Step |
Hyp |
Ref |
Expression |
1 |
|
elfzoel1 |
|- ( C e. ( A ..^ B ) -> A e. ZZ ) |
2 |
|
uzid |
|- ( A e. ZZ -> A e. ( ZZ>= ` A ) ) |
3 |
|
peano2uz |
|- ( A e. ( ZZ>= ` A ) -> ( A + 1 ) e. ( ZZ>= ` A ) ) |
4 |
|
fzoss1 |
|- ( ( A + 1 ) e. ( ZZ>= ` A ) -> ( ( A + 1 ) ..^ ( B + 1 ) ) C_ ( A ..^ ( B + 1 ) ) ) |
5 |
1 2 3 4
|
4syl |
|- ( C e. ( A ..^ B ) -> ( ( A + 1 ) ..^ ( B + 1 ) ) C_ ( A ..^ ( B + 1 ) ) ) |
6 |
|
1z |
|- 1 e. ZZ |
7 |
|
fzoaddel |
|- ( ( C e. ( A ..^ B ) /\ 1 e. ZZ ) -> ( C + 1 ) e. ( ( A + 1 ) ..^ ( B + 1 ) ) ) |
8 |
6 7
|
mpan2 |
|- ( C e. ( A ..^ B ) -> ( C + 1 ) e. ( ( A + 1 ) ..^ ( B + 1 ) ) ) |
9 |
5 8
|
sseldd |
|- ( C e. ( A ..^ B ) -> ( C + 1 ) e. ( A ..^ ( B + 1 ) ) ) |
10 |
|
elfzoel2 |
|- ( C e. ( A ..^ B ) -> B e. ZZ ) |
11 |
|
fzval3 |
|- ( B e. ZZ -> ( A ... B ) = ( A ..^ ( B + 1 ) ) ) |
12 |
10 11
|
syl |
|- ( C e. ( A ..^ B ) -> ( A ... B ) = ( A ..^ ( B + 1 ) ) ) |
13 |
9 12
|
eleqtrrd |
|- ( C e. ( A ..^ B ) -> ( C + 1 ) e. ( A ... B ) ) |