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Theorem gsumwmhm 15641
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 6207 . . . . 5  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( M  gsumg  (/) ) )
2 eqid 2454 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
32gsum0 15628 . . . . 5  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
41, 3syl6eq 2511 . . . 4  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( 0g `  M ) )
54fveq2d 5802 . . 3  |-  ( W  =  (/)  ->  ( H `
 ( M  gsumg  W ) )  =  ( H `
 ( 0g `  M ) ) )
6 coeq2 5105 . . . . . 6  |-  ( W  =  (/)  ->  ( H  o.  W )  =  ( H  o.  (/) ) )
7 co02 5458 . . . . . 6  |-  ( H  o.  (/) )  =  (/)
86, 7syl6eq 2511 . . . . 5  |-  ( W  =  (/)  ->  ( H  o.  W )  =  (/) )
98oveq2d 6215 . . . 4  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( N  gsumg  (/) ) )
10 eqid 2454 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
1110gsum0 15628 . . . 4  |-  ( N 
gsumg  (/) )  =  ( 0g
`  N )
129, 11syl6eq 2511 . . 3  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( 0g `  N ) )
135, 12eqeq12d 2476 . 2  |-  ( W  =  (/)  ->  ( ( H `  ( M 
gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) )  <->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) ) )
14 mhmrcl1 15585 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  M  e.  Mnd )
1514ad2antrr 725 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  M  e.  Mnd )
16 gsumwmhm.b . . . . . . 7  |-  B  =  ( Base `  M
)
17 eqid 2454 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
1816, 17mndcl 15538 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
19183expb 1189 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
2015, 19sylan 471 . . . 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 12357 . . . . . . 7  |-  ( W  e. Word  B  ->  W : ( 0..^ (
# `  W )
) --> B )
2221ad2antlr 726 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W : ( 0..^ (
# `  W )
) --> B )
23 wrdfin 12365 . . . . . . . . . . . 12  |-  ( W  e. Word  B  ->  W  e.  Fin )
2423adantl 466 . . . . . . . . . . 11  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  W  e.  Fin )
25 hashnncl 12250 . . . . . . . . . . 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 485 . . . . . . . . 9  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  NN )
2827nnzd 10856 . . . . . . . 8  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  ZZ )
29 fzoval 11670 . . . . . . . 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 5654 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( W : ( 0..^ ( # `  W
) ) --> B  <->  W :
( 0 ... (
( # `  W )  -  1 ) ) --> B ) )
3222, 31mpbid 210 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B )
3332ffvelrnda 5951 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( W `  x )  e.  B
)
34 nnm1nn0 10731 . . . . . 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 11005 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
3735, 36syl6eleq 2552 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( # `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
38 simpll 753 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  H  e.  ( M MndHom  N ) )
39 eqid 2454 . . . . . . 7  |-  ( +g  `  N )  =  ( +g  `  N )
4016, 17, 39mhmlin 15589 . . . . . 6  |-  ( ( H  e.  ( M MndHom  N )  /\  x  e.  B  /\  y  e.  B )  ->  ( H `  ( x
( +g  `  M ) y ) )  =  ( ( H `  x ) ( +g  `  N ) ( H `
 y ) ) )
41403expb 1189 . . . . 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 471 . . . 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 5666 . . . . . . 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 5874 . . . . . 6  |-  ( ( W  Fn  ( 0 ... ( ( # `  W )  -  1 ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
4644, 45sylan 471 . . . . 5  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
4746eqcomd 2462 . . . 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 11969 . . 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 15631 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( M  gsumg  W )  =  (  seq 0 ( ( +g  `  M ) ,  W ) `  ( ( # `  W
)  -  1 ) ) )
5049fveq2d 5802 . . 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 2454 . . . 4  |-  ( Base `  N )  =  (
Base `  N )
52 mhmrcl2 15586 . . . . 5  |-  ( H  e.  ( M MndHom  N
)  ->  N  e.  Mnd )
5352ad2antrr 725 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  N  e.  Mnd )
5416, 51mhmf 15587 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  H : B
--> ( Base `  N
) )
5554ad2antrr 725 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  H : B --> ( Base `  N ) )
56 fco 5675 . . . . 5  |-  ( ( H : B --> ( Base `  N )  /\  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B )  -> 
( H  o.  W
) : ( 0 ... ( ( # `  W )  -  1 ) ) --> ( Base `  N ) )
5755, 32, 56syl2anc 661 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H  o.  W
) : ( 0 ... ( ( # `  W )  -  1 ) ) --> ( Base `  N ) )
5851, 39, 53, 37, 57gsumval2 15631 . . 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 2505 . 2  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )
602, 10mhm0 15590 . . 3  |-  ( H  e.  ( M MndHom  N
)  ->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) )
6160adantr 465 . 2  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H `  ( 0g `  M ) )  =  ( 0g `  N
) )
6213, 59, 61pm2.61ne 2766 1  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   (/)c0 3744    o. ccom 4951    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6199   Fincfn 7419   0cc0 9392   1c1 9393    - cmin 9705   NNcn 10432   NN0cn0 10689   ZZcz 10756   ZZ>=cuz 10971   ...cfz 11553  ..^cfzo 11664    seqcseq 11922   #chash 12219  Word cword 12338   Basecbs 14291   +g cplusg 14356   0gc0g 14496    gsumg cgsu 14497   Mndcmnd 15527   MndHom cmhm 15580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-seq 11923  df-hash 12220  df-word 12346  df-0g 14498  df-gsum 14499  df-mnd 15533  df-mhm 15582
This theorem is referenced by:  frmdup3  15662  symgtrinv  16096  frgpup3lem  16394
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