Step |
Hyp |
Ref |
Expression |
1 |
|
elfzelz |
|- ( x e. ( M ... N ) -> x e. ZZ ) |
2 |
|
elfzelz |
|- ( K e. ( M ... N ) -> K e. ZZ ) |
3 |
|
zltlem1 |
|- ( ( x e. ZZ /\ K e. ZZ ) -> ( x < K <-> x <_ ( K - 1 ) ) ) |
4 |
1 2 3
|
syl2anr |
|- ( ( K e. ( M ... N ) /\ x e. ( M ... N ) ) -> ( x < K <-> x <_ ( K - 1 ) ) ) |
5 |
|
elfzuz |
|- ( x e. ( M ... N ) -> x e. ( ZZ>= ` M ) ) |
6 |
|
peano2zm |
|- ( K e. ZZ -> ( K - 1 ) e. ZZ ) |
7 |
2 6
|
syl |
|- ( K e. ( M ... N ) -> ( K - 1 ) e. ZZ ) |
8 |
|
elfz5 |
|- ( ( x e. ( ZZ>= ` M ) /\ ( K - 1 ) e. ZZ ) -> ( x e. ( M ... ( K - 1 ) ) <-> x <_ ( K - 1 ) ) ) |
9 |
5 7 8
|
syl2anr |
|- ( ( K e. ( M ... N ) /\ x e. ( M ... N ) ) -> ( x e. ( M ... ( K - 1 ) ) <-> x <_ ( K - 1 ) ) ) |
10 |
4 9
|
bitr4d |
|- ( ( K e. ( M ... N ) /\ x e. ( M ... N ) ) -> ( x < K <-> x e. ( M ... ( K - 1 ) ) ) ) |
11 |
10
|
pm5.32da |
|- ( K e. ( M ... N ) -> ( ( x e. ( M ... N ) /\ x < K ) <-> ( x e. ( M ... N ) /\ x e. ( M ... ( K - 1 ) ) ) ) ) |
12 |
|
vex |
|- x e. _V |
13 |
12
|
elpred |
|- ( K e. ( M ... N ) -> ( x e. Pred ( < , ( M ... N ) , K ) <-> ( x e. ( M ... N ) /\ x < K ) ) ) |
14 |
|
elfzuz3 |
|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) ) |
15 |
2
|
zcnd |
|- ( K e. ( M ... N ) -> K e. CC ) |
16 |
|
ax-1cn |
|- 1 e. CC |
17 |
|
npcan |
|- ( ( K e. CC /\ 1 e. CC ) -> ( ( K - 1 ) + 1 ) = K ) |
18 |
15 16 17
|
sylancl |
|- ( K e. ( M ... N ) -> ( ( K - 1 ) + 1 ) = K ) |
19 |
18
|
fveq2d |
|- ( K e. ( M ... N ) -> ( ZZ>= ` ( ( K - 1 ) + 1 ) ) = ( ZZ>= ` K ) ) |
20 |
14 19
|
eleqtrrd |
|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` ( ( K - 1 ) + 1 ) ) ) |
21 |
|
peano2uzr |
|- ( ( ( K - 1 ) e. ZZ /\ N e. ( ZZ>= ` ( ( K - 1 ) + 1 ) ) ) -> N e. ( ZZ>= ` ( K - 1 ) ) ) |
22 |
7 20 21
|
syl2anc |
|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` ( K - 1 ) ) ) |
23 |
|
fzss2 |
|- ( N e. ( ZZ>= ` ( K - 1 ) ) -> ( M ... ( K - 1 ) ) C_ ( M ... N ) ) |
24 |
22 23
|
syl |
|- ( K e. ( M ... N ) -> ( M ... ( K - 1 ) ) C_ ( M ... N ) ) |
25 |
24
|
sseld |
|- ( K e. ( M ... N ) -> ( x e. ( M ... ( K - 1 ) ) -> x e. ( M ... N ) ) ) |
26 |
25
|
pm4.71rd |
|- ( K e. ( M ... N ) -> ( x e. ( M ... ( K - 1 ) ) <-> ( x e. ( M ... N ) /\ x e. ( M ... ( K - 1 ) ) ) ) ) |
27 |
11 13 26
|
3bitr4d |
|- ( K e. ( M ... N ) -> ( x e. Pred ( < , ( M ... N ) , K ) <-> x e. ( M ... ( K - 1 ) ) ) ) |
28 |
27
|
eqrdv |
|- ( K e. ( M ... N ) -> Pred ( < , ( M ... N ) , K ) = ( M ... ( K - 1 ) ) ) |