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Theorem gsumccat 16136
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 ++  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 6303 . . . 4  |-  ( W  =  (/)  ->  ( W ++  X )  =  (
(/) ++  X ) )
21oveq2d 6312 . . 3  |-  ( W  =  (/)  ->  ( G 
gsumg  ( W ++  X )
)  =  ( G 
gsumg  ( (/) ++  X ) ) )
3 oveq2 6304 . . . . 5  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( G  gsumg  (/) ) )
4 eqid 2457 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
54gsum0 16032 . . . . 5  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
63, 5syl6eq 2514 . . . 4  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( 0g `  G ) )
76oveq1d 6311 . . 3  |-  ( W  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) )
82, 7eqeq12d 2479 . 2  |-  ( W  =  (/)  ->  ( ( G  gsumg  ( W ++  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  <->  ( G  gsumg  (
(/) ++  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) ) )
9 oveq2 6304 . . . . 5  |-  ( X  =  (/)  ->  ( W ++  X )  =  ( W ++  (/) ) )
109oveq2d 6312 . . . 4  |-  ( X  =  (/)  ->  ( G 
gsumg  ( W ++  X )
)  =  ( G 
gsumg  ( W ++  (/) ) ) )
11 oveq2 6304 . . . . . 6  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( G  gsumg  (/) ) )
1211, 5syl6eq 2514 . . . . 5  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( 0g `  G ) )
1312oveq2d 6312 . . . 4  |-  ( X  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( G  gsumg  W ) 
.+  ( 0g `  G ) ) )
1410, 13eqeq12d 2479 . . 3  |-  ( X  =  (/)  ->  ( ( G  gsumg  ( W ++  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  <->  ( G  gsumg  ( W ++  (/) ) )  =  ( ( G  gsumg  W ) 
.+  ( 0g `  G ) ) ) )
15 gsumwcl.b . . . . . 6  |-  B  =  ( Base `  G
)
16 gsumccat.p . . . . . 6  |-  .+  =  ( +g  `  G )
17 simpl1 999 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  G  e.  Mnd )
18 lennncl 12570 . . . . . . . . . . 11  |-  ( ( W  e. Word  B  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
19183ad2antl2 1159 . . . . . . . . . 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 12570 . . . . . . . . . . 11  |-  ( ( X  e. Word  B  /\  X  =/=  (/) )  ->  ( # `
 X )  e.  NN )
22213ad2antl3 1160 . . . . . . . . . 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 10603 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  NN )
25 nnm1nn0 10858 . . . . . . . 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 11140 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2826, 27syl6eleq 2555 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  ( ZZ>= `  0
) )
29 simpl2 1000 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  W  e. Word  B )
30 simpl3 1001 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  X  e. Word  B )
31 ccatcl 12602 . . . . . . . . 9  |-  ( ( W  e. Word  B  /\  X  e. Word  B )  ->  ( W ++  X )  e. Word  B )
3229, 30, 31syl2anc 661 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( W ++  X )  e. Word  B )
33 wrdf 12558 . . . . . . . 8  |-  ( ( W ++  X )  e. Word  B  ->  ( W ++  X
) : ( 0..^ ( # `  ( W ++  X ) ) ) --> B )
3432, 33syl 16 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( W ++  X ) : ( 0..^ (
# `  ( W ++  X ) ) ) --> B )
35 ccatlen 12603 . . . . . . . . . . 11  |-  ( ( W  e. Word  B  /\  X  e. Word  B )  ->  ( # `  ( W ++  X ) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3629, 30, 35syl2anc 661 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  ( W ++  X ) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3736oveq2d 6312 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W ++  X ) ) )  =  ( 0..^ ( ( # `  W )  +  (
# `  X )
) ) )
3820nnzd 10989 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  ZZ )
3923nnzd 10989 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  ZZ )
4038, 39zaddcld 10994 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  ZZ )
41 fzoval 11827 . . . . . . . . . 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 2498 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W ++  X ) ) )  =  ( 0 ... ( ( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )
4443feq2d 5724 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( W ++  X
) : ( 0..^ ( # `  ( W ++  X ) ) ) --> B  <->  ( W ++  X
) : ( 0 ... ( ( (
# `  W )  +  ( # `  X
) )  -  1 ) ) --> B ) )
4534, 44mpbid 210 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( W ++  X ) : ( 0 ... ( ( ( # `  W )  +  (
# `  X )
)  -  1 ) ) --> B )
4615, 16, 17, 28, 45gsumval2 16034 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  ( W ++  X ) )  =  (  seq 0 (  .+  , 
( W ++  X ) ) `  ( ( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )
47 nnm1nn0 10858 . . . . . . . . . 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 2555 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
50 wrdf 12558 . . . . . . . . . 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 11827 . . . . . . . . . . 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 5724 . . . . . . . . 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 16034 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  W )  =  (  seq 0 (  .+  ,  W ) `  (
( # `  W )  -  1 ) ) )
57 nnm1nn0 10858 . . . . . . . . . 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 2555 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  X
)  -  1 )  e.  ( ZZ>= `  0
) )
60 wrdf 12558 . . . . . . . . . 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 11827 . . . . . . . . . . 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 5724 . . . . . . . . 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 16034 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  X )  =  (  seq 0 (  .+  ,  X ) `  (
( # `  X )  -  1 ) ) )
6756, 66oveq12d 6314 . . . . . 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 16056 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
69683expb 1197 . . . . . . . . 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 16057 . . . . . . . . 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 11120 . . . . . . . . . . 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 11162 . . . . . . . . . 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 10572 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  CC )
7823nncnd 10572 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  CC )
79 1cnd 9629 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
1  e.  CC )
8077, 78, 79addsubassd 9970 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( # `  W )  +  ( ( # `  X
)  -  1 ) ) )
81 ax-1cn 9567 . . . . . . . . . . 11  |-  1  e.  CC
82 npcan 9848 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( # `  W )  -  1 )  +  1 )  =  ( # `  W
) )
8377, 81, 82sylancl 662 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  -  1 )  +  1 )  =  ( # `  W
) )
8483fveq2d 5876 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ZZ>= `  ( (
( # `  W )  -  1 )  +  1 ) )  =  ( ZZ>= `  ( # `  W
) ) )
8576, 80, 843eltr4d 2560 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  ( ZZ>= `  (
( ( # `  W
)  -  1 )  +  1 ) ) )
8645ffvelrnda 6032 . . . . . . . 8  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )  ->  ( ( W ++  X ) `  x
)  e.  B )
8770, 72, 85, 49, 86seqsplit 12143 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq 0 (  .+  ,  ( W ++  X
) ) `  (
( ( # `  W
)  +  ( # `  X ) )  - 
1 ) )  =  ( (  seq 0
(  .+  ,  ( W ++  X ) ) `  ( ( # `  W
)  -  1 ) )  .+  (  seq ( ( ( # `  W )  -  1 )  +  1 ) (  .+  ,  ( W ++  X ) ) `
 ( ( (
# `  W )  +  ( # `  X
) )  -  1 ) ) ) )
88 simpll2 1036 . . . . . . . . . 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 1037 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  X  e. Word  B )
9053eleq2d 2527 . . . . . . . . . . 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 12604 . . . . . . . . . 10  |-  ( ( W  e. Word  B  /\  X  e. Word  B  /\  x  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W ++  X ) `  x
)  =  ( W `
 x ) )
9388, 89, 91, 92syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  ( ( W ++  X ) `  x
)  =  ( W `
 x ) )
9449, 93seqfveq 12134 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq 0 (  .+  ,  ( W ++  X
) ) `  (
( # `  W )  -  1 ) )  =  (  seq 0
(  .+  ,  W
) `  ( ( # `
 W )  - 
1 ) ) )
9577addid2d 9798 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0  +  (
# `  W )
)  =  ( # `  W ) )
9683, 95eqtr4d 2501 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  -  1 )  +  1 )  =  ( 0  +  ( # `  W
) ) )
9796seqeq1d 12116 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  seq ( ( ( # `  W )  -  1 )  +  1 ) (  .+  ,  ( W ++  X ) )  =  seq ( 0  +  ( # `  W
) ) (  .+  ,  ( W ++  X
) ) )
9877, 78addcomd 9799 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  =  ( ( # `  X
)  +  ( # `  W ) ) )
9998oveq1d 6311 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  +  ( # `  W
) )  -  1 ) )
10078, 77, 79addsubd 9971 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  X )  +  (
# `  W )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10199, 100eqtrd 2498 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10297, 101fveq12d 5878 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq ( ( (
# `  W )  -  1 )  +  1 ) (  .+  ,  ( W ++  X
) ) `  (
( ( # `  W
)  +  ( # `  X ) )  - 
1 ) )  =  (  seq ( 0  +  ( # `  W
) ) (  .+  ,  ( W ++  X
) ) `  (
( ( # `  X
)  -  1 )  +  ( # `  W
) ) ) )
103 simpll2 1036 . . . . . . . . . . . 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 1037 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  X  e. Word  B )
10563eleq2d 2527 . . . . . . . . . . . . 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 12606 . . . . . . . . . . . 12  |-  ( ( W  e. Word  B  /\  X  e. Word  B  /\  x  e.  ( 0..^ ( # `  X ) ) )  ->  ( ( W ++  X ) `  (
x  +  ( # `  W ) ) )  =  ( X `  x ) )
108103, 104, 106, 107syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  ( ( W ++  X ) `  (
x  +  ( # `  W ) ) )  =  ( X `  x ) )
109108eqcomd 2465 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  ( X `  x )  =  ( ( W ++  X ) `
 ( x  +  ( # `  W ) ) ) )
11059, 38, 109seqshft2 12136 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq 0 (  .+  ,  X ) `  (
( # `  X )  -  1 ) )  =  (  seq (
0  +  ( # `  W ) ) ( 
.+  ,  ( W ++  X ) ) `  ( ( ( # `  X )  -  1 )  +  ( # `  W ) ) ) )
111102, 110eqtr4d 2501 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq ( ( (
# `  W )  -  1 )  +  1 ) (  .+  ,  ( W ++  X
) ) `  (
( ( # `  W
)  +  ( # `  X ) )  - 
1 ) )  =  (  seq 0 ( 
.+  ,  X ) `
 ( ( # `  X )  -  1 ) ) )
11294, 111oveq12d 6314 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( (  seq 0
(  .+  ,  ( W ++  X ) ) `  ( ( # `  W
)  -  1 ) )  .+  (  seq ( ( ( # `  W )  -  1 )  +  1 ) (  .+  ,  ( W ++  X ) ) `
 ( ( (
# `  W )  +  ( # `  X
) )  -  1 ) ) )  =  ( (  seq 0
(  .+  ,  W
) `  ( ( # `
 W )  - 
1 ) )  .+  (  seq 0 (  .+  ,  X ) `  (
( # `  X )  -  1 ) ) ) )
11387, 112eqtrd 2498 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
(  seq 0 (  .+  ,  ( W ++  X
) ) `  (
( ( # `  W
)  +  ( # `  X ) )  - 
1 ) )  =  ( (  seq 0
(  .+  ,  W
) `  ( ( # `
 W )  - 
1 ) )  .+  (  seq 0 (  .+  ,  X ) `  (
( # `  X )  -  1 ) ) ) )
11467, 113eqtr4d 2501 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( G  gsumg  W ) 
.+  ( G  gsumg  X ) )  =  (  seq 0 (  .+  , 
( W ++  X ) ) `  ( ( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )
11546, 114eqtr4d 2501 . . . 4  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  ( W ++  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 ++  X )
)  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) ) )
117 simpl2 1000 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  ->  W  e. Word  B )
118 ccatrid 12613 . . . . . 6  |-  ( W  e. Word  B  ->  ( W ++  (/) )  =  W )
119117, 118syl 16 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( W ++  (/) )  =  W )
120119oveq2d 6312 . . . 4  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  ( W ++  (/) ) )  =  ( G  gsumg  W ) )
121 simpl1 999 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  ->  G  e.  Mnd )
12215gsumwcl 16135 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  W  e. Word  B )  ->  ( G  gsumg  W )  e.  B
)
1231223adant3 1016 . . . . . 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 16069 . . . . 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 2501 . . 3  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  ( W ++  (/) ) )  =  ( ( G 
gsumg  W )  .+  ( 0g `  G ) ) )
12814, 116, 127pm2.61ne 2772 . 2  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  ( W ++  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) ) )
129 ccatlid 12612 . . . . 5  |-  ( X  e. Word  B  ->  ( (/) ++  X )  =  X )
1301293ad2ant3 1019 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( (/) ++  X )  =  X )
131130oveq2d 6312 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  ( (/) ++  X )
)  =  ( G 
gsumg  X ) )
132 simp1 996 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  G  e.  Mnd )
13315gsumwcl 16135 . . . . 5  |-  ( ( G  e.  Mnd  /\  X  e. Word  B )  ->  ( G  gsumg  X )  e.  B
)
1341333adant2 1015 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  X )  e.  B
)
13515, 16, 4mndlid 16068 . . . 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 2501 . 2  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  ( (/) ++  X )
)  =  ( ( 0g `  G ) 
.+  ( G  gsumg  X ) ) )
1388, 128, 137pm2.61ne 2772 1  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  ( W ++  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   (/)c0 3793   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    - cmin 9824   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697  ..^cfzo 11821    seqcseq 12110   #chash 12408  Word cword 12538   ++ cconcat 12540   Basecbs 14644   +g cplusg 14712   0gc0g 14857    gsumg cgsu 14858   Mndcmnd 16046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-word 12546  df-concat 12548  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-0g 14859  df-gsum 14860  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094
This theorem is referenced by:  gsumws2  16137  gsumccatsn  16138  gsumspl  16139  gsumwspan  16141  frmdgsum  16157  frmdup1  16159  gsumwrev  16528  psgnunilem5  16646  psgnuni  16651  frgpuplem  16917  frgpup1  16920  psgnghm  18743  mrsubccat  29075
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