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Theorem gsumccat 14707
Description: Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
gsumwcl.b  |-  B  =  ( Base `  G
)
gsumccat.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
gsumccat  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) ) )

Proof of Theorem gsumccat
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6020 . . . 4  |-  ( W  =  (/)  ->  ( W concat  X )  =  (
(/) concat  X ) )
21oveq2d 6029 . . 3  |-  ( W  =  (/)  ->  ( G 
gsumg  ( W concat  X ) )  =  ( G  gsumg  ( (/) concat  X ) ) )
3 oveq2 6021 . . . . 5  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( G  gsumg  (/) ) )
4 eqid 2380 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
54gsum0 14700 . . . . 5  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
63, 5syl6eq 2428 . . . 4  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( 0g `  G ) )
76oveq1d 6028 . . 3  |-  ( W  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) )
82, 7eqeq12d 2394 . 2  |-  ( W  =  (/)  ->  ( ( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  <->  ( G  gsumg  (
(/) concat  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) ) )
9 oveq2 6021 . . . . 5  |-  ( X  =  (/)  ->  ( W concat  X )  =  ( W concat  (/) ) )
109oveq2d 6029 . . . 4  |-  ( X  =  (/)  ->  ( G 
gsumg  ( W concat  X ) )  =  ( G  gsumg  ( W concat  (/) ) ) )
11 oveq2 6021 . . . . . 6  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( G  gsumg  (/) ) )
1211, 5syl6eq 2428 . . . . 5  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( 0g `  G ) )
1312oveq2d 6029 . . . 4  |-  ( X  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( G  gsumg  W ) 
.+  ( 0g `  G ) ) )
1410, 13eqeq12d 2394 . . 3  |-  ( X  =  (/)  ->  ( ( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  <->  ( G  gsumg  ( W concat  (/) ) )  =  ( ( G  gsumg  W ) 
.+  ( 0g `  G ) ) ) )
15 gsumwcl.b . . . . . 6  |-  B  =  ( Base `  G
)
16 gsumccat.p . . . . . 6  |-  .+  =  ( +g  `  G )
17 simpl1 960 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  G  e.  Mnd )
18 lennncl 11656 . . . . . . . . . . 11  |-  ( ( W  e. Word  B  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
19183ad2antl2 1120 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  NN )
2019adantrr 698 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  NN )
21 lennncl 11656 . . . . . . . . . . 11  |-  ( ( X  e. Word  B  /\  X  =/=  (/) )  ->  ( # `
 X )  e.  NN )
22213ad2antl3 1121 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  X  =/=  (/) )  -> 
( # `  X )  e.  NN )
2322adantrl 697 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  NN )
2420, 23nnaddcld 9971 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  NN )
25 nnm1nn0 10186 . . . . . . . 8  |-  ( ( ( # `  W
)  +  ( # `  X ) )  e.  NN  ->  ( (
( # `  W )  +  ( # `  X
) )  -  1 )  e.  NN0 )
2624, 25syl 16 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  NN0 )
27 nn0uz 10445 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2826, 27syl6eleq 2470 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  ( ZZ>= `  0
) )
29 simpl2 961 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  W  e. Word  B )
30 simpl3 962 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  X  e. Word  B )
31 ccatcl 11663 . . . . . . . . 9  |-  ( ( W  e. Word  B  /\  X  e. Word  B )  ->  ( W concat  X )  e. Word  B )
3229, 30, 31syl2anc 643 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( W concat  X )  e. Word  B )
33 wrdf 11653 . . . . . . . 8  |-  ( ( W concat  X )  e. Word  B  ->  ( W concat  X
) : ( 0..^ ( # `  ( W concat  X ) ) ) --> B )
3432, 33syl 16 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( W concat  X ) : ( 0..^ (
# `  ( W concat  X ) ) ) --> B )
35 ccatlen 11664 . . . . . . . . . . 11  |-  ( ( W  e. Word  B  /\  X  e. Word  B )  ->  ( # `  ( W concat  X ) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3629, 30, 35syl2anc 643 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  ( W concat  X ) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3736oveq2d 6029 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W concat  X ) ) )  =  ( 0..^ ( ( # `  W )  +  (
# `  X )
) ) )
3820nnzd 10299 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  ZZ )
3923nnzd 10299 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  ZZ )
4038, 39zaddcld 10304 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  ZZ )
41 fzoval 11064 . . . . . . . . . 10  |-  ( ( ( # `  W
)  +  ( # `  X ) )  e.  ZZ  ->  ( 0..^ ( ( # `  W
)  +  ( # `  X ) ) )  =  ( 0 ... ( ( ( # `  W )  +  (
# `  X )
)  -  1 ) ) )
4240, 41syl 16 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( (
# `  W )  +  ( # `  X
) ) )  =  ( 0 ... (
( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )
4337, 42eqtrd 2412 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W concat  X ) ) )  =  ( 0 ... ( ( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )
4443feq2d 5514 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( W concat  X
) : ( 0..^ ( # `  ( W concat  X ) ) ) --> B  <->  ( W concat  X
) : ( 0 ... ( ( (
# `  W )  +  ( # `  X
) )  -  1 ) ) --> B ) )
4534, 44mpbid 202 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( W concat  X ) : ( 0 ... ( ( ( # `  W )  +  (
# `  X )
)  -  1 ) ) --> B )
4615, 16, 17, 28, 45gsumval2 14703 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  ( W concat  X ) )  =  (  seq  0 (  .+  , 
( W concat  X )
) `  ( (
( # `  W )  +  ( # `  X
) )  -  1 ) ) )
47 nnm1nn0 10186 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  NN0 )
4820, 47syl 16 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  -  1 )  e.  NN0 )
4948, 27syl6eleq 2470 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
50 wrdf 11653 . . . . . . . . . 10  |-  ( W  e. Word  B  ->  W : ( 0..^ (
# `  W )
) --> B )
5129, 50syl 16 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  W : ( 0..^ (
# `  W )
) --> B )
52 fzoval 11064 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  ZZ  ->  ( 0..^ ( # `  W
) )  =  ( 0 ... ( (
# `  W )  -  1 ) ) )
5338, 52syl 16 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  W ) )  =  ( 0 ... (
( # `  W )  -  1 ) ) )
5453feq2d 5514 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( W : ( 0..^ ( # `  W
) ) --> B  <->  W :
( 0 ... (
( # `  W )  -  1 ) ) --> B ) )
5551, 54mpbid 202 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B )
5615, 16, 17, 49, 55gsumval2 14703 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  W )  =  (  seq  0 (  .+  ,  W ) `  (
( # `  W )  -  1 ) ) )
57 nnm1nn0 10186 . . . . . . . . . 10  |-  ( (
# `  X )  e.  NN  ->  ( ( # `
 X )  - 
1 )  e.  NN0 )
5823, 57syl 16 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  X
)  -  1 )  e.  NN0 )
5958, 27syl6eleq 2470 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  X
)  -  1 )  e.  ( ZZ>= `  0
) )
60 wrdf 11653 . . . . . . . . . 10  |-  ( X  e. Word  B  ->  X : ( 0..^ (
# `  X )
) --> B )
6130, 60syl 16 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  X : ( 0..^ (
# `  X )
) --> B )
62 fzoval 11064 . . . . . . . . . . 11  |-  ( (
# `  X )  e.  ZZ  ->  ( 0..^ ( # `  X
) )  =  ( 0 ... ( (
# `  X )  -  1 ) ) )
6339, 62syl 16 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  X ) )  =  ( 0 ... (
( # `  X )  -  1 ) ) )
6463feq2d 5514 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( X : ( 0..^ ( # `  X
) ) --> B  <->  X :
( 0 ... (
( # `  X )  -  1 ) ) --> B ) )
6561, 64mpbid 202 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  X : ( 0 ... ( ( # `  X
)  -  1 ) ) --> B )
6615, 16, 17, 59, 65gsumval2 14703 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  X )  =  (  seq  0 (  .+  ,  X ) `  (
( # `  X )  -  1 ) ) )
6756, 66oveq12d 6031 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( G  gsumg  W ) 
.+  ( G  gsumg  X ) )  =  ( (  seq  0 (  .+  ,  W ) `  (
( # `  W )  -  1 ) ) 
.+  (  seq  0
(  .+  ,  X
) `  ( ( # `
 X )  - 
1 ) ) ) )
6815, 16mndcl 14615 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
69683expb 1154 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
7017, 69sylan 458 . . . . . . . 8  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x  .+  y )  e.  B
)
7115, 16mndass 14616 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )  ->  (
( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )
7217, 71sylan 458 . . . . . . . 8  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  (
x  e.  B  /\  y  e.  B  /\  z  e.  B )
)  ->  ( (
x  .+  y )  .+  z )  =  ( x  .+  ( y 
.+  z ) ) )
73 uzid 10425 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  ZZ  ->  ( # `  W
)  e.  ( ZZ>= `  ( # `  W ) ) )
7438, 73syl 16 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  ( ZZ>= `  ( # `
 W ) ) )
75 uzaddcl 10458 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  ( ZZ>= `  ( # `  W ) )  /\  ( (
# `  X )  -  1 )  e. 
NN0 )  ->  (
( # `  W )  +  ( ( # `  X )  -  1 ) )  e.  (
ZZ>= `  ( # `  W
) ) )
7674, 58, 75syl2anc 643 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( (
# `  X )  -  1 ) )  e.  ( ZZ>= `  ( # `
 W ) ) )
7720nncnd 9941 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  CC )
7823nncnd 9941 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  CC )
79 ax-1cn 8974 . . . . . . . . . . 11  |-  1  e.  CC
8079a1i 11 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
1  e.  CC )
8177, 78, 80addsubassd 9356 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( # `  W )  +  ( ( # `  X
)  -  1 ) ) )
82 npcan 9239 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( # `  W )  -  1 )  +  1 )  =  ( # `  W
) )
8377, 79, 82sylancl 644 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  -  1 )  +  1 )  =  ( # `  W
) )
8483fveq2d 5665 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ZZ>= `  ( (
( # `  W )  -  1 )  +  1 ) )  =  ( ZZ>= `  ( # `  W
) ) )
8576, 81, 843eltr4d 2461 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  ( ZZ>= `  (
( ( # `  W
)  -  1 )  +  1 ) ) )
8645ffvelrnda 5802 . . . . . . . 8  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )  ->  ( ( W concat  X ) `  x
)  e.  B )
8770, 72, 85, 49, 86seqsplit 11276 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq  0 ( 
.+  ,  ( W concat  X ) ) `  ( ( ( # `  W )  +  (
# `  X )
)  -  1 ) )  =  ( (  seq  0 (  .+  ,  ( W concat  X
) ) `  (
( # `  W )  -  1 ) ) 
.+  (  seq  (
( ( # `  W
)  -  1 )  +  1 ) ( 
.+  ,  ( W concat  X ) ) `  ( ( ( # `  W )  +  (
# `  X )
)  -  1 ) ) ) )
88 simpll2 997 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  W  e. Word  B )
89 simpll3 998 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  X  e. Word  B )
9053eleq2d 2447 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( x  e.  ( 0..^ ( # `  W
) )  <->  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) ) )
9190biimpar 472 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  x  e.  ( 0..^ ( # `  W
) ) )
92 ccatval1 11665 . . . . . . . . . 10  |-  ( ( W  e. Word  B  /\  X  e. Word  B  /\  x  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W concat  X ) `  x
)  =  ( W `
 x ) )
9388, 89, 91, 92syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( W concat  X ) `  x
)  =  ( W `
 x ) )
9449, 93seqfveq 11267 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq  0 ( 
.+  ,  ( W concat  X ) ) `  ( ( # `  W
)  -  1 ) )  =  (  seq  0 (  .+  ,  W ) `  (
( # `  W )  -  1 ) ) )
9577addid2d 9192 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0  +  (
# `  W )
)  =  ( # `  W ) )
9683, 95eqtr4d 2415 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  -  1 )  +  1 )  =  ( 0  +  ( # `  W
) ) )
9796seqeq1d 11249 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  seq  ( ( ( # `  W )  -  1 )  +  1 ) (  .+  ,  ( W concat  X ) )  =  seq  ( 0  +  ( # `  W
) ) (  .+  ,  ( W concat  X
) ) )
9877, 78addcomd 9193 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  =  ( ( # `  X
)  +  ( # `  W ) ) )
9998oveq1d 6028 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  +  ( # `  W
) )  -  1 ) )
10078, 77, 80addsubd 9357 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  X )  +  (
# `  W )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10199, 100eqtrd 2412 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10297, 101fveq12d 5667 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq  ( (
( # `  W )  -  1 )  +  1 ) (  .+  ,  ( W concat  X
) ) `  (
( ( # `  W
)  +  ( # `  X ) )  - 
1 ) )  =  (  seq  ( 0  +  ( # `  W
) ) (  .+  ,  ( W concat  X
) ) `  (
( ( # `  X
)  -  1 )  +  ( # `  W
) ) ) )
103 simpll2 997 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  W  e. Word  B )
104 simpll3 998 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  X  e. Word  B )
10563eleq2d 2447 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( x  e.  ( 0..^ ( # `  X
) )  <->  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) ) )
106105biimpar 472 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  x  e.  ( 0..^ ( # `  X
) ) )
107 ccatval3 11667 . . . . . . . . . . . 12  |-  ( ( W  e. Word  B  /\  X  e. Word  B  /\  x  e.  ( 0..^ ( # `  X ) ) )  ->  ( ( W concat  X ) `  (
x  +  ( # `  W ) ) )  =  ( X `  x ) )
108103, 104, 106, 107syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  ( ( W concat  X ) `  (
x  +  ( # `  W ) ) )  =  ( X `  x ) )
109108eqcomd 2385 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  ( X `  x )  =  ( ( W concat  X ) `
 ( x  +  ( # `  W ) ) ) )
11059, 38, 109seqshft2 11269 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq  0 ( 
.+  ,  X ) `
 ( ( # `  X )  -  1 ) )  =  (  seq  ( 0  +  ( # `  W
) ) (  .+  ,  ( W concat  X
) ) `  (
( ( # `  X
)  -  1 )  +  ( # `  W
) ) ) )
111102, 110eqtr4d 2415 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq  ( (
( # `  W )  -  1 )  +  1 ) (  .+  ,  ( W concat  X
) ) `  (
( ( # `  W
)  +  ( # `  X ) )  - 
1 ) )  =  (  seq  0 ( 
.+  ,  X ) `
 ( ( # `  X )  -  1 ) ) )
11294, 111oveq12d 6031 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( (  seq  0
(  .+  ,  ( W concat  X ) ) `  ( ( # `  W
)  -  1 ) )  .+  (  seq  ( ( ( # `  W )  -  1 )  +  1 ) (  .+  ,  ( W concat  X ) ) `
 ( ( (
# `  W )  +  ( # `  X
) )  -  1 ) ) )  =  ( (  seq  0
(  .+  ,  W
) `  ( ( # `
 W )  - 
1 ) )  .+  (  seq  0 (  .+  ,  X ) `  (
( # `  X )  -  1 ) ) ) )
11387, 112eqtrd 2412 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq  0 ( 
.+  ,  ( W concat  X ) ) `  ( ( ( # `  W )  +  (
# `  X )
)  -  1 ) )  =  ( (  seq  0 (  .+  ,  W ) `  (
( # `  W )  -  1 ) ) 
.+  (  seq  0
(  .+  ,  X
) `  ( ( # `
 X )  - 
1 ) ) ) )
11467, 113eqtr4d 2415 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( G  gsumg  W ) 
.+  ( G  gsumg  X ) )  =  (  seq  0 (  .+  , 
( W concat  X )
) `  ( (
( # `  W )  +  ( # `  X
) )  -  1 ) ) )
11546, 114eqtr4d 2415 . . . 4  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) ) )
116115anassrs 630 . . 3  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  W  =/=  (/) )  /\  X  =/=  (/) )  ->  ( G 
gsumg  ( W concat  X ) )  =  ( ( G 
gsumg  W )  .+  ( G  gsumg  X ) ) )
117 simpl2 961 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  ->  W  e. Word  B )
118 ccatrid 11669 . . . . . 6  |-  ( W  e. Word  B  ->  ( W concat 
(/) )  =  W )
119117, 118syl 16 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( W concat  (/) )  =  W )
120119oveq2d 6029 . . . 4  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  ( W concat  (/) ) )  =  ( G  gsumg  W ) )
121 simpl1 960 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  ->  G  e.  Mnd )
12215gsumwcl 14706 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  W  e. Word  B )  ->  ( G  gsumg  W )  e.  B
)
1231223adant3 977 . . . . . 6  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  W )  e.  B
)
124123adantr 452 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  W )  e.  B
)
12515, 16, 4mndrid 14637 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( G  gsumg  W )  e.  B
)  ->  ( ( G  gsumg  W )  .+  ( 0g `  G ) )  =  ( G  gsumg  W ) )
126121, 124, 125syl2anc 643 . . . 4  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( G  gsumg  W ) 
.+  ( 0g `  G ) )  =  ( G  gsumg  W ) )
127120, 126eqtr4d 2415 . . 3  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  ( W concat  (/) ) )  =  ( ( G 
gsumg  W )  .+  ( 0g `  G ) ) )
12814, 116, 127pm2.61ne 2618 . 2  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) ) )
129 ccatlid 11668 . . . . 5  |-  ( X  e. Word  B  ->  ( (/) concat  X )  =  X )
1301293ad2ant3 980 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( (/) concat  X )  =  X )
131130oveq2d 6029 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  ( (/) concat  X ) )  =  ( G  gsumg  X ) )
132 simp1 957 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  G  e.  Mnd )
13315gsumwcl 14706 . . . . 5  |-  ( ( G  e.  Mnd  /\  X  e. Word  B )  ->  ( G  gsumg  X )  e.  B
)
1341333adant2 976 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  X )  e.  B
)
13515, 16, 4mndlid 14636 . . . 4  |-  ( ( G  e.  Mnd  /\  ( G  gsumg  X )  e.  B
)  ->  ( ( 0g `  G )  .+  ( G  gsumg  X ) )  =  ( G  gsumg  X ) )
136132, 134, 135syl2anc 643 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  (
( 0g `  G
)  .+  ( G  gsumg  X ) )  =  ( G  gsumg  X ) )
137131, 136eqtr4d 2415 . 2  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  ( (/) concat  X ) )  =  ( ( 0g
`  G )  .+  ( G  gsumg  X ) ) )
1388, 128, 137pm2.61ne 2618 1  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   (/)c0 3564   -->wf 5383   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    - cmin 9216   NNcn 9925   NN0cn0 10146   ZZcz 10207   ZZ>=cuz 10413   ...cfz 10968  ..^cfzo 11058    seq cseq 11243   #chash 11538  Word cword 11637   concat cconcat 11638   Basecbs 13389   +g cplusg 13449   0gc0g 13643    gsumg cgsu 13644   Mndcmnd 14604
This theorem is referenced by:  gsumws2  14708  gsumspl  14709  gsumwspan  14711  frmdgsum  14727  frmdup1  14729  gsumwrev  15082  frgpuplem  15324  frgpup1  15327  psgnunilem5  27079  psgnuni  27084  psgnghm  27099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969  df-fzo 11059  df-seq 11244  df-hash 11539  df-word 11643  df-concat 11644  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-0g 13647  df-gsum 13648  df-mnd 14610  df-submnd 14659
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