Step |
Hyp |
Ref |
Expression |
1 |
|
evenz |
|- ( M e. Even -> M e. ZZ ) |
2 |
|
evenz |
|- ( N e. Even -> N e. ZZ ) |
3 |
|
zltp1le |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) ) |
4 |
1 2 3
|
syl2anr |
|- ( ( N e. Even /\ M e. Even ) -> ( M < N <-> ( M + 1 ) <_ N ) ) |
5 |
1
|
zred |
|- ( M e. Even -> M e. RR ) |
6 |
|
peano2re |
|- ( M e. RR -> ( M + 1 ) e. RR ) |
7 |
5 6
|
syl |
|- ( M e. Even -> ( M + 1 ) e. RR ) |
8 |
2
|
zred |
|- ( N e. Even -> N e. RR ) |
9 |
|
leloe |
|- ( ( ( M + 1 ) e. RR /\ N e. RR ) -> ( ( M + 1 ) <_ N <-> ( ( M + 1 ) < N \/ ( M + 1 ) = N ) ) ) |
10 |
7 8 9
|
syl2anr |
|- ( ( N e. Even /\ M e. Even ) -> ( ( M + 1 ) <_ N <-> ( ( M + 1 ) < N \/ ( M + 1 ) = N ) ) ) |
11 |
1
|
peano2zd |
|- ( M e. Even -> ( M + 1 ) e. ZZ ) |
12 |
|
zltp1le |
|- ( ( ( M + 1 ) e. ZZ /\ N e. ZZ ) -> ( ( M + 1 ) < N <-> ( ( M + 1 ) + 1 ) <_ N ) ) |
13 |
11 2 12
|
syl2anr |
|- ( ( N e. Even /\ M e. Even ) -> ( ( M + 1 ) < N <-> ( ( M + 1 ) + 1 ) <_ N ) ) |
14 |
1
|
zcnd |
|- ( M e. Even -> M e. CC ) |
15 |
14
|
adantl |
|- ( ( N e. Even /\ M e. Even ) -> M e. CC ) |
16 |
|
add1p1 |
|- ( M e. CC -> ( ( M + 1 ) + 1 ) = ( M + 2 ) ) |
17 |
15 16
|
syl |
|- ( ( N e. Even /\ M e. Even ) -> ( ( M + 1 ) + 1 ) = ( M + 2 ) ) |
18 |
17
|
breq1d |
|- ( ( N e. Even /\ M e. Even ) -> ( ( ( M + 1 ) + 1 ) <_ N <-> ( M + 2 ) <_ N ) ) |
19 |
18
|
biimpd |
|- ( ( N e. Even /\ M e. Even ) -> ( ( ( M + 1 ) + 1 ) <_ N -> ( M + 2 ) <_ N ) ) |
20 |
13 19
|
sylbid |
|- ( ( N e. Even /\ M e. Even ) -> ( ( M + 1 ) < N -> ( M + 2 ) <_ N ) ) |
21 |
|
evenp1odd |
|- ( M e. Even -> ( M + 1 ) e. Odd ) |
22 |
|
zneoALTV |
|- ( ( N e. Even /\ ( M + 1 ) e. Odd ) -> N =/= ( M + 1 ) ) |
23 |
|
eqneqall |
|- ( N = ( M + 1 ) -> ( N =/= ( M + 1 ) -> ( M + 2 ) <_ N ) ) |
24 |
23
|
eqcoms |
|- ( ( M + 1 ) = N -> ( N =/= ( M + 1 ) -> ( M + 2 ) <_ N ) ) |
25 |
22 24
|
syl5com |
|- ( ( N e. Even /\ ( M + 1 ) e. Odd ) -> ( ( M + 1 ) = N -> ( M + 2 ) <_ N ) ) |
26 |
21 25
|
sylan2 |
|- ( ( N e. Even /\ M e. Even ) -> ( ( M + 1 ) = N -> ( M + 2 ) <_ N ) ) |
27 |
20 26
|
jaod |
|- ( ( N e. Even /\ M e. Even ) -> ( ( ( M + 1 ) < N \/ ( M + 1 ) = N ) -> ( M + 2 ) <_ N ) ) |
28 |
10 27
|
sylbid |
|- ( ( N e. Even /\ M e. Even ) -> ( ( M + 1 ) <_ N -> ( M + 2 ) <_ N ) ) |
29 |
4 28
|
sylbid |
|- ( ( N e. Even /\ M e. Even ) -> ( M < N -> ( M + 2 ) <_ N ) ) |
30 |
29
|
3impia |
|- ( ( N e. Even /\ M e. Even /\ M < N ) -> ( M + 2 ) <_ N ) |