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Theorem gsumwmhm 13070
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  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2193 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
2 eqid 2193 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
31, 2mhm0 13040 . . . 4  |-  ( H  e.  ( M MndHom  N
)  ->  ( H `  ( 0g `  M
) )  =  ( 0g `  N ) )
43ad2antrr 488 . . 3  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  -> 
( H `  ( 0g `  M ) )  =  ( 0g `  N ) )
5 oveq2 5926 . . . . . 6  |-  ( W  =  (/)  ->  ( M 
gsumg  W )  =  ( M  gsumg  (/) ) )
65adantl 277 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  -> 
( M  gsumg  W )  =  ( M  gsumg  (/) ) )
7 mhmrcl1 13035 . . . . . . 7  |-  ( H  e.  ( M MndHom  N
)  ->  M  e.  Mnd )
87ad2antrr 488 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  ->  M  e.  Mnd )
91gsum0g 12979 . . . . . 6  |-  ( M  e.  Mnd  ->  ( M  gsumg  (/) )  =  ( 0g `  M ) )
108, 9syl 14 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  -> 
( M  gsumg  (/) )  =  ( 0g `  M ) )
116, 10eqtrd 2226 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  -> 
( M  gsumg  W )  =  ( 0g `  M ) )
1211fveq2d 5558 . . 3  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( H `  ( 0g `  M ) ) )
13 coeq2 4820 . . . . . . 7  |-  ( W  =  (/)  ->  ( H  o.  W )  =  ( H  o.  (/) ) )
14 co02 5179 . . . . . . 7  |-  ( H  o.  (/) )  =  (/)
1513, 14eqtrdi 2242 . . . . . 6  |-  ( W  =  (/)  ->  ( H  o.  W )  =  (/) )
1615oveq2d 5934 . . . . 5  |-  ( W  =  (/)  ->  ( N 
gsumg  ( H  o.  W
) )  =  ( N  gsumg  (/) ) )
1716adantl 277 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  -> 
( N  gsumg  ( H  o.  W
) )  =  ( N  gsumg  (/) ) )
18 mhmrcl2 13036 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  N  e.  Mnd )
1918ad2antrr 488 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  ->  N  e.  Mnd )
202gsum0g 12979 . . . . 5  |-  ( N  e.  Mnd  ->  ( N  gsumg  (/) )  =  ( 0g `  N ) )
2119, 20syl 14 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  -> 
( N  gsumg  (/) )  =  ( 0g `  N ) )
2217, 21eqtrd 2226 . . 3  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  -> 
( N  gsumg  ( H  o.  W
) )  =  ( 0g `  N ) )
234, 12, 223eqtr4d 2236 . 2  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )
247ad2antrr 488 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  M  e.  Mnd )
25 gsumwmhm.b . . . . . . 7  |-  B  =  ( Base `  M
)
26 eqid 2193 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
2725, 26mndcl 13004 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
28273expb 1206 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
2924, 28sylan 283 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  M ) y )  e.  B )
30 wrdf 10920 . . . . . . 7  |-  ( W  e. Word  B  ->  W : ( 0..^ ( `  W ) ) --> B )
3130ad2antlr 489 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W : ( 0..^ ( `  W ) ) --> B )
32 wrdfin 10933 . . . . . . . . . . . 12  |-  ( W  e. Word  B  ->  W  e.  Fin )
3332adantl 277 . . . . . . . . . . 11  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  W  e.  Fin )
34 hashnncl 10866 . . . . . . . . . . 11  |-  ( W  e.  Fin  ->  (
( `  W )  e.  NN  <->  W  =/=  (/) ) )
3533, 34syl 14 . . . . . . . . . 10  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  (
( `  W )  e.  NN  <->  W  =/=  (/) ) )
3635biimpar 297 . . . . . . . . 9  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( `  W )  e.  NN )
3736nnzd 9438 . . . . . . . 8  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( `  W )  e.  ZZ )
38 fzoval 10214 . . . . . . . 8  |-  ( ( `  W )  e.  ZZ  ->  ( 0..^ ( `  W
) )  =  ( 0 ... ( ( `  W )  -  1 ) ) )
3937, 38syl 14 . . . . . . 7  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( 0..^ ( `  W
) )  =  ( 0 ... ( ( `  W )  -  1 ) ) )
4039feq2d 5391 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( W : ( 0..^ ( `  W
) ) --> B  <->  W :
( 0 ... (
( `  W )  - 
1 ) ) --> B ) )
4131, 40mpbid 147 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W : ( 0 ... ( ( `  W
)  -  1 ) ) --> B )
4241ffvelcdmda 5693 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( `  W )  - 
1 ) ) )  ->  ( W `  x )  e.  B
)
43 nnm1nn0 9281 . . . . . 6  |-  ( ( `  W )  e.  NN  ->  ( ( `  W
)  -  1 )  e.  NN0 )
4436, 43syl 14 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( `  W
)  -  1 )  e.  NN0 )
45 nn0uz 9627 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
4644, 45eleqtrdi 2286 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
47 eqid 2193 . . . . . . 7  |-  ( +g  `  N )  =  ( +g  `  N )
4825, 26, 47mhmlin 13039 . . . . . 6  |-  ( ( H  e.  ( M MndHom  N )  /\  x  e.  B  /\  y  e.  B )  ->  ( H `  ( x
( +g  `  M ) y ) )  =  ( ( H `  x ) ( +g  `  N ) ( H `
 y ) ) )
49483expb 1206 . . . . 5  |-  ( ( H  e.  ( M MndHom  N )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( H `  ( x ( +g  `  M ) y ) )  =  ( ( H `  x ) ( +g  `  N
) ( H `  y ) ) )
5049ad4ant14 514 . . . 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
) ) )
5141ffnd 5404 . . . . . 6  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W  Fn  ( 0 ... ( ( `  W
)  -  1 ) ) )
52 fvco2 5626 . . . . . 6  |-  ( ( W  Fn  ( 0 ... ( ( `  W
)  -  1 ) )  /\  x  e.  ( 0 ... (
( `  W )  - 
1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
5351, 52sylan 283 . . . . 5  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( `  W )  - 
1 ) ) )  ->  ( ( H  o.  W ) `  x )  =  ( H `  ( W `
 x ) ) )
5453eqcomd 2199 . . . 4  |-  ( ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B
)  /\  W  =/=  (/) )  /\  x  e.  ( 0 ... (
( `  W )  - 
1 ) ) )  ->  ( H `  ( W `  x ) )  =  ( ( H  o.  W ) `
 x ) )
55 simplr 528 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  W  e. Word  B )
56 coexg 5210 . . . . 5  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H  o.  W )  e.  _V )
5756adantr 276 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H  o.  W
)  e.  _V )
58 plusgslid 12730 . . . . . . 7  |-  ( +g  = Slot  ( +g  `  ndx )  /\  ( +g  `  ndx )  e.  NN )
5958slotex 12645 . . . . . 6  |-  ( M  e.  Mnd  ->  ( +g  `  M )  e. 
_V )
607, 59syl 14 . . . . 5  |-  ( H  e.  ( M MndHom  N
)  ->  ( +g  `  M )  e.  _V )
6160ad2antrr 488 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( +g  `  M )  e.  _V )
6258slotex 12645 . . . . . 6  |-  ( N  e.  Mnd  ->  ( +g  `  N )  e. 
_V )
6318, 62syl 14 . . . . 5  |-  ( H  e.  ( M MndHom  N
)  ->  ( +g  `  N )  e.  _V )
6463ad2antrr 488 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( +g  `  N )  e.  _V )
6529, 42, 46, 50, 54, 55, 57, 61, 64seqhomog 10601 . . 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 ) ) )
6625, 26, 24, 46, 41gsumval2 12980 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( M  gsumg  W )  =  (  seq 0 ( ( +g  `  M ) ,  W ) `  ( ( `  W )  -  1 ) ) )
6766fveq2d 5558 . . 3  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( H `  (  seq 0 ( ( +g  `  M ) ,  W
) `  ( ( `  W )  -  1 ) ) ) )
68 eqid 2193 . . . 4  |-  ( Base `  N )  =  (
Base `  N )
6918ad2antrr 488 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  N  e.  Mnd )
7025, 68mhmf 13037 . . . . . 6  |-  ( H  e.  ( M MndHom  N
)  ->  H : B
--> ( Base `  N
) )
7170ad2antrr 488 . . . . 5  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  ->  H : B --> ( Base `  N ) )
72 fco 5419 . . . . 5  |-  ( ( H : B --> ( Base `  N )  /\  W : ( 0 ... ( ( `  W
)  -  1 ) ) --> B )  -> 
( H  o.  W
) : ( 0 ... ( ( `  W
)  -  1 ) ) --> ( Base `  N
) )
7371, 41, 72syl2anc 411 . . . 4  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H  o.  W
) : ( 0 ... ( ( `  W
)  -  1 ) ) --> ( Base `  N
) )
7468, 47, 69, 46, 73gsumval2 12980 . . 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 ) ) )
7565, 67, 743eqtr4d 2236 . 2  |-  ( ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  /\  W  =/=  (/) )  -> 
( H `  ( M  gsumg  W ) )  =  ( N  gsumg  ( H  o.  W
) ) )
76 fin0or 6942 . . . 4  |-  ( W  e.  Fin  ->  ( W  =  (/)  \/  E. j  j  e.  W
) )
77 n0r 3460 . . . . 5  |-  ( E. j  j  e.  W  ->  W  =/=  (/) )
7877orim2i 762 . . . 4  |-  ( ( W  =  (/)  \/  E. j  j  e.  W
)  ->  ( W  =  (/)  \/  W  =/=  (/) ) )
7976, 78syl 14 . . 3  |-  ( W  e.  Fin  ->  ( W  =  (/)  \/  W  =/=  (/) ) )
8033, 79syl 14 . 2  |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( W  =  (/)  \/  W  =/=  (/) ) )
8123, 75, 80mpjaodan 799 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    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364   E.wex 1503    e. wcel 2164    =/= wne 2364   _Vcvv 2760   (/)c0 3446    o. ccom 4663    Fn wfn 5249   -->wf 5250   ` cfv 5254  (class class class)co 5918   Fincfn 6794   0cc0 7872   1c1 7873    - cmin 8190   NNcn 8982   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074  ..^cfzo 10208    seqcseq 10518  ♯chash 10846  Word cword 10914   Basecbs 12618   +g cplusg 12695   0gc0g 12867    gsumg cgsu 12868   Mndcmnd 12997   MndHom cmhm 13029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-map 6704  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-ihash 10847  df-word 10915  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-igsum 12870  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mhm 13031
This theorem is referenced by: (None)
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