Step |
Hyp |
Ref |
Expression |
1 |
|
elxr |
|- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) ) |
2 |
|
elxr |
|- ( C e. RR* <-> ( C e. RR \/ C = +oo \/ C = -oo ) ) |
3 |
|
elxr |
|- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) ) |
4 |
|
lttr |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
5 |
4
|
3expa |
|- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
6 |
5
|
an32s |
|- ( ( ( A e. RR /\ C e. RR ) /\ B e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
7 |
|
rexr |
|- ( C e. RR -> C e. RR* ) |
8 |
|
pnfnlt |
|- ( C e. RR* -> -. +oo < C ) |
9 |
7 8
|
syl |
|- ( C e. RR -> -. +oo < C ) |
10 |
9
|
adantr |
|- ( ( C e. RR /\ B = +oo ) -> -. +oo < C ) |
11 |
|
breq1 |
|- ( B = +oo -> ( B < C <-> +oo < C ) ) |
12 |
11
|
adantl |
|- ( ( C e. RR /\ B = +oo ) -> ( B < C <-> +oo < C ) ) |
13 |
10 12
|
mtbird |
|- ( ( C e. RR /\ B = +oo ) -> -. B < C ) |
14 |
13
|
pm2.21d |
|- ( ( C e. RR /\ B = +oo ) -> ( B < C -> A < C ) ) |
15 |
14
|
adantll |
|- ( ( ( A e. RR /\ C e. RR ) /\ B = +oo ) -> ( B < C -> A < C ) ) |
16 |
15
|
adantld |
|- ( ( ( A e. RR /\ C e. RR ) /\ B = +oo ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
17 |
|
rexr |
|- ( A e. RR -> A e. RR* ) |
18 |
|
nltmnf |
|- ( A e. RR* -> -. A < -oo ) |
19 |
17 18
|
syl |
|- ( A e. RR -> -. A < -oo ) |
20 |
19
|
adantr |
|- ( ( A e. RR /\ B = -oo ) -> -. A < -oo ) |
21 |
|
breq2 |
|- ( B = -oo -> ( A < B <-> A < -oo ) ) |
22 |
21
|
adantl |
|- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) ) |
23 |
20 22
|
mtbird |
|- ( ( A e. RR /\ B = -oo ) -> -. A < B ) |
24 |
23
|
pm2.21d |
|- ( ( A e. RR /\ B = -oo ) -> ( A < B -> A < C ) ) |
25 |
24
|
adantlr |
|- ( ( ( A e. RR /\ C e. RR ) /\ B = -oo ) -> ( A < B -> A < C ) ) |
26 |
25
|
adantrd |
|- ( ( ( A e. RR /\ C e. RR ) /\ B = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
27 |
6 16 26
|
3jaodan |
|- ( ( ( A e. RR /\ C e. RR ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
28 |
3 27
|
sylan2b |
|- ( ( ( A e. RR /\ C e. RR ) /\ B e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
29 |
28
|
an32s |
|- ( ( ( A e. RR /\ B e. RR* ) /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
30 |
|
ltpnf |
|- ( A e. RR -> A < +oo ) |
31 |
30
|
adantr |
|- ( ( A e. RR /\ C = +oo ) -> A < +oo ) |
32 |
|
breq2 |
|- ( C = +oo -> ( A < C <-> A < +oo ) ) |
33 |
32
|
adantl |
|- ( ( A e. RR /\ C = +oo ) -> ( A < C <-> A < +oo ) ) |
34 |
31 33
|
mpbird |
|- ( ( A e. RR /\ C = +oo ) -> A < C ) |
35 |
34
|
adantlr |
|- ( ( ( A e. RR /\ B e. RR* ) /\ C = +oo ) -> A < C ) |
36 |
35
|
a1d |
|- ( ( ( A e. RR /\ B e. RR* ) /\ C = +oo ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
37 |
|
nltmnf |
|- ( B e. RR* -> -. B < -oo ) |
38 |
37
|
adantr |
|- ( ( B e. RR* /\ C = -oo ) -> -. B < -oo ) |
39 |
|
breq2 |
|- ( C = -oo -> ( B < C <-> B < -oo ) ) |
40 |
39
|
adantl |
|- ( ( B e. RR* /\ C = -oo ) -> ( B < C <-> B < -oo ) ) |
41 |
38 40
|
mtbird |
|- ( ( B e. RR* /\ C = -oo ) -> -. B < C ) |
42 |
41
|
pm2.21d |
|- ( ( B e. RR* /\ C = -oo ) -> ( B < C -> A < C ) ) |
43 |
42
|
adantld |
|- ( ( B e. RR* /\ C = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
44 |
43
|
adantll |
|- ( ( ( A e. RR /\ B e. RR* ) /\ C = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
45 |
29 36 44
|
3jaodan |
|- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
46 |
45
|
anasss |
|- ( ( A e. RR /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
47 |
|
pnfnlt |
|- ( B e. RR* -> -. +oo < B ) |
48 |
47
|
adantl |
|- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B ) |
49 |
|
breq1 |
|- ( A = +oo -> ( A < B <-> +oo < B ) ) |
50 |
49
|
adantr |
|- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) ) |
51 |
48 50
|
mtbird |
|- ( ( A = +oo /\ B e. RR* ) -> -. A < B ) |
52 |
51
|
pm2.21d |
|- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> A < C ) ) |
53 |
52
|
adantrd |
|- ( ( A = +oo /\ B e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
54 |
53
|
adantrr |
|- ( ( A = +oo /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
55 |
|
mnflt |
|- ( C e. RR -> -oo < C ) |
56 |
55
|
adantl |
|- ( ( A = -oo /\ C e. RR ) -> -oo < C ) |
57 |
|
breq1 |
|- ( A = -oo -> ( A < C <-> -oo < C ) ) |
58 |
57
|
adantr |
|- ( ( A = -oo /\ C e. RR ) -> ( A < C <-> -oo < C ) ) |
59 |
56 58
|
mpbird |
|- ( ( A = -oo /\ C e. RR ) -> A < C ) |
60 |
59
|
a1d |
|- ( ( A = -oo /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
61 |
60
|
adantlr |
|- ( ( ( A = -oo /\ B e. RR* ) /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
62 |
|
mnfltpnf |
|- -oo < +oo |
63 |
|
breq12 |
|- ( ( A = -oo /\ C = +oo ) -> ( A < C <-> -oo < +oo ) ) |
64 |
62 63
|
mpbiri |
|- ( ( A = -oo /\ C = +oo ) -> A < C ) |
65 |
64
|
a1d |
|- ( ( A = -oo /\ C = +oo ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
66 |
65
|
adantlr |
|- ( ( ( A = -oo /\ B e. RR* ) /\ C = +oo ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
67 |
43
|
adantll |
|- ( ( ( A = -oo /\ B e. RR* ) /\ C = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
68 |
61 66 67
|
3jaodan |
|- ( ( ( A = -oo /\ B e. RR* ) /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
69 |
68
|
anasss |
|- ( ( A = -oo /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
70 |
46 54 69
|
3jaoian |
|- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
71 |
70
|
3impb |
|- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
72 |
2 71
|
syl3an3b |
|- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) ) |
73 |
1 72
|
syl3an1b |
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) ) |