Step |
Hyp |
Ref |
Expression |
1 |
|
simp2 |
|- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> M e. NN0 ) |
2 |
|
nn0z |
|- ( M e. NN0 -> M e. ZZ ) |
3 |
|
nn0z |
|- ( N e. NN0 -> N e. ZZ ) |
4 |
|
znnsub |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) ) |
5 |
2 3 4
|
syl2anr |
|- ( ( N e. NN0 /\ M e. NN0 ) -> ( M < N <-> ( N - M ) e. NN ) ) |
6 |
5
|
biimp3a |
|- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> ( N - M ) e. NN ) |
7 |
|
fmtnodvds |
|- ( ( M e. NN0 /\ ( N - M ) e. NN ) -> ( FermatNo ` M ) || ( ( FermatNo ` ( M + ( N - M ) ) ) - 2 ) ) |
8 |
1 6 7
|
syl2anc |
|- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> ( FermatNo ` M ) || ( ( FermatNo ` ( M + ( N - M ) ) ) - 2 ) ) |
9 |
|
nn0cn |
|- ( N e. NN0 -> N e. CC ) |
10 |
|
nn0cn |
|- ( M e. NN0 -> M e. CC ) |
11 |
9 10
|
anim12ci |
|- ( ( N e. NN0 /\ M e. NN0 ) -> ( M e. CC /\ N e. CC ) ) |
12 |
11
|
3adant3 |
|- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> ( M e. CC /\ N e. CC ) ) |
13 |
|
pncan3 |
|- ( ( M e. CC /\ N e. CC ) -> ( M + ( N - M ) ) = N ) |
14 |
12 13
|
syl |
|- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> ( M + ( N - M ) ) = N ) |
15 |
14
|
eqcomd |
|- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> N = ( M + ( N - M ) ) ) |
16 |
15
|
fveq2d |
|- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> ( FermatNo ` N ) = ( FermatNo ` ( M + ( N - M ) ) ) ) |
17 |
16
|
oveq1d |
|- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> ( ( FermatNo ` N ) - 2 ) = ( ( FermatNo ` ( M + ( N - M ) ) ) - 2 ) ) |
18 |
8 17
|
breqtrrd |
|- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> ( FermatNo ` M ) || ( ( FermatNo ` N ) - 2 ) ) |