Description: Lemma for axcont . Given the separation assumption, all values of F over A are less than or equal to all values of F over B . (Contributed by Scott Fenton, 20-Jun-2013)
Ref | Expression | ||
---|---|---|---|
Hypotheses | axcontlem9.1 | |
|
axcontlem9.2 | |
||
Assertion | axcontlem9 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axcontlem9.1 | |
|
2 | axcontlem9.2 | |
|
3 | simpll | |
|
4 | simprl1 | |
|
5 | simplr1 | |
|
6 | simprl2 | |
|
7 | 5 6 | sseldd | |
8 | simprr | |
|
9 | 1 2 | axcontlem2 | |
10 | 3 4 7 8 9 | syl31anc | |
11 | f1ofun | |
|
12 | fvelima | |
|
13 | 12 | ex | |
14 | 10 11 13 | 3syl | |
15 | fvelima | |
|
16 | 15 | ex | |
17 | 10 11 16 | 3syl | |
18 | reeanv | |
|
19 | simplr3 | |
|
20 | breq1 | |
|
21 | opeq2 | |
|
22 | 21 | breq2d | |
23 | 20 22 | rspc2v | |
24 | 19 23 | mpan9 | |
25 | simplll | |
|
26 | 4 | adantr | |
27 | 7 | adantr | |
28 | 25 26 27 | 3jca | |
29 | simplrr | |
|
30 | 1 | axcontlem4 | |
31 | 30 | sseld | |
32 | simpl | |
|
33 | 1 | axcontlem3 | |
34 | 32 4 6 8 33 | syl13anc | |
35 | 34 | sseld | |
36 | 31 35 | anim12d | |
37 | 36 | imp | |
38 | 1 2 | axcontlem7 | |
39 | 28 29 37 38 | syl21anc | |
40 | 24 39 | mpbid | |
41 | breq12 | |
|
42 | 40 41 | syl5ibcom | |
43 | 42 | rexlimdvva | |
44 | 18 43 | biimtrrid | |
45 | 14 17 44 | syl2and | |
46 | 45 | ralrimivv | |