Step |
Hyp |
Ref |
Expression |
1 |
|
climadd.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
climadd.2 |
|- ( ph -> M e. ZZ ) |
3 |
|
climadd.4 |
|- ( ph -> F ~~> A ) |
4 |
|
climle.5 |
|- ( ph -> G ~~> B ) |
5 |
|
climle.6 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) |
6 |
|
climle.7 |
|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR ) |
7 |
|
climle.8 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) ) |
8 |
1
|
fvexi |
|- Z e. _V |
9 |
8
|
mptex |
|- ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) e. _V |
10 |
9
|
a1i |
|- ( ph -> ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) e. _V ) |
11 |
6
|
recnd |
|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC ) |
12 |
5
|
recnd |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) |
13 |
|
fveq2 |
|- ( j = k -> ( G ` j ) = ( G ` k ) ) |
14 |
|
fveq2 |
|- ( j = k -> ( F ` j ) = ( F ` k ) ) |
15 |
13 14
|
oveq12d |
|- ( j = k -> ( ( G ` j ) - ( F ` j ) ) = ( ( G ` k ) - ( F ` k ) ) ) |
16 |
|
eqid |
|- ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) = ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) |
17 |
|
ovex |
|- ( ( G ` k ) - ( F ` k ) ) e. _V |
18 |
15 16 17
|
fvmpt |
|- ( k e. Z -> ( ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) ) |
19 |
18
|
adantl |
|- ( ( ph /\ k e. Z ) -> ( ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) ) |
20 |
1 2 4 10 3 11 12 19
|
climsub |
|- ( ph -> ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ~~> ( B - A ) ) |
21 |
6 5
|
resubcld |
|- ( ( ph /\ k e. Z ) -> ( ( G ` k ) - ( F ` k ) ) e. RR ) |
22 |
19 21
|
eqeltrd |
|- ( ( ph /\ k e. Z ) -> ( ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ` k ) e. RR ) |
23 |
6 5
|
subge0d |
|- ( ( ph /\ k e. Z ) -> ( 0 <_ ( ( G ` k ) - ( F ` k ) ) <-> ( F ` k ) <_ ( G ` k ) ) ) |
24 |
7 23
|
mpbird |
|- ( ( ph /\ k e. Z ) -> 0 <_ ( ( G ` k ) - ( F ` k ) ) ) |
25 |
24 19
|
breqtrrd |
|- ( ( ph /\ k e. Z ) -> 0 <_ ( ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ` k ) ) |
26 |
1 2 20 22 25
|
climge0 |
|- ( ph -> 0 <_ ( B - A ) ) |
27 |
1 2 4 6
|
climrecl |
|- ( ph -> B e. RR ) |
28 |
1 2 3 5
|
climrecl |
|- ( ph -> A e. RR ) |
29 |
27 28
|
subge0d |
|- ( ph -> ( 0 <_ ( B - A ) <-> A <_ B ) ) |
30 |
26 29
|
mpbid |
|- ( ph -> A <_ B ) |