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

Theorem gsumccat 14742
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 6047 . . . 4  |-  ( W  =  (/)  ->  ( W concat  X )  =  (
(/) concat  X ) )
21oveq2d 6056 . . 3  |-  ( W  =  (/)  ->  ( G 
gsumg  ( W concat  X ) )  =  ( G  gsumg  ( (/) concat  X ) ) )
3 oveq2 6048 . . . . 5  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( G  gsumg  (/) ) )
4 eqid 2404 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
54gsum0 14735 . . . . 5  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
63, 5syl6eq 2452 . . . 4  |-  ( W  =  (/)  ->  ( G 
gsumg  W )  =  ( 0g `  G ) )
76oveq1d 6055 . . 3  |-  ( W  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) )
82, 7eqeq12d 2418 . 2  |-  ( W  =  (/)  ->  ( ( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  <->  ( G  gsumg  (
(/) concat  X ) )  =  ( ( 0g `  G )  .+  ( G  gsumg  X ) ) ) )
9 oveq2 6048 . . . . 5  |-  ( X  =  (/)  ->  ( W concat  X )  =  ( W concat  (/) ) )
109oveq2d 6056 . . . 4  |-  ( X  =  (/)  ->  ( G 
gsumg  ( W concat  X ) )  =  ( G  gsumg  ( W concat  (/) ) ) )
11 oveq2 6048 . . . . . 6  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( G  gsumg  (/) ) )
1211, 5syl6eq 2452 . . . . 5  |-  ( X  =  (/)  ->  ( G 
gsumg  X )  =  ( 0g `  G ) )
1312oveq2d 6056 . . . 4  |-  ( X  =  (/)  ->  ( ( G  gsumg  W )  .+  ( G  gsumg  X ) )  =  ( ( G  gsumg  W ) 
.+  ( 0g `  G ) ) )
1410, 13eqeq12d 2418 . . 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 11691 . . . . . . . . . . 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 11691 . . . . . . . . . . 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 10002 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  NN )
25 nnm1nn0 10217 . . . . . . . 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 10476 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2826, 27syl6eleq 2494 . . . . . 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 11698 . . . . . . . . 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 11688 . . . . . . . 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 11699 . . . . . . . . . . 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 6056 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W concat  X ) ) )  =  ( 0..^ ( ( # `  W )  +  (
# `  X )
) ) )
3820nnzd 10330 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  ZZ )
3923nnzd 10330 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  ZZ )
4038, 39zaddcld 10335 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  e.  ZZ )
41 fzoval 11096 . . . . . . . . . 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 2436 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0..^ ( # `  ( W concat  X ) ) )  =  ( 0 ... ( ( ( # `  W
)  +  ( # `  X ) )  - 
1 ) ) )
4443feq2d 5540 . . . . . . 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 14738 . . . . 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 10217 . . . . . . . . . 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 2494 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  -  1 )  e.  ( ZZ>= `  0
) )
50 wrdf 11688 . . . . . . . . . 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 11096 . . . . . . . . . . 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 5540 . . . . . . . . 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 14738 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  W )  =  (  seq  0 (  .+  ,  W ) `  (
( # `  W )  -  1 ) ) )
57 nnm1nn0 10217 . . . . . . . . . 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 2494 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  X
)  -  1 )  e.  ( ZZ>= `  0
) )
60 wrdf 11688 . . . . . . . . . 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 11096 . . . . . . . . . . 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 5540 . . . . . . . . 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 14738 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( G  gsumg  X )  =  (  seq  0 (  .+  ,  X ) `  (
( # `  X )  -  1 ) ) )
6756, 66oveq12d 6058 . . . . . 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 14650 . . . . . . . . . 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 14651 . . . . . . . . 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 10456 . . . . . . . . . . 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 10489 . . . . . . . . . 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 9972 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  W )  e.  CC )
7823nncnd 9972 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( # `  X )  e.  CC )
79 ax-1cn 9004 . . . . . . . . . . 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 9387 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( # `  W )  +  ( ( # `  X
)  -  1 ) ) )
82 npcan 9270 . . . . . . . . . . 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 5691 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ZZ>= `  ( (
( # `  W )  -  1 )  +  1 ) )  =  ( ZZ>= `  ( # `  W
) ) )
8576, 81, 843eltr4d 2485 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  e.  ( ZZ>= `  (
( ( # `  W
)  -  1 )  +  1 ) ) )
8645ffvelrnda 5829 . . . . . . . 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 11311 . . . . . . 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 2471 . . . . . . . . . . 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 11700 . . . . . . . . . 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 11302 . . . . . . . 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 9223 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( 0  +  (
# `  W )
)  =  ( # `  W ) )
9683, 95eqtr4d 2439 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  -  1 )  +  1 )  =  ( 0  +  ( # `  W
) ) )
9796seqeq1d 11284 . . . . . . . . . 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 9224 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( # `  W
)  +  ( # `  X ) )  =  ( ( # `  X
)  +  ( # `  W ) ) )
9998oveq1d 6055 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  +  ( # `  W
) )  -  1 ) )
10078, 77, 80addsubd 9388 . . . . . . . . . . 11  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  X )  +  (
# `  W )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10199, 100eqtrd 2436 . . . . . . . . . 10  |-  ( ( ( G  e.  Mnd  /\  W  e. Word  B  /\  X  e. Word  B )  /\  ( W  =/=  (/)  /\  X  =/=  (/) ) )  -> 
( ( ( # `  W )  +  (
# `  X )
)  -  1 )  =  ( ( (
# `  X )  -  1 )  +  ( # `  W
) ) )
10297, 101fveq12d 5693 . . . . . . . . 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 2471 . . . . . . . . . . . . 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 11702 . . . . . . . . . . . 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 2409 . . . . . . . . . 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 11304 . . . . . . . . 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 2439 . . . . . . . 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 6058 . . . . . . 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 2436 . . . . . 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 2439 . . . . 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 2439 . . . 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 11704 . . . . . 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 6056 . . . 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 14741 . . . . . . 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 14672 . . . . 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 2439 . . 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 2642 . 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 11703 . . . . 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 6056 . . 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 14741 . . . . 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 14671 . . . 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 2439 . 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 2642 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 1721    =/= wne 2567   (/)c0 3588   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    - cmin 9247   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  ..^cfzo 11090    seq cseq 11278   #chash 11573  Word cword 11672   concat cconcat 11673   Basecbs 13424   +g cplusg 13484   0gc0g 13678    gsumg cgsu 13679   Mndcmnd 14639
This theorem is referenced by:  gsumws2  14743  gsumspl  14744  gsumwspan  14746  frmdgsum  14762  frmdup1  14764  gsumwrev  15117  frgpuplem  15359  frgpup1  15362  psgnunilem5  27285  psgnuni  27290  psgnghm  27305
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-word 11678  df-concat 11679  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-gsum 13683  df-mnd 14645  df-submnd 14694
  Copyright terms: Public domain W3C validator