MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumwmhm Structured version   Unicode version

Theorem gsumwmhm 15514
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 6094 . . . . 5  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( M  gsumg  (/) ) )
2 eqid 2438 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
32gsum0 15501 . . . . 5  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
41, 3syl6eq 2486 . . . 4  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( 0g `  M ) )
54fveq2d 5690 . . 3  |-  ( W  =  (/)  ->  ( H `
 ( M  gsumg  W ) )  =  ( H `
 ( 0g `  M ) ) )
6 coeq2 4993 . . . . . 6  |-  ( W  =  (/)  ->  ( H  o.  W )  =  ( H  o.  (/) ) )
7 co02 5346 . . . . . 6  |-  ( H  o.  (/) )  =  (/)
86, 7syl6eq 2486 . . . . 5  |-  ( W  =  (/)  ->  ( H  o.  W )  =  (/) )
98oveq2d 6102 . . . 4  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( N  gsumg  (/) ) )
10 eqid 2438 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
1110gsum0 15501 . . . 4  |-  ( N 
gsumg  (/) )  =  ( 0g
`  N )
129, 11syl6eq 2486 . . 3  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( 0g `  N ) )
135, 12eqeq12d 2452 . 2  |-  ( W  =  (/)  ->  ( ( H `  ( M 
gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) )  <->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) ) )
14 mhmrcl1 15459 . . . . . 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 2438 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
1816, 17mndcl 15412 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
19183expb 1188 . . . . 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 12232 . . . . . . 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 12240 . . . . . . . . . . . 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 12126 . . . . . . . . . . 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 10738 . . . . . . . 8  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  ZZ )
29 fzoval 11546 . . . . . . . 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 5542 . . . . . 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 5838 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( W `  x )  e.  B
)
34 nnm1nn0 10613 . . . . . 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 10887 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
3735, 36syl6eleq 2528 . . . 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 2438 . . . . . . 7  |-  ( +g  `  N )  =  ( +g  `  N )
4016, 17, 39mhmlin 15463 . . . . . 6  |-  ( ( H  e.  ( M MndHom  N )  /\  x  e.  B  /\  y  e.  B )  ->  ( H `  ( x
( +g  `  M ) y ) )  =  ( ( H `  x ) ( +g  `  N ) ( H `
 y ) ) )
41403expb 1188 . . . . 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 5554 . . . . . . 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 5761 . . . . . 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 2443 . . . 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 11845 . . 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 15504 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( M  gsumg  W )  =  (  seq 0 ( ( +g  `  M ) ,  W ) `  ( ( # `  W
)  -  1 ) ) )
5049fveq2d 5690 . . 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 2438 . . . 4  |-  ( Base `  N )  =  (
Base `  N )
52 mhmrcl2 15460 . . . . 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 15461 . . . . . 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 5563 . . . . 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 15504 . . 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 2480 . 2  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )
602, 10mhm0 15464 . . 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 2681 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 1369    e. wcel 1756    =/= wne 2601   (/)c0 3632    o. ccom 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   Fincfn 7302   0cc0 9274   1c1 9275    - cmin 9587   NNcn 10314   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429  ..^cfzo 11540    seqcseq 11798   #chash 12095  Word cword 12213   Basecbs 14166   +g cplusg 14230   0gc0g 14370    gsumg cgsu 14371   Mndcmnd 15401   MndHom cmhm 15454
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-seq 11799  df-hash 12096  df-word 12221  df-0g 14372  df-gsum 14373  df-mnd 15407  df-mhm 15456
This theorem is referenced by:  frmdup3  15535  symgtrinv  15969  frgpup3lem  16265
  Copyright terms: Public domain W3C validator