Step |
Hyp |
Ref |
Expression |
1 |
|
ltord.1 |
|- ( x = y -> A = B ) |
2 |
|
ltord.2 |
|- ( x = C -> A = M ) |
3 |
|
ltord.3 |
|- ( x = D -> A = N ) |
4 |
|
ltord.4 |
|- S C_ RR |
5 |
|
ltord.5 |
|- ( ( ph /\ x e. S ) -> A e. RR ) |
6 |
|
ltord.6 |
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) ) |
7 |
1 3 2 4 5 6
|
ltord1 |
|- ( ( ph /\ ( D e. S /\ C e. S ) ) -> ( D < C <-> N < M ) ) |
8 |
7
|
ancom2s |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( D < C <-> N < M ) ) |
9 |
8
|
notbid |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( -. D < C <-> -. N < M ) ) |
10 |
4
|
sseli |
|- ( C e. S -> C e. RR ) |
11 |
4
|
sseli |
|- ( D e. S -> D e. RR ) |
12 |
|
lenlt |
|- ( ( C e. RR /\ D e. RR ) -> ( C <_ D <-> -. D < C ) ) |
13 |
10 11 12
|
syl2an |
|- ( ( C e. S /\ D e. S ) -> ( C <_ D <-> -. D < C ) ) |
14 |
13
|
adantl |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C <_ D <-> -. D < C ) ) |
15 |
5
|
ralrimiva |
|- ( ph -> A. x e. S A e. RR ) |
16 |
2
|
eleq1d |
|- ( x = C -> ( A e. RR <-> M e. RR ) ) |
17 |
16
|
rspccva |
|- ( ( A. x e. S A e. RR /\ C e. S ) -> M e. RR ) |
18 |
15 17
|
sylan |
|- ( ( ph /\ C e. S ) -> M e. RR ) |
19 |
18
|
adantrr |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. RR ) |
20 |
3
|
eleq1d |
|- ( x = D -> ( A e. RR <-> N e. RR ) ) |
21 |
20
|
rspccva |
|- ( ( A. x e. S A e. RR /\ D e. S ) -> N e. RR ) |
22 |
15 21
|
sylan |
|- ( ( ph /\ D e. S ) -> N e. RR ) |
23 |
22
|
adantrl |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. RR ) |
24 |
19 23
|
lenltd |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M <_ N <-> -. N < M ) ) |
25 |
9 14 24
|
3bitr4d |
|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C <_ D <-> M <_ N ) ) |