, 33, 37, 42, 47seqhomo 11362 . . 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 ) ) )
4916, 17, 15, 37, 32gsumval2 14775 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( M  gsumg  W )  =  (  seq  0 ( ( +g  `  M ) ,  W ) `  ( ( # `  W
)  -  1 ) ) )
5049fveq2d 5724 . . 3  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( H `  (  seq  0 ( ( +g  `  M ) ,  W
) `  ( ( # `
 W )  - 
1 ) ) ) )
51 eqid 2435 . . . 4  |-  ( Base `  N )  =  (
Base `  N )
52 mhmrcl2 14734 . . . . 5  |-  ( H  e.  ( M MndHom  N
)  ->  N  e.  Mnd )
5352ad2antrr 707 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  N  e.  Mnd )
5416, 51mhmf 14735 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  H : B
--> ( Base `  N
) )
5554ad2antrr 707 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  H : B --> ( Base `  N ) )
56 fco 5592 . . . . 5  |-  ( ( H : B --> ( Base `  N )  /\  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B )  -> 
( H  o.  W
) : ( 0 ... ( ( # `  W )  -  1 ) ) --> ( Base `  N ) )
5755, 32, 56syl2anc 643 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H  o.  W
) : ( 0 ... ( ( # `  W )  -  1 ) ) --> ( Base `  N ) )
5851, 39, 53, 37, 57gsumval2 14775 . . 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 ) ) )
5948, 50, 583eqtr4d 2477 . 2  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )
602, 10mhm0 14738 . . 3  |-  ( H  e.  ( M MndHom  N
)  ->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) )
6160adantr 452 . 2  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H `  ( 0g `  M ) )  =  ( 0g `  N
) )
6213, 59, 61pm2.61ne 2673 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   (/)c0 3620    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Fincfn 7101   0cc0 8982   1c1 8983    - cmin 9283   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035  ..^cfzo 11127    seq cseq 11315   #chash 11610  Word cword 11709   Basecbs 13461   +g cplusg 13521   0gc0g 13715    gsumg cgsu 13716   Mndcmnd 14676   MndHom cmhm 14728
This theorem is referenced by:  frmdup3  14803  frgpup3lem  15401  symgtrinv  27381
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-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-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-word 11715  df-0g 13719  df-gsum 13720  df-mnd 14682  df-mhm 14730
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