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

Theorem gsumwmhm 15503
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 6088 . . . . 5  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( M  gsumg  (/) ) )
2 eqid 2433 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
32gsum0 15490 . . . . 5  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
41, 3syl6eq 2481 . . . 4  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( 0g `  M ) )
54fveq2d 5683 . . 3  |-  ( W  =  (/)  ->  ( H `
 ( M  gsumg  W ) )  =  ( H `
 ( 0g `  M ) ) )
6 coeq2 4985 . . . . . 6  |-  ( W  =  (/)  ->  ( H  o.  W )  =  ( H  o.  (/) ) )
7 co02 5339 . . . . . 6  |-  ( H  o.  (/) )  =  (/)
86, 7syl6eq 2481 . . . . 5  |-  ( W  =  (/)  ->  ( H  o.  W )  =  (/) )
98oveq2d 6096 . . . 4  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( N  gsumg  (/) ) )
10 eqid 2433 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
1110gsum0 15490 . . . 4  |-  ( N 
gsumg  (/) )  =  ( 0g
`  N )
129, 11syl6eq 2481 . . 3  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( 0g `  N ) )
135, 12eqeq12d 2447 . 2  |-  ( W  =  (/)  ->  ( ( H `  ( M 
gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) )  <->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) ) )
14 mhmrcl1 15450 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  M  e.  Mnd )
1514ad2antrr 718 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  M  e.  Mnd )
16 gsumwmhm.b . . . . . . 7  |-  B  =  ( Base `  M
)
17 eqid 2433 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
1816, 17mndcl 15403 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
19183expb 1181 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
2015, 19sylan 468 . . . 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 12224 . . . . . . 7  |-  ( W  e. Word  B  ->  W : ( 0..^ (
# `  W )
) --> B )
2221ad2antlr 719 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W : ( 0..^ (
# `  W )
) --> B )
23 wrdfin 12232 . . . . . . . . . . . 12  |-  ( W  e. Word  B  ->  W  e.  Fin )
2423adantl 463 . . . . . . . . . . 11  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  W  e.  Fin )
25 hashnncl 12118 . . . . . . . . . . 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 482 . . . . . . . . 9  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  NN )
2827nnzd 10734 . . . . . . . 8  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  ZZ )
29 fzoval 11538 . . . . . . . 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 5535 . . . . . 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 5831 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( W `  x )  e.  B
)
34 nnm1nn0 10609 . . . . . 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 10883 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
3735, 36syl6eleq 2523 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( # `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
38 simpll 746 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  H  e.  ( M MndHom  N ) )
39 eqid 2433 . . . . . . 7  |-  ( +g  `  N )  =  ( +g  `  N )
4016, 17, 39mhmlin 15454 . . . . . 6  |-  ( ( H  e.  ( M MndHom  N )  /\  x  e.  B  /\  y  e.  B )  ->  ( H `  ( x
( +g  `  M ) y ) )  =  ( ( H `  x ) ( +g  `  N ) ( H `
 y ) ) )
41403expb 1181 . . . . 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 468 . . . 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 5547 . . . . . . 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 5754 . . . . . 6  |-  ( ( W  Fn  ( 0 ... ( ( # `  W )  -  1 ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
4644, 45sylan 468 . . . . 5  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
4746eqcomd 2438 . . . 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 11837 . . 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 15493 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( M  gsumg  W )  =  (  seq 0 ( ( +g  `  M ) ,  W ) `  ( ( # `  W
)  -  1 ) ) )
5049fveq2d 5683 . . 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 2433 . . . 4  |-  ( Base `  N )  =  (
Base `  N )
52 mhmrcl2 15451 . . . . 5  |-  ( H  e.  ( M MndHom  N
)  ->  N  e.  Mnd )
5352ad2antrr 718 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  N  e.  Mnd )
5416, 51mhmf 15452 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  H : B
--> ( Base `  N
) )
5554ad2antrr 718 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  H : B --> ( Base `  N ) )
56 fco 5556 . . . . 5  |-  ( ( H : B --> ( Base `  N )  /\  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B )  -> 
( H  o.  W
) : ( 0 ... ( ( # `  W )  -  1 ) ) --> ( Base `  N ) )
5755, 32, 56syl2anc 654 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H  o.  W
) : ( 0 ... ( ( # `  W )  -  1 ) ) --> ( Base `  N ) )
5851, 39, 53, 37, 57gsumval2 15493 . . 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 2475 . 2  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )
602, 10mhm0 15455 . . 3  |-  ( H  e.  ( M MndHom  N
)  ->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) )
6160adantr 462 . 2  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H `  ( 0g `  M ) )  =  ( 0g `  N
) )
6213, 59, 61pm2.61ne 2676 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 1362    e. wcel 1755    =/= wne 2596   (/)c0 3625    o. ccom 4831    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080   Fincfn 7298   0cc0 9270   1c1 9271    - cmin 9583   NNcn 10310   NN0cn0 10567   ZZcz 10634   ZZ>=cuz 10849   ...cfz 11424  ..^cfzo 11532    seqcseq 11790   #chash 12087  Word cword 12205   Basecbs 14157   +g cplusg 14221   0gc0g 14361    gsumg cgsu 14362   Mndcmnd 15392   MndHom cmhm 15445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-fzo 11533  df-seq 11791  df-hash 12088  df-word 12213  df-0g 14363  df-gsum 14364  df-mnd 15398  df-mhm 15447
This theorem is referenced by:  frmdup3  15524  symgtrinv  15958  frgpup3lem  16254
  Copyright terms: Public domain W3C validator