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

Theorem gsumccat 15512
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 6097 . . . 4  |-  ( W  =  (/)  ->  ( W concat  X )  =  (
(/) concat  X ) )
21oveq2d 6106 . . 3  |-  ( W  =  (/)  ->  ( G 
gsumg  ( W concat  X ) )  =  ( G  gsumg  ( (/) concat  X ) ) )
3 oveq2 6098 . . . . 5  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( G  gsumg  (/) ) )
4 eqid 2441 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
54gsum0 15503 . . . . 5  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
63, 5syl6eq 2489 . . . 4  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( 0g `  G ) )
76oveq1d 6105 . . 3  |-  ( W  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) )
82, 7eqeq12d 2455 . 2  |-  ( W  =  (/)  ->  ( ( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  <->  ( G  gsumg  (
(/) concat  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) ) )
9 oveq2 6098 . . . . 5  |-  ( X  =  (/)  ->  ( W concat  X )  =  ( W concat  (/) ) )
109oveq2d 6106 . . . 4  |-  ( X  =  (/)  ->  ( G 
gsumg  ( W concat  X ) )  =  ( G  gsumg  ( W concat  (/) ) ) )
11 oveq2 6098 . . . . . 6  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( G  gsumg  (/) ) )
1211, 5syl6eq 2489 . . . . 5  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( 0g `  G ) )
1312oveq2d 6106 . . . 4  |-  ( X  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( G  gsumg  W ) 
.+  ( 0g `  G ) ) )
1410, 13eqeq12d 2455 . . 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 986 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  G  e.  Mnd )
18 lennncl 12246 . . . . . . . . . . 11  |-  ( ( W  e. Word  B  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
19183ad2antl2 1146 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( # `  W )  e.  NN )
2019adantrr 711 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  NN )
21 lennncl 12246 . . . . . . . . . . 11  |-  ( ( X  e. Word  B  /\  X  =/=  (/) )  ->  ( # `
 X )  e.  NN )
22213ad2antl3 1147 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  X  =/=  (/) )  -> 
( # `  X )  e.  NN )
2322adantrl 710 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  NN )
2420, 23nnaddcld 10364 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  NN )
25 nnm1nn0 10617 . . . . . . . 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 10891 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2826, 27syl6eleq 2531 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  ( ZZ>= `  0
) )
29 simpl2 987 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  W  e. Word  B )
30 simpl3 988 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  ->  X  e. Word  B )
31 ccatcl 12270 . . . . . . . . 9  |-  ( ( W  e. Word  B  /\  X  e. Word  B )  ->  ( W concat  X )  e. Word  B )
3229, 30, 31syl2anc 656 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( W concat  X )  e. Word  B )
33 wrdf 12236 . . . . . . . 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 12271 . . . . . . . . . . 11  |-  ( ( W  e. Word  B  /\  X  e. Word  B )  ->  ( # `  ( W concat  X ) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3629, 30, 35syl2anc 656 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  ( W concat  X ) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3736oveq2d 6106 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W concat  X ) ) )  =  ( 0..^ ( ( # `  W )  +  (
# `  X )
) ) )
3820nnzd 10742 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  ZZ )
3923nnzd 10742 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  ZZ )
4038, 39zaddcld 10747 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  ZZ )
41 fzoval 11550 . . . . . . . . . 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 2473 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W concat  X ) ) )  =  ( 0 ... ( ( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )
4443feq2d 5544 . . . . . . 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 15506 . . . . 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 10617 . . . . . . . . . 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 2531 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
50 wrdf 12236 . . . . . . . . . 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 11550 . . . . . . . . . . 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 5544 . . . . . . . . 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 15506 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  W )  =  (  seq 0 (  .+  ,  W ) `  (
( # `  W )  -  1 ) ) )
57 nnm1nn0 10617 . . . . . . . . . 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 2531 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  X
)  -  1 )  e.  ( ZZ>= `  0
) )
60 wrdf 12236 . . . . . . . . . 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 11550 . . . . . . . . . . 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 5544 . . . . . . . . 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 15506 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  X )  =  (  seq 0 (  .+  ,  X ) `  (
( # `  X )  -  1 ) ) )
6756, 66oveq12d 6108 . . . . . 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 15416 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
69683expb 1183 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
7017, 69sylan 468 . . . . . . . 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 15417 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )  ->  (
( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )
7217, 71sylan 468 . . . . . . . 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 10871 . . . . . . . . . . 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 10907 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  ( ZZ>= `  ( # `  W ) )  /\  ( (
# `  X )  -  1 )  e. 
NN0 )  ->  (
( # `  W )  +  ( ( # `  X )  -  1 ) )  e.  (
ZZ>= `  ( # `  W
) ) )
7674, 58, 75syl2anc 656 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( (
# `  X )  -  1 ) )  e.  ( ZZ>= `  ( # `
 W ) ) )
7720nncnd 10334 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  CC )
7823nncnd 10334 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  CC )
79 ax-1cn 9336 . . . . . . . . . . 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 9735 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( # `  W )  +  ( ( # `  X
)  -  1 ) ) )
82 npcan 9615 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( # `  W )  -  1 )  +  1 )  =  ( # `  W
) )
8377, 79, 82sylancl 657 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  -  1 )  +  1 )  =  ( # `  W
) )
8483fveq2d 5692 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ZZ>= `  ( (
( # `  W )  -  1 )  +  1 ) )  =  ( ZZ>= `  ( # `  W
) ) )
8576, 81, 843eltr4d 2522 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  ( ZZ>= `  (
( ( # `  W
)  -  1 )  +  1 ) ) )
8645ffvelrnda 5840 . . . . . . . 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 11835 . . . . . . 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 1023 . . . . . . . . . 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 1024 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) )  ->  X  e. Word  B )
9053eleq2d 2508 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( x  e.  ( 0..^ ( # `  W
) )  <->  x  e.  ( 0 ... (
( # `  W )  -  1 ) ) ) )
9190biimpar 482 . . . . . . . . . 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 12272 . . . . . . . . . 10  |-  ( ( W  e. Word  B  /\  X  e. Word  B  /\  x  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W concat  X ) `  x
)  =  ( W `
 x ) )
9388, 89, 91, 92syl3anc 1213 . . . . . . . . 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 11826 . . . . . . . 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 9566 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0  +  (
# `  W )
)  =  ( # `  W ) )
9683, 95eqtr4d 2476 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  -  1 )  +  1 )  =  ( 0  +  ( # `  W
) ) )
9796seqeq1d 11808 . . . . . . . . . 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 9567 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  =  ( ( # `  X
)  +  ( # `  W ) ) )
9998oveq1d 6105 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  +  ( # `  W
) )  -  1 ) )
10078, 77, 80addsubd 9736 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  X )  +  (
# `  W )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10199, 100eqtrd 2473 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10297, 101fveq12d 5694 . . . . . . . . 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 1023 . . . . . . . . . . . 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 1024 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Mnd  /\  W  e. Word  B  /\  X  e. Word  B
)  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  /\  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) )  ->  X  e. Word  B )
10563eleq2d 2508 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( x  e.  ( 0..^ ( # `  X
) )  <->  x  e.  ( 0 ... (
( # `  X )  -  1 ) ) ) )
106105biimpar 482 . . . . . . . . . . . 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 12274 . . . . . . . . . . . 12  |-  ( ( W  e. Word  B  /\  X  e. Word  B  /\  x  e.  ( 0..^ ( # `  X ) ) )  ->  ( ( W concat  X ) `  (
x  +  ( # `  W ) ) )  =  ( X `  x ) )
108103, 104, 106, 107syl3anc 1213 . . . . . . . . . . 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 2446 . . . . . . . . . 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 11828 . . . . . . . . 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 2476 . . . . . . . 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 6108 . . . . . . 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 2473 . . . . . 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 2476 . . . . 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 2476 . . . 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 643 . . 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 987 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  ->  W  e. Word  B )
118 ccatrid 12281 . . . . . 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 6106 . . . 4  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  ( W concat  (/) ) )  =  ( G  gsumg  W ) )
121 simpl1 986 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  ->  G  e.  Mnd )
12215gsumwcl 15511 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  W  e. Word  B )  ->  ( G  gsumg  W )  e.  B
)
1231223adant3 1003 . . . . . 6  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  W )  e.  B
)
124123adantr 462 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( G  gsumg  W )  e.  B
)
12515, 16, 4mndrid 15438 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( G  gsumg  W )  e.  B
)  ->  ( ( G  gsumg  W )  .+  ( 0g `  G ) )  =  ( G  gsumg  W ) )
126121, 124, 125syl2anc 656 . . . 4  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  W  =/=  (/) )  -> 
( ( G  gsumg  W ) 
.+  ( 0g `  G ) )  =  ( G  gsumg  W ) )
127120, 126eqtr4d 2476 . . 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 2684 . 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 12280 . . . . 5  |-  ( X  e. Word  B  ->  ( (/) concat  X )  =  X )
1301293ad2ant3 1006 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( (/) concat  X )  =  X )
131130oveq2d 6106 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  ( (/) concat  X ) )  =  ( G  gsumg  X ) )
132 simp1 983 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  G  e.  Mnd )
13315gsumwcl 15511 . . . . 5  |-  ( ( G  e.  Mnd  /\  X  e. Word  B )  ->  ( G  gsumg  X )  e.  B
)
1341333adant2 1002 . . . 4  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  ( G  gsumg  X )  e.  B
)
13515, 16, 4mndlid 15437 . . . 4  |-  ( ( G  e.  Mnd  /\  ( G  gsumg  X )  e.  B
)  ->  ( ( 0g `  G )  .+  ( G  gsumg  X ) )  =  ( G  gsumg  X ) )
136132, 134, 135syl2anc 656 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  ->  (
( 0g `  G
)  .+  ( G  gsumg  X ) )  =  ( G  gsumg  X ) )
137131, 136eqtr4d 2476 . 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 2684 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   (/)c0 3634   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    - cmin 9591   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433  ..^cfzo 11544    seqcseq 11802   #chash 12099  Word cword 12217   concat cconcat 12219   Basecbs 14170   +g cplusg 14234   0gc0g 14374    gsumg cgsu 14375   Mndcmnd 15405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-word 12225  df-concat 12227  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-gsum 14377  df-mnd 15411  df-submnd 15461
This theorem is referenced by:  gsumws2  15513  gsumccatsn  15514  gsumspl  15515  gsumwspan  15517  frmdgsum  15533  frmdup1  15535  gsumwrev  15874  psgnunilem5  15993  psgnuni  15998  frgpuplem  16262  frgpup1  16265  psgnghm  17969
  Copyright terms: Public domain W3C validator