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Theorem gsumccat 15524
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 6103 . . . 4  |-  ( W  =  (/)  ->  ( W concat  X )  =  (
(/) concat  X ) )
21oveq2d 6112 . . 3  |-  ( W  =  (/)  ->  ( G 
gsumg  ( W concat  X ) )  =  ( G  gsumg  ( (/) concat  X ) ) )
3 oveq2 6104 . . . . 5  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( G  gsumg  (/) ) )
4 eqid 2443 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
54gsum0 15515 . . . . 5  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
63, 5syl6eq 2491 . . . 4  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( 0g `  G ) )
76oveq1d 6111 . . 3  |-  ( W  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) )
82, 7eqeq12d 2457 . 2  |-  ( W  =  (/)  ->  ( ( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  <->  ( G  gsumg  (
(/) concat  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) ) )
9 oveq2 6104 . . . . 5  |-  ( X  =  (/)  ->  ( W concat  X )  =  ( W concat  (/) ) )
109oveq2d 6112 . . . 4  |-  ( X  =  (/)  ->  ( G 
gsumg  ( W concat  X ) )  =  ( G  gsumg  ( W concat  (/) ) ) )
11 oveq2 6104 . . . . . 6  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( G  gsumg  (/) ) )
1211, 5syl6eq 2491 . . . . 5  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( 0g `  G ) )
1312oveq2d 6112 . . . 4  |-  ( X  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( G  gsumg  W ) 
.+  ( 0g `  G ) ) )
1410, 13eqeq12d 2457 . . 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 991 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  G  e.  Mnd )
18 lennncl 12255 . . . . . . . . . . 11  |-  ( ( W  e. Word  B  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
19183ad2antl2 1151 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  NN )
2019adantrr 716 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  NN )
21 lennncl 12255 . . . . . . . . . . 11  |-  ( ( X  e. Word  B  /\  X  =/=  (/) )  ->  ( # `
 X )  e.  NN )
22213ad2antl3 1152 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  X  =/=  (/) )  -> 
( # `  X )  e.  NN )
2322adantrl 715 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  NN )
2420, 23nnaddcld 10373 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  NN )
25 nnm1nn0 10626 . . . . . . . 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 10900 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2826, 27syl6eleq 2533 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  ( ZZ>= `  0
) )
29 simpl2 992 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  W  e. Word  B )
30 simpl3 993 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  X  e. Word  B )
31 ccatcl 12279 . . . . . . . . 9  |-  ( ( W  e. Word  B  /\  X  e. Word  B )  ->  ( W concat  X )  e. Word  B )
3229, 30, 31syl2anc 661 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( W concat  X )  e. Word  B )
33 wrdf 12245 . . . . . . . 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 12280 . . . . . . . . . . 11  |-  ( ( W  e. Word  B  /\  X  e. Word  B )  ->  ( # `  ( W concat  X ) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3629, 30, 35syl2anc 661 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  ( W concat  X ) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3736oveq2d 6112 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W concat  X ) ) )  =  ( 0..^ ( ( # `  W )  +  (
# `  X )
) ) )
3820nnzd 10751 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  ZZ )
3923nnzd 10751 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  ZZ )
4038, 39zaddcld 10756 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  ZZ )
41 fzoval 11559 . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W concat  X ) ) )  =  ( 0 ... ( ( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )
4443feq2d 5552 . . . . . . 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 210 . . . . . 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 15518 . . . . 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 10626 . . . . . . . . . 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 2533 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
50 wrdf 12245 . . . . . . . . . 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 11559 . . . . . . . . . . 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 5552 . . . . . . . . 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 210 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  W : ( 0 ... ( ( # `  W
)  -  1 ) ) --> B )
5615, 16, 17, 49, 55gsumval2 15518 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  W )  =  (  seq 0 (  .+  ,  W ) `  (
( # `  W )  -  1 ) ) )
57 nnm1nn0 10626 . . . . . . . . . 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 2533 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  X
)  -  1 )  e.  ( ZZ>= `  0
) )
60 wrdf 12245 . . . . . . . . . 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 11559 . . . . . . . . . . 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 5552 . . . . . . . . 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 210 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  X : ( 0 ... ( ( # `  X
)  -  1 ) ) --> B )
6615, 16, 17, 59, 65gsumval2 15518 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  X )  =  (  seq 0 (  .+  ,  X ) `  (
( # `  X )  -  1 ) ) )
6756, 66oveq12d 6114 . . . . . 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 15425 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
69683expb 1188 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
7017, 69sylan 471 . . . . . . . 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 15426 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )  ->  (
( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )
7217, 71sylan 471 . . . . . . . 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 10880 . . . . . . . . . . 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 10916 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  ( ZZ>= `  ( # `  W ) )  /\  ( (
# `  X )  -  1 )  e. 
NN0 )  ->  (
( # `  W )  +  ( ( # `  X )  -  1 ) )  e.  (
ZZ>= `  ( # `  W
) ) )
7674, 58, 75syl2anc 661 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( (
# `  X )  -  1 ) )  e.  ( ZZ>= `  ( # `
 W ) ) )
7720nncnd 10343 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  CC )
7823nncnd 10343 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  CC )
79 ax-1cn 9345 . . . . . . . . . . 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 9744 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( # `  W )  +  ( ( # `  X
)  -  1 ) ) )
82 npcan 9624 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( # `  W )  -  1 )  +  1 )  =  ( # `  W
) )
8377, 79, 82sylancl 662 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  -  1 )  +  1 )  =  ( # `  W
) )
8483fveq2d 5700 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ZZ>= `  ( (
( # `  W )  -  1 )  +  1 ) )  =  ( ZZ>= `  ( # `  W
) ) )
8576, 81, 843eltr4d 2524 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  ( ZZ>= `  (
( ( # `  W
)  -  1 )  +  1 ) ) )
8645ffvelrnda 5848 . . . . . . . 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 11844 . . . . . . 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 1028 . . . . . . . . . 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 1029 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  X  e. Word  B )
9053eleq2d 2510 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( x  e.  ( 0..^ ( # `  W
) )  <->  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) ) )
9190biimpar 485 . . . . . . . . . 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 12281 . . . . . . . . . 10  |-  ( ( W  e. Word  B  /\  X  e. Word  B  /\  x  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W concat  X ) `  x
)  =  ( W `
 x ) )
9388, 89, 91, 92syl3anc 1218 . . . . . . . . 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 11835 . . . . . . . 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 9575 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0  +  (
# `  W )
)  =  ( # `  W ) )
9683, 95eqtr4d 2478 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  -  1 )  +  1 )  =  ( 0  +  ( # `  W
) ) )
9796seqeq1d 11817 . . . . . . . . . 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 9576 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  =  ( ( # `  X
)  +  ( # `  W ) ) )
9998oveq1d 6111 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  +  ( # `  W
) )  -  1 ) )
10078, 77, 80addsubd 9745 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  X )  +  (
# `  W )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10199, 100eqtrd 2475 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10297, 101fveq12d 5702 . . . . . . . . 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 1028 . . . . . . . . . . . 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 1029 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  X  e. Word  B )
10563eleq2d 2510 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( x  e.  ( 0..^ ( # `  X
) )  <->  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) ) )
106105biimpar 485 . . . . . . . . . . . 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 12283 . . . . . . . . . . . 12  |-  ( ( W  e. Word  B  /\  X  e. Word  B  /\  x  e.  ( 0..^ ( # `  X ) ) )  ->  ( ( W concat  X ) `  (
x  +  ( # `  W ) ) )  =  ( X `  x ) )
108103, 104, 106, 107syl3anc 1218 . . . . . . . . . . 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 2448 . . . . . . . . . 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 11837 . . . . . . . . 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 2478 . . . . . . . 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 6114 . . . . . . 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 2475 . . . . . 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 2478 . . . . 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 2478 . . . 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 648 . . 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 992 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  ->  W  e. Word  B )
118 ccatrid 12290 . . . . . 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 6112 . . . 4  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  ( W concat  (/) ) )  =  ( G  gsumg  W ) )
121 simpl1 991 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  ->  G  e.  Mnd )
12215gsumwcl 15523 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  W  e. Word  B )  ->  ( G  gsumg  W )  e.  B
)
1231223adant3 1008 . . . . . 6  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  W )  e.  B
)
124123adantr 465 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  W )  e.  B
)
12515, 16, 4mndrid 15447 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( G  gsumg  W )  e.  B
)  ->  ( ( G  gsumg  W )  .+  ( 0g `  G ) )  =  ( G  gsumg  W ) )
126121, 124, 125syl2anc 661 . . . 4  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( G  gsumg  W ) 
.+  ( 0g `  G ) )  =  ( G  gsumg  W ) )
127120, 126eqtr4d 2478 . . 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 2691 . 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 12289 . . . . 5  |-  ( X  e. Word  B  ->  ( (/) concat  X )  =  X )
1301293ad2ant3 1011 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( (/) concat  X )  =  X )
131130oveq2d 6112 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  ( (/) concat  X ) )  =  ( G  gsumg  X ) )
132 simp1 988 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  G  e.  Mnd )
13315gsumwcl 15523 . . . . 5  |-  ( ( G  e.  Mnd  /\  X  e. Word  B )  ->  ( G  gsumg  X )  e.  B
)
1341333adant2 1007 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  X )  e.  B
)
13515, 16, 4mndlid 15446 . . . 4  |-  ( ( G  e.  Mnd  /\  ( G  gsumg  X )  e.  B
)  ->  ( ( 0g `  G )  .+  ( G  gsumg  X ) )  =  ( G  gsumg  X ) )
136132, 134, 135syl2anc 661 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  (
( 0g `  G
)  .+  ( G  gsumg  X ) )  =  ( G  gsumg  X ) )
137131, 136eqtr4d 2478 . 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 2691 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   (/)c0 3642   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288    + caddc 9290    - cmin 9600   NNcn 10327   NN0cn0 10584   ZZcz 10651   ZZ>=cuz 10866   ...cfz 11442  ..^cfzo 11553    seqcseq 11811   #chash 12108  Word cword 12226   concat cconcat 12228   Basecbs 14179   +g cplusg 14243   0gc0g 14383    gsumg cgsu 14384   Mndcmnd 15414
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-word 12234  df-concat 12236  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-gsum 14386  df-mnd 15420  df-submnd 15470
This theorem is referenced by:  gsumws2  15525  gsumccatsn  15526  gsumspl  15527  gsumwspan  15529  frmdgsum  15545  frmdup1  15547  gsumwrev  15886  psgnunilem5  16005  psgnuni  16010  frgpuplem  16274  frgpup1  16277  psgnghm  18015
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