Step |
Hyp |
Ref |
Expression |
1 |
|
infxr.x |
|- F/ x ph |
2 |
|
infxr.y |
|- F/ y ph |
3 |
|
infxr.a |
|- ( ph -> A C_ RR* ) |
4 |
|
infxr.b |
|- ( ph -> B e. RR* ) |
5 |
|
infxr.n |
|- ( ph -> A. x e. A -. x < B ) |
6 |
|
infxr.e |
|- ( ph -> A. x e. RR ( B < x -> E. y e. A y < x ) ) |
7 |
6
|
r19.21bi |
|- ( ( ph /\ x e. RR ) -> ( B < x -> E. y e. A y < x ) ) |
8 |
7
|
adantlr |
|- ( ( ( ph /\ x e. RR* ) /\ x e. RR ) -> ( B < x -> E. y e. A y < x ) ) |
9 |
|
simplll |
|- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> ph ) |
10 |
|
simpllr |
|- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> x e. RR* ) |
11 |
|
simplr |
|- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> -. x e. RR ) |
12 |
|
mnfxr |
|- -oo e. RR* |
13 |
12
|
a1i |
|- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> -oo e. RR* ) |
14 |
|
simplr |
|- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> x e. RR* ) |
15 |
4
|
ad2antrr |
|- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> B e. RR* ) |
16 |
|
mnfle |
|- ( B e. RR* -> -oo <_ B ) |
17 |
4 16
|
syl |
|- ( ph -> -oo <_ B ) |
18 |
17
|
ad2antrr |
|- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> -oo <_ B ) |
19 |
|
simpr |
|- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> B < x ) |
20 |
13 15 14 18 19
|
xrlelttrd |
|- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> -oo < x ) |
21 |
13 14 20
|
xrgtned |
|- ( ( ( ph /\ x e. RR* ) /\ B < x ) -> x =/= -oo ) |
22 |
21
|
adantlr |
|- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> x =/= -oo ) |
23 |
10 11 22
|
xrnmnfpnf |
|- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> x = +oo ) |
24 |
|
simpr |
|- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> B < x ) |
25 |
|
simpl |
|- ( ( ph /\ B = -oo ) -> ph ) |
26 |
|
id |
|- ( B = -oo -> B = -oo ) |
27 |
|
1re |
|- 1 e. RR |
28 |
|
mnflt |
|- ( 1 e. RR -> -oo < 1 ) |
29 |
27 28
|
ax-mp |
|- -oo < 1 |
30 |
26 29
|
eqbrtrdi |
|- ( B = -oo -> B < 1 ) |
31 |
30
|
adantl |
|- ( ( ph /\ B = -oo ) -> B < 1 ) |
32 |
|
1red |
|- ( ph -> 1 e. RR ) |
33 |
|
breq2 |
|- ( x = 1 -> ( B < x <-> B < 1 ) ) |
34 |
|
breq2 |
|- ( x = 1 -> ( y < x <-> y < 1 ) ) |
35 |
34
|
rexbidv |
|- ( x = 1 -> ( E. y e. A y < x <-> E. y e. A y < 1 ) ) |
36 |
33 35
|
imbi12d |
|- ( x = 1 -> ( ( B < x -> E. y e. A y < x ) <-> ( B < 1 -> E. y e. A y < 1 ) ) ) |
37 |
36
|
rspcva |
|- ( ( 1 e. RR /\ A. x e. RR ( B < x -> E. y e. A y < x ) ) -> ( B < 1 -> E. y e. A y < 1 ) ) |
38 |
32 6 37
|
syl2anc |
|- ( ph -> ( B < 1 -> E. y e. A y < 1 ) ) |
39 |
25 31 38
|
sylc |
|- ( ( ph /\ B = -oo ) -> E. y e. A y < 1 ) |
40 |
39
|
adantlr |
|- ( ( ( ph /\ x = +oo ) /\ B = -oo ) -> E. y e. A y < 1 ) |
41 |
|
nfv |
|- F/ y x = +oo |
42 |
2 41
|
nfan |
|- F/ y ( ph /\ x = +oo ) |
43 |
3
|
sselda |
|- ( ( ph /\ y e. A ) -> y e. RR* ) |
44 |
43
|
ad4ant13 |
|- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> y e. RR* ) |
45 |
|
1xr |
|- 1 e. RR* |
46 |
45
|
a1i |
|- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> 1 e. RR* ) |
47 |
|
id |
|- ( x = +oo -> x = +oo ) |
48 |
|
pnfxr |
|- +oo e. RR* |
49 |
47 48
|
eqeltrdi |
|- ( x = +oo -> x e. RR* ) |
50 |
49
|
adantl |
|- ( ( ph /\ x = +oo ) -> x e. RR* ) |
51 |
50
|
ad2antrr |
|- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> x e. RR* ) |
52 |
|
simpr |
|- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> y < 1 ) |
53 |
|
ltpnf |
|- ( 1 e. RR -> 1 < +oo ) |
54 |
27 53
|
ax-mp |
|- 1 < +oo |
55 |
54
|
a1i |
|- ( x = +oo -> 1 < +oo ) |
56 |
47
|
eqcomd |
|- ( x = +oo -> +oo = x ) |
57 |
55 56
|
breqtrd |
|- ( x = +oo -> 1 < x ) |
58 |
57
|
adantl |
|- ( ( ph /\ x = +oo ) -> 1 < x ) |
59 |
58
|
ad2antrr |
|- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> 1 < x ) |
60 |
44 46 51 52 59
|
xrlttrd |
|- ( ( ( ( ph /\ x = +oo ) /\ y e. A ) /\ y < 1 ) -> y < x ) |
61 |
60
|
ex |
|- ( ( ( ph /\ x = +oo ) /\ y e. A ) -> ( y < 1 -> y < x ) ) |
62 |
61
|
ex |
|- ( ( ph /\ x = +oo ) -> ( y e. A -> ( y < 1 -> y < x ) ) ) |
63 |
42 62
|
reximdai |
|- ( ( ph /\ x = +oo ) -> ( E. y e. A y < 1 -> E. y e. A y < x ) ) |
64 |
63
|
adantr |
|- ( ( ( ph /\ x = +oo ) /\ B = -oo ) -> ( E. y e. A y < 1 -> E. y e. A y < x ) ) |
65 |
40 64
|
mpd |
|- ( ( ( ph /\ x = +oo ) /\ B = -oo ) -> E. y e. A y < x ) |
66 |
65
|
3adantl3 |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ B = -oo ) -> E. y e. A y < x ) |
67 |
4
|
adantr |
|- ( ( ph /\ -. B = -oo ) -> B e. RR* ) |
68 |
67
|
3ad2antl1 |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B e. RR* ) |
69 |
26
|
necon3bi |
|- ( -. B = -oo -> B =/= -oo ) |
70 |
69
|
adantl |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B =/= -oo ) |
71 |
48
|
a1i |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> +oo e. RR* ) |
72 |
|
simpr |
|- ( ( x = +oo /\ B < x ) -> B < x ) |
73 |
|
simpl |
|- ( ( x = +oo /\ B < x ) -> x = +oo ) |
74 |
72 73
|
breqtrd |
|- ( ( x = +oo /\ B < x ) -> B < +oo ) |
75 |
74
|
3adant1 |
|- ( ( ph /\ x = +oo /\ B < x ) -> B < +oo ) |
76 |
75
|
adantr |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B < +oo ) |
77 |
68 71 76
|
xrltned |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B =/= +oo ) |
78 |
68 70 77
|
xrred |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B e. RR ) |
79 |
27
|
a1i |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> 1 e. RR ) |
80 |
78 79
|
readdcld |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( B + 1 ) e. RR ) |
81 |
6
|
adantr |
|- ( ( ph /\ -. B = -oo ) -> A. x e. RR ( B < x -> E. y e. A y < x ) ) |
82 |
81
|
3ad2antl1 |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> A. x e. RR ( B < x -> E. y e. A y < x ) ) |
83 |
80 82
|
jca |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( ( B + 1 ) e. RR /\ A. x e. RR ( B < x -> E. y e. A y < x ) ) ) |
84 |
78
|
ltp1d |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> B < ( B + 1 ) ) |
85 |
|
breq2 |
|- ( x = ( B + 1 ) -> ( B < x <-> B < ( B + 1 ) ) ) |
86 |
|
breq2 |
|- ( x = ( B + 1 ) -> ( y < x <-> y < ( B + 1 ) ) ) |
87 |
86
|
rexbidv |
|- ( x = ( B + 1 ) -> ( E. y e. A y < x <-> E. y e. A y < ( B + 1 ) ) ) |
88 |
85 87
|
imbi12d |
|- ( x = ( B + 1 ) -> ( ( B < x -> E. y e. A y < x ) <-> ( B < ( B + 1 ) -> E. y e. A y < ( B + 1 ) ) ) ) |
89 |
88
|
rspcva |
|- ( ( ( B + 1 ) e. RR /\ A. x e. RR ( B < x -> E. y e. A y < x ) ) -> ( B < ( B + 1 ) -> E. y e. A y < ( B + 1 ) ) ) |
90 |
83 84 89
|
sylc |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> E. y e. A y < ( B + 1 ) ) |
91 |
|
nfv |
|- F/ y B < x |
92 |
2 41 91
|
nf3an |
|- F/ y ( ph /\ x = +oo /\ B < x ) |
93 |
|
nfv |
|- F/ y -. B = -oo |
94 |
92 93
|
nfan |
|- F/ y ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) |
95 |
43
|
3ad2antl1 |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ y e. A ) -> y e. RR* ) |
96 |
95
|
ad4ant13 |
|- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> y e. RR* ) |
97 |
80
|
adantr |
|- ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) -> ( B + 1 ) e. RR ) |
98 |
97
|
rexrd |
|- ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) -> ( B + 1 ) e. RR* ) |
99 |
98
|
adantr |
|- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> ( B + 1 ) e. RR* ) |
100 |
50
|
3adant3 |
|- ( ( ph /\ x = +oo /\ B < x ) -> x e. RR* ) |
101 |
100
|
ad3antrrr |
|- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> x e. RR* ) |
102 |
|
simpr |
|- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> y < ( B + 1 ) ) |
103 |
80
|
ltpnfd |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( B + 1 ) < +oo ) |
104 |
56
|
adantr |
|- ( ( x = +oo /\ -. B = -oo ) -> +oo = x ) |
105 |
104
|
3ad2antl2 |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> +oo = x ) |
106 |
103 105
|
breqtrd |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( B + 1 ) < x ) |
107 |
106
|
ad2antrr |
|- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> ( B + 1 ) < x ) |
108 |
96 99 101 102 107
|
xrlttrd |
|- ( ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) /\ y < ( B + 1 ) ) -> y < x ) |
109 |
108
|
ex |
|- ( ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) /\ y e. A ) -> ( y < ( B + 1 ) -> y < x ) ) |
110 |
109
|
ex |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( y e. A -> ( y < ( B + 1 ) -> y < x ) ) ) |
111 |
94 110
|
reximdai |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> ( E. y e. A y < ( B + 1 ) -> E. y e. A y < x ) ) |
112 |
90 111
|
mpd |
|- ( ( ( ph /\ x = +oo /\ B < x ) /\ -. B = -oo ) -> E. y e. A y < x ) |
113 |
66 112
|
pm2.61dan |
|- ( ( ph /\ x = +oo /\ B < x ) -> E. y e. A y < x ) |
114 |
9 23 24 113
|
syl3anc |
|- ( ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) /\ B < x ) -> E. y e. A y < x ) |
115 |
114
|
ex |
|- ( ( ( ph /\ x e. RR* ) /\ -. x e. RR ) -> ( B < x -> E. y e. A y < x ) ) |
116 |
8 115
|
pm2.61dan |
|- ( ( ph /\ x e. RR* ) -> ( B < x -> E. y e. A y < x ) ) |
117 |
116
|
ex |
|- ( ph -> ( x e. RR* -> ( B < x -> E. y e. A y < x ) ) ) |
118 |
1 117
|
ralrimi |
|- ( ph -> A. x e. RR* ( B < x -> E. y e. A y < x ) ) |
119 |
|
xrltso |
|- < Or RR* |
120 |
119
|
a1i |
|- ( T. -> < Or RR* ) |
121 |
120
|
eqinf |
|- ( T. -> ( ( B e. RR* /\ A. x e. A -. x < B /\ A. x e. RR* ( B < x -> E. y e. A y < x ) ) -> inf ( A , RR* , < ) = B ) ) |
122 |
121
|
mptru |
|- ( ( B e. RR* /\ A. x e. A -. x < B /\ A. x e. RR* ( B < x -> E. y e. A y < x ) ) -> inf ( A , RR* , < ) = B ) |
123 |
4 5 118 122
|
syl3anc |
|- ( ph -> inf ( A , RR* , < ) = B ) |