Step |
Hyp |
Ref |
Expression |
1 |
|
ccatlen |
|- ( ( A e. Word V /\ B e. Word V ) -> ( # ` ( A ++ B ) ) = ( ( # ` A ) + ( # ` B ) ) ) |
2 |
1
|
oveq1d |
|- ( ( A e. Word V /\ B e. Word V ) -> ( ( # ` ( A ++ B ) ) - 1 ) = ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) |
3 |
2
|
3adant3 |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( # ` ( A ++ B ) ) - 1 ) = ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) |
4 |
|
lencl |
|- ( A e. Word V -> ( # ` A ) e. NN0 ) |
5 |
4
|
nn0zd |
|- ( A e. Word V -> ( # ` A ) e. ZZ ) |
6 |
|
lennncl |
|- ( ( B e. Word V /\ B =/= (/) ) -> ( # ` B ) e. NN ) |
7 |
|
simpl |
|- ( ( ( # ` A ) e. ZZ /\ ( # ` B ) e. NN ) -> ( # ` A ) e. ZZ ) |
8 |
|
nnz |
|- ( ( # ` B ) e. NN -> ( # ` B ) e. ZZ ) |
9 |
|
zaddcl |
|- ( ( ( # ` A ) e. ZZ /\ ( # ` B ) e. ZZ ) -> ( ( # ` A ) + ( # ` B ) ) e. ZZ ) |
10 |
8 9
|
sylan2 |
|- ( ( ( # ` A ) e. ZZ /\ ( # ` B ) e. NN ) -> ( ( # ` A ) + ( # ` B ) ) e. ZZ ) |
11 |
|
zre |
|- ( ( # ` A ) e. ZZ -> ( # ` A ) e. RR ) |
12 |
|
nnrp |
|- ( ( # ` B ) e. NN -> ( # ` B ) e. RR+ ) |
13 |
|
ltaddrp |
|- ( ( ( # ` A ) e. RR /\ ( # ` B ) e. RR+ ) -> ( # ` A ) < ( ( # ` A ) + ( # ` B ) ) ) |
14 |
11 12 13
|
syl2an |
|- ( ( ( # ` A ) e. ZZ /\ ( # ` B ) e. NN ) -> ( # ` A ) < ( ( # ` A ) + ( # ` B ) ) ) |
15 |
7 10 14
|
3jca |
|- ( ( ( # ` A ) e. ZZ /\ ( # ` B ) e. NN ) -> ( ( # ` A ) e. ZZ /\ ( ( # ` A ) + ( # ` B ) ) e. ZZ /\ ( # ` A ) < ( ( # ` A ) + ( # ` B ) ) ) ) |
16 |
5 6 15
|
syl2an |
|- ( ( A e. Word V /\ ( B e. Word V /\ B =/= (/) ) ) -> ( ( # ` A ) e. ZZ /\ ( ( # ` A ) + ( # ` B ) ) e. ZZ /\ ( # ` A ) < ( ( # ` A ) + ( # ` B ) ) ) ) |
17 |
16
|
3impb |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( # ` A ) e. ZZ /\ ( ( # ` A ) + ( # ` B ) ) e. ZZ /\ ( # ` A ) < ( ( # ` A ) + ( # ` B ) ) ) ) |
18 |
|
fzolb |
|- ( ( # ` A ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) <-> ( ( # ` A ) e. ZZ /\ ( ( # ` A ) + ( # ` B ) ) e. ZZ /\ ( # ` A ) < ( ( # ` A ) + ( # ` B ) ) ) ) |
19 |
17 18
|
sylibr |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( # ` A ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) |
20 |
|
fzoend |
|- ( ( # ` A ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) -> ( ( ( # ` A ) + ( # ` B ) ) - 1 ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) |
21 |
19 20
|
syl |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( ( # ` A ) + ( # ` B ) ) - 1 ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) |
22 |
3 21
|
eqeltrd |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( # ` ( A ++ B ) ) - 1 ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) |
23 |
|
ccatval2 |
|- ( ( A e. Word V /\ B e. Word V /\ ( ( # ` ( A ++ B ) ) - 1 ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) -> ( ( A ++ B ) ` ( ( # ` ( A ++ B ) ) - 1 ) ) = ( B ` ( ( ( # ` ( A ++ B ) ) - 1 ) - ( # ` A ) ) ) ) |
24 |
22 23
|
syld3an3 |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( A ++ B ) ` ( ( # ` ( A ++ B ) ) - 1 ) ) = ( B ` ( ( ( # ` ( A ++ B ) ) - 1 ) - ( # ` A ) ) ) ) |
25 |
2
|
oveq1d |
|- ( ( A e. Word V /\ B e. Word V ) -> ( ( ( # ` ( A ++ B ) ) - 1 ) - ( # ` A ) ) = ( ( ( ( # ` A ) + ( # ` B ) ) - 1 ) - ( # ` A ) ) ) |
26 |
4
|
nn0cnd |
|- ( A e. Word V -> ( # ` A ) e. CC ) |
27 |
|
lencl |
|- ( B e. Word V -> ( # ` B ) e. NN0 ) |
28 |
27
|
nn0cnd |
|- ( B e. Word V -> ( # ` B ) e. CC ) |
29 |
|
addcl |
|- ( ( ( # ` A ) e. CC /\ ( # ` B ) e. CC ) -> ( ( # ` A ) + ( # ` B ) ) e. CC ) |
30 |
|
1cnd |
|- ( ( ( # ` A ) e. CC /\ ( # ` B ) e. CC ) -> 1 e. CC ) |
31 |
|
simpl |
|- ( ( ( # ` A ) e. CC /\ ( # ` B ) e. CC ) -> ( # ` A ) e. CC ) |
32 |
29 30 31
|
sub32d |
|- ( ( ( # ` A ) e. CC /\ ( # ` B ) e. CC ) -> ( ( ( ( # ` A ) + ( # ` B ) ) - 1 ) - ( # ` A ) ) = ( ( ( ( # ` A ) + ( # ` B ) ) - ( # ` A ) ) - 1 ) ) |
33 |
|
pncan2 |
|- ( ( ( # ` A ) e. CC /\ ( # ` B ) e. CC ) -> ( ( ( # ` A ) + ( # ` B ) ) - ( # ` A ) ) = ( # ` B ) ) |
34 |
33
|
oveq1d |
|- ( ( ( # ` A ) e. CC /\ ( # ` B ) e. CC ) -> ( ( ( ( # ` A ) + ( # ` B ) ) - ( # ` A ) ) - 1 ) = ( ( # ` B ) - 1 ) ) |
35 |
32 34
|
eqtrd |
|- ( ( ( # ` A ) e. CC /\ ( # ` B ) e. CC ) -> ( ( ( ( # ` A ) + ( # ` B ) ) - 1 ) - ( # ` A ) ) = ( ( # ` B ) - 1 ) ) |
36 |
26 28 35
|
syl2an |
|- ( ( A e. Word V /\ B e. Word V ) -> ( ( ( ( # ` A ) + ( # ` B ) ) - 1 ) - ( # ` A ) ) = ( ( # ` B ) - 1 ) ) |
37 |
25 36
|
eqtrd |
|- ( ( A e. Word V /\ B e. Word V ) -> ( ( ( # ` ( A ++ B ) ) - 1 ) - ( # ` A ) ) = ( ( # ` B ) - 1 ) ) |
38 |
37
|
3adant3 |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( ( # ` ( A ++ B ) ) - 1 ) - ( # ` A ) ) = ( ( # ` B ) - 1 ) ) |
39 |
38
|
fveq2d |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( B ` ( ( ( # ` ( A ++ B ) ) - 1 ) - ( # ` A ) ) ) = ( B ` ( ( # ` B ) - 1 ) ) ) |
40 |
24 39
|
eqtrd |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( A ++ B ) ` ( ( # ` ( A ++ B ) ) - 1 ) ) = ( B ` ( ( # ` B ) - 1 ) ) ) |
41 |
|
ovex |
|- ( A ++ B ) e. _V |
42 |
|
lsw |
|- ( ( A ++ B ) e. _V -> ( lastS ` ( A ++ B ) ) = ( ( A ++ B ) ` ( ( # ` ( A ++ B ) ) - 1 ) ) ) |
43 |
41 42
|
mp1i |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( lastS ` ( A ++ B ) ) = ( ( A ++ B ) ` ( ( # ` ( A ++ B ) ) - 1 ) ) ) |
44 |
|
lsw |
|- ( B e. Word V -> ( lastS ` B ) = ( B ` ( ( # ` B ) - 1 ) ) ) |
45 |
44
|
3ad2ant2 |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( lastS ` B ) = ( B ` ( ( # ` B ) - 1 ) ) ) |
46 |
40 43 45
|
3eqtr4d |
|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( lastS ` ( A ++ B ) ) = ( lastS ` B ) ) |