Step |
Hyp |
Ref |
Expression |
1 |
|
eluzel2 |
|- ( B e. ( ZZ>= ` A ) -> A e. ZZ ) |
2 |
|
eluzelz |
|- ( B e. ( ZZ>= ` A ) -> B e. ZZ ) |
3 |
|
1z |
|- 1 e. ZZ |
4 |
|
zsubcl |
|- ( ( 1 e. ZZ /\ A e. ZZ ) -> ( 1 - A ) e. ZZ ) |
5 |
3 1 4
|
sylancr |
|- ( B e. ( ZZ>= ` A ) -> ( 1 - A ) e. ZZ ) |
6 |
|
fzen |
|- ( ( A e. ZZ /\ B e. ZZ /\ ( 1 - A ) e. ZZ ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) ) |
7 |
1 2 5 6
|
syl3anc |
|- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) ) |
8 |
1
|
zcnd |
|- ( B e. ( ZZ>= ` A ) -> A e. CC ) |
9 |
|
ax-1cn |
|- 1 e. CC |
10 |
|
pncan3 |
|- ( ( A e. CC /\ 1 e. CC ) -> ( A + ( 1 - A ) ) = 1 ) |
11 |
8 9 10
|
sylancl |
|- ( B e. ( ZZ>= ` A ) -> ( A + ( 1 - A ) ) = 1 ) |
12 |
|
1cnd |
|- ( B e. ( ZZ>= ` A ) -> 1 e. CC ) |
13 |
2
|
zcnd |
|- ( B e. ( ZZ>= ` A ) -> B e. CC ) |
14 |
13 8
|
subcld |
|- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC ) |
15 |
13 12 8
|
addsub12d |
|- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) ) |
16 |
12 14 15
|
comraddd |
|- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) ) |
17 |
11 16
|
oveq12d |
|- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) ) |
18 |
7 17
|
breqtrd |
|- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) ) |
19 |
|
hasheni |
|- ( ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) -> ( # ` ( A ... B ) ) = ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) ) |
20 |
18 19
|
syl |
|- ( B e. ( ZZ>= ` A ) -> ( # ` ( A ... B ) ) = ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) ) |
21 |
|
uznn0sub |
|- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 ) |
22 |
|
peano2nn0 |
|- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 ) |
23 |
|
hashfz1 |
|- ( ( ( B - A ) + 1 ) e. NN0 -> ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) ) |
24 |
21 22 23
|
3syl |
|- ( B e. ( ZZ>= ` A ) -> ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) ) |
25 |
20 24
|
eqtrd |
|- ( B e. ( ZZ>= ` A ) -> ( # ` ( A ... B ) ) = ( ( B - A ) + 1 ) ) |