Step |
Hyp |
Ref |
Expression |
1 |
|
sge0prle.a |
|- ( ph -> A e. V ) |
2 |
|
sge0prle.b |
|- ( ph -> B e. W ) |
3 |
|
sge0prle.d |
|- ( ph -> D e. ( 0 [,] +oo ) ) |
4 |
|
sge0prle.e |
|- ( ph -> E e. ( 0 [,] +oo ) ) |
5 |
|
sge0prle.cd |
|- ( k = A -> C = D ) |
6 |
|
sge0prle.ce |
|- ( k = B -> C = E ) |
7 |
|
preq1 |
|- ( A = B -> { A , B } = { B , B } ) |
8 |
|
dfsn2 |
|- { B } = { B , B } |
9 |
8
|
eqcomi |
|- { B , B } = { B } |
10 |
9
|
a1i |
|- ( A = B -> { B , B } = { B } ) |
11 |
7 10
|
eqtrd |
|- ( A = B -> { A , B } = { B } ) |
12 |
11
|
mpteq1d |
|- ( A = B -> ( k e. { A , B } |-> C ) = ( k e. { B } |-> C ) ) |
13 |
12
|
fveq2d |
|- ( A = B -> ( sum^ ` ( k e. { A , B } |-> C ) ) = ( sum^ ` ( k e. { B } |-> C ) ) ) |
14 |
13
|
adantl |
|- ( ( ph /\ A = B ) -> ( sum^ ` ( k e. { A , B } |-> C ) ) = ( sum^ ` ( k e. { B } |-> C ) ) ) |
15 |
2 4 6
|
sge0snmpt |
|- ( ph -> ( sum^ ` ( k e. { B } |-> C ) ) = E ) |
16 |
15
|
adantr |
|- ( ( ph /\ A = B ) -> ( sum^ ` ( k e. { B } |-> C ) ) = E ) |
17 |
14 16
|
eqtrd |
|- ( ( ph /\ A = B ) -> ( sum^ ` ( k e. { A , B } |-> C ) ) = E ) |
18 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
19 |
18 4
|
sselid |
|- ( ph -> E e. RR* ) |
20 |
19
|
xaddid2d |
|- ( ph -> ( 0 +e E ) = E ) |
21 |
20
|
eqcomd |
|- ( ph -> E = ( 0 +e E ) ) |
22 |
|
0xr |
|- 0 e. RR* |
23 |
22
|
a1i |
|- ( ph -> 0 e. RR* ) |
24 |
18 3
|
sselid |
|- ( ph -> D e. RR* ) |
25 |
|
pnfxr |
|- +oo e. RR* |
26 |
25
|
a1i |
|- ( ph -> +oo e. RR* ) |
27 |
|
iccgelb |
|- ( ( 0 e. RR* /\ +oo e. RR* /\ D e. ( 0 [,] +oo ) ) -> 0 <_ D ) |
28 |
23 26 3 27
|
syl3anc |
|- ( ph -> 0 <_ D ) |
29 |
23 24 19 28
|
xleadd1d |
|- ( ph -> ( 0 +e E ) <_ ( D +e E ) ) |
30 |
21 29
|
eqbrtrd |
|- ( ph -> E <_ ( D +e E ) ) |
31 |
30
|
adantr |
|- ( ( ph /\ A = B ) -> E <_ ( D +e E ) ) |
32 |
17 31
|
eqbrtrd |
|- ( ( ph /\ A = B ) -> ( sum^ ` ( k e. { A , B } |-> C ) ) <_ ( D +e E ) ) |
33 |
1
|
adantr |
|- ( ( ph /\ -. A = B ) -> A e. V ) |
34 |
2
|
adantr |
|- ( ( ph /\ -. A = B ) -> B e. W ) |
35 |
3
|
adantr |
|- ( ( ph /\ -. A = B ) -> D e. ( 0 [,] +oo ) ) |
36 |
4
|
adantr |
|- ( ( ph /\ -. A = B ) -> E e. ( 0 [,] +oo ) ) |
37 |
|
neqne |
|- ( -. A = B -> A =/= B ) |
38 |
37
|
adantl |
|- ( ( ph /\ -. A = B ) -> A =/= B ) |
39 |
33 34 35 36 5 6 38
|
sge0pr |
|- ( ( ph /\ -. A = B ) -> ( sum^ ` ( k e. { A , B } |-> C ) ) = ( D +e E ) ) |
40 |
24 19
|
xaddcld |
|- ( ph -> ( D +e E ) e. RR* ) |
41 |
40
|
xrleidd |
|- ( ph -> ( D +e E ) <_ ( D +e E ) ) |
42 |
41
|
adantr |
|- ( ( ph /\ -. A = B ) -> ( D +e E ) <_ ( D +e E ) ) |
43 |
39 42
|
eqbrtrd |
|- ( ( ph /\ -. A = B ) -> ( sum^ ` ( k e. { A , B } |-> C ) ) <_ ( D +e E ) ) |
44 |
32 43
|
pm2.61dan |
|- ( ph -> ( sum^ ` ( k e. { A , B } |-> C ) ) <_ ( D +e E ) ) |