Step |
Hyp |
Ref |
Expression |
1 |
|
supxrgere.xph |
|- F/ x ph |
2 |
|
supxrgere.a |
|- ( ph -> A C_ RR* ) |
3 |
|
supxrgere.b |
|- ( ph -> B e. RR ) |
4 |
|
supxrgere.y |
|- ( ( ph /\ x e. RR+ ) -> E. y e. A ( B - x ) < y ) |
5 |
|
rexr |
|- ( B e. RR -> B e. RR* ) |
6 |
|
pnfxr |
|- +oo e. RR* |
7 |
6
|
a1i |
|- ( B e. RR -> +oo e. RR* ) |
8 |
|
ltpnf |
|- ( B e. RR -> B < +oo ) |
9 |
5 7 8
|
xrltled |
|- ( B e. RR -> B <_ +oo ) |
10 |
3 9
|
syl |
|- ( ph -> B <_ +oo ) |
11 |
10
|
adantr |
|- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> B <_ +oo ) |
12 |
|
id |
|- ( sup ( A , RR* , < ) = +oo -> sup ( A , RR* , < ) = +oo ) |
13 |
12
|
eqcomd |
|- ( sup ( A , RR* , < ) = +oo -> +oo = sup ( A , RR* , < ) ) |
14 |
13
|
adantl |
|- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> +oo = sup ( A , RR* , < ) ) |
15 |
11 14
|
breqtrd |
|- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> B <_ sup ( A , RR* , < ) ) |
16 |
|
simpl |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ph ) |
17 |
|
1rp |
|- 1 e. RR+ |
18 |
|
nfcv |
|- F/_ x 1 |
19 |
|
nfv |
|- F/ x 1 e. RR+ |
20 |
1 19
|
nfan |
|- F/ x ( ph /\ 1 e. RR+ ) |
21 |
|
nfv |
|- F/ x E. y e. A ( B - 1 ) < y |
22 |
20 21
|
nfim |
|- F/ x ( ( ph /\ 1 e. RR+ ) -> E. y e. A ( B - 1 ) < y ) |
23 |
|
eleq1 |
|- ( x = 1 -> ( x e. RR+ <-> 1 e. RR+ ) ) |
24 |
23
|
anbi2d |
|- ( x = 1 -> ( ( ph /\ x e. RR+ ) <-> ( ph /\ 1 e. RR+ ) ) ) |
25 |
|
oveq2 |
|- ( x = 1 -> ( B - x ) = ( B - 1 ) ) |
26 |
25
|
breq1d |
|- ( x = 1 -> ( ( B - x ) < y <-> ( B - 1 ) < y ) ) |
27 |
26
|
rexbidv |
|- ( x = 1 -> ( E. y e. A ( B - x ) < y <-> E. y e. A ( B - 1 ) < y ) ) |
28 |
24 27
|
imbi12d |
|- ( x = 1 -> ( ( ( ph /\ x e. RR+ ) -> E. y e. A ( B - x ) < y ) <-> ( ( ph /\ 1 e. RR+ ) -> E. y e. A ( B - 1 ) < y ) ) ) |
29 |
18 22 28 4
|
vtoclgf |
|- ( 1 e. RR+ -> ( ( ph /\ 1 e. RR+ ) -> E. y e. A ( B - 1 ) < y ) ) |
30 |
17 29
|
ax-mp |
|- ( ( ph /\ 1 e. RR+ ) -> E. y e. A ( B - 1 ) < y ) |
31 |
17 30
|
mpan2 |
|- ( ph -> E. y e. A ( B - 1 ) < y ) |
32 |
31
|
adantr |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> E. y e. A ( B - 1 ) < y ) |
33 |
|
mnfxr |
|- -oo e. RR* |
34 |
33
|
a1i |
|- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> -oo e. RR* ) |
35 |
2
|
sselda |
|- ( ( ph /\ y e. A ) -> y e. RR* ) |
36 |
35
|
3adant3 |
|- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> y e. RR* ) |
37 |
|
supxrcl |
|- ( A C_ RR* -> sup ( A , RR* , < ) e. RR* ) |
38 |
2 37
|
syl |
|- ( ph -> sup ( A , RR* , < ) e. RR* ) |
39 |
38
|
3ad2ant1 |
|- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> sup ( A , RR* , < ) e. RR* ) |
40 |
|
peano2rem |
|- ( B e. RR -> ( B - 1 ) e. RR ) |
41 |
3 40
|
syl |
|- ( ph -> ( B - 1 ) e. RR ) |
42 |
41
|
rexrd |
|- ( ph -> ( B - 1 ) e. RR* ) |
43 |
42
|
adantr |
|- ( ( ph /\ -. -oo < y ) -> ( B - 1 ) e. RR* ) |
44 |
43
|
3ad2antl1 |
|- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> ( B - 1 ) e. RR* ) |
45 |
36
|
adantr |
|- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> y e. RR* ) |
46 |
33
|
a1i |
|- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> -oo e. RR* ) |
47 |
|
simpl3 |
|- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> ( B - 1 ) < y ) |
48 |
|
simpr |
|- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> -. -oo < y ) |
49 |
35
|
adantr |
|- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> y e. RR* ) |
50 |
33
|
a1i |
|- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> -oo e. RR* ) |
51 |
|
xrlenlt |
|- ( ( y e. RR* /\ -oo e. RR* ) -> ( y <_ -oo <-> -. -oo < y ) ) |
52 |
49 50 51
|
syl2anc |
|- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> ( y <_ -oo <-> -. -oo < y ) ) |
53 |
48 52
|
mpbird |
|- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> y <_ -oo ) |
54 |
53
|
3adantl3 |
|- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> y <_ -oo ) |
55 |
44 45 46 47 54
|
xrltletrd |
|- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> ( B - 1 ) < -oo ) |
56 |
|
nltmnf |
|- ( ( B - 1 ) e. RR* -> -. ( B - 1 ) < -oo ) |
57 |
42 56
|
syl |
|- ( ph -> -. ( B - 1 ) < -oo ) |
58 |
57
|
adantr |
|- ( ( ph /\ -. -oo < y ) -> -. ( B - 1 ) < -oo ) |
59 |
58
|
3ad2antl1 |
|- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> -. ( B - 1 ) < -oo ) |
60 |
55 59
|
condan |
|- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> -oo < y ) |
61 |
2
|
adantr |
|- ( ( ph /\ y e. A ) -> A C_ RR* ) |
62 |
|
simpr |
|- ( ( ph /\ y e. A ) -> y e. A ) |
63 |
|
supxrub |
|- ( ( A C_ RR* /\ y e. A ) -> y <_ sup ( A , RR* , < ) ) |
64 |
61 62 63
|
syl2anc |
|- ( ( ph /\ y e. A ) -> y <_ sup ( A , RR* , < ) ) |
65 |
64
|
3adant3 |
|- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> y <_ sup ( A , RR* , < ) ) |
66 |
34 36 39 60 65
|
xrltletrd |
|- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> -oo < sup ( A , RR* , < ) ) |
67 |
66
|
3exp |
|- ( ph -> ( y e. A -> ( ( B - 1 ) < y -> -oo < sup ( A , RR* , < ) ) ) ) |
68 |
67
|
adantr |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( y e. A -> ( ( B - 1 ) < y -> -oo < sup ( A , RR* , < ) ) ) ) |
69 |
68
|
rexlimdv |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( E. y e. A ( B - 1 ) < y -> -oo < sup ( A , RR* , < ) ) ) |
70 |
32 69
|
mpd |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> -oo < sup ( A , RR* , < ) ) |
71 |
|
simpr |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> -. sup ( A , RR* , < ) = +oo ) |
72 |
|
nltpnft |
|- ( sup ( A , RR* , < ) e. RR* -> ( sup ( A , RR* , < ) = +oo <-> -. sup ( A , RR* , < ) < +oo ) ) |
73 |
38 72
|
syl |
|- ( ph -> ( sup ( A , RR* , < ) = +oo <-> -. sup ( A , RR* , < ) < +oo ) ) |
74 |
73
|
adantr |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( sup ( A , RR* , < ) = +oo <-> -. sup ( A , RR* , < ) < +oo ) ) |
75 |
71 74
|
mtbid |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> -. -. sup ( A , RR* , < ) < +oo ) |
76 |
75
|
notnotrd |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) < +oo ) |
77 |
70 76
|
jca |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( -oo < sup ( A , RR* , < ) /\ sup ( A , RR* , < ) < +oo ) ) |
78 |
38
|
adantr |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) e. RR* ) |
79 |
|
xrrebnd |
|- ( sup ( A , RR* , < ) e. RR* -> ( sup ( A , RR* , < ) e. RR <-> ( -oo < sup ( A , RR* , < ) /\ sup ( A , RR* , < ) < +oo ) ) ) |
80 |
78 79
|
syl |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( sup ( A , RR* , < ) e. RR <-> ( -oo < sup ( A , RR* , < ) /\ sup ( A , RR* , < ) < +oo ) ) ) |
81 |
77 80
|
mpbird |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) e. RR ) |
82 |
|
simpl |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> ( ph /\ sup ( A , RR* , < ) e. RR ) ) |
83 |
|
simpr |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> -. B <_ sup ( A , RR* , < ) ) |
84 |
82
|
simprd |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> sup ( A , RR* , < ) e. RR ) |
85 |
3
|
ad2antrr |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> B e. RR ) |
86 |
84 85
|
ltnled |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> ( sup ( A , RR* , < ) < B <-> -. B <_ sup ( A , RR* , < ) ) ) |
87 |
83 86
|
mpbird |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> sup ( A , RR* , < ) < B ) |
88 |
|
simpll |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ph ) |
89 |
3
|
adantr |
|- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> B e. RR ) |
90 |
|
simpr |
|- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> sup ( A , RR* , < ) e. RR ) |
91 |
89 90
|
resubcld |
|- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> ( B - sup ( A , RR* , < ) ) e. RR ) |
92 |
91
|
adantr |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( B - sup ( A , RR* , < ) ) e. RR ) |
93 |
|
simpr |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> sup ( A , RR* , < ) < B ) |
94 |
90
|
adantr |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> sup ( A , RR* , < ) e. RR ) |
95 |
88 3
|
syl |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> B e. RR ) |
96 |
94 95
|
posdifd |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( sup ( A , RR* , < ) < B <-> 0 < ( B - sup ( A , RR* , < ) ) ) ) |
97 |
93 96
|
mpbid |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> 0 < ( B - sup ( A , RR* , < ) ) ) |
98 |
92 97
|
elrpd |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( B - sup ( A , RR* , < ) ) e. RR+ ) |
99 |
|
ovex |
|- ( B - sup ( A , RR* , < ) ) e. _V |
100 |
|
nfcv |
|- F/_ x ( B - sup ( A , RR* , < ) ) |
101 |
|
nfv |
|- F/ x ( B - sup ( A , RR* , < ) ) e. RR+ |
102 |
1 101
|
nfan |
|- F/ x ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) |
103 |
|
nfv |
|- F/ x E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y |
104 |
102 103
|
nfim |
|- F/ x ( ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y ) |
105 |
|
eleq1 |
|- ( x = ( B - sup ( A , RR* , < ) ) -> ( x e. RR+ <-> ( B - sup ( A , RR* , < ) ) e. RR+ ) ) |
106 |
105
|
anbi2d |
|- ( x = ( B - sup ( A , RR* , < ) ) -> ( ( ph /\ x e. RR+ ) <-> ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) ) ) |
107 |
|
oveq2 |
|- ( x = ( B - sup ( A , RR* , < ) ) -> ( B - x ) = ( B - ( B - sup ( A , RR* , < ) ) ) ) |
108 |
107
|
breq1d |
|- ( x = ( B - sup ( A , RR* , < ) ) -> ( ( B - x ) < y <-> ( B - ( B - sup ( A , RR* , < ) ) ) < y ) ) |
109 |
108
|
rexbidv |
|- ( x = ( B - sup ( A , RR* , < ) ) -> ( E. y e. A ( B - x ) < y <-> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y ) ) |
110 |
106 109
|
imbi12d |
|- ( x = ( B - sup ( A , RR* , < ) ) -> ( ( ( ph /\ x e. RR+ ) -> E. y e. A ( B - x ) < y ) <-> ( ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y ) ) ) |
111 |
100 104 110 4
|
vtoclgf |
|- ( ( B - sup ( A , RR* , < ) ) e. _V -> ( ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y ) ) |
112 |
99 111
|
ax-mp |
|- ( ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y ) |
113 |
88 98 112
|
syl2anc |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y ) |
114 |
3
|
recnd |
|- ( ph -> B e. CC ) |
115 |
114
|
ad3antrrr |
|- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> B e. CC ) |
116 |
90
|
recnd |
|- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> sup ( A , RR* , < ) e. CC ) |
117 |
116
|
ad2antrr |
|- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> sup ( A , RR* , < ) e. CC ) |
118 |
115 117
|
nncand |
|- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> ( B - ( B - sup ( A , RR* , < ) ) ) = sup ( A , RR* , < ) ) |
119 |
118
|
eqcomd |
|- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> sup ( A , RR* , < ) = ( B - ( B - sup ( A , RR* , < ) ) ) ) |
120 |
|
simpr |
|- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> ( B - ( B - sup ( A , RR* , < ) ) ) < y ) |
121 |
119 120
|
eqbrtrd |
|- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> sup ( A , RR* , < ) < y ) |
122 |
121
|
ex |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( ( B - ( B - sup ( A , RR* , < ) ) ) < y -> sup ( A , RR* , < ) < y ) ) |
123 |
122
|
adantr |
|- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ y e. A ) -> ( ( B - ( B - sup ( A , RR* , < ) ) ) < y -> sup ( A , RR* , < ) < y ) ) |
124 |
123
|
reximdva |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y -> E. y e. A sup ( A , RR* , < ) < y ) ) |
125 |
113 124
|
mpd |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> E. y e. A sup ( A , RR* , < ) < y ) |
126 |
82 87 125
|
syl2anc |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> E. y e. A sup ( A , RR* , < ) < y ) |
127 |
61 37
|
syl |
|- ( ( ph /\ y e. A ) -> sup ( A , RR* , < ) e. RR* ) |
128 |
35 127
|
xrlenltd |
|- ( ( ph /\ y e. A ) -> ( y <_ sup ( A , RR* , < ) <-> -. sup ( A , RR* , < ) < y ) ) |
129 |
64 128
|
mpbid |
|- ( ( ph /\ y e. A ) -> -. sup ( A , RR* , < ) < y ) |
130 |
129
|
ralrimiva |
|- ( ph -> A. y e. A -. sup ( A , RR* , < ) < y ) |
131 |
|
ralnex |
|- ( A. y e. A -. sup ( A , RR* , < ) < y <-> -. E. y e. A sup ( A , RR* , < ) < y ) |
132 |
130 131
|
sylib |
|- ( ph -> -. E. y e. A sup ( A , RR* , < ) < y ) |
133 |
132
|
ad2antrr |
|- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> -. E. y e. A sup ( A , RR* , < ) < y ) |
134 |
126 133
|
condan |
|- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> B <_ sup ( A , RR* , < ) ) |
135 |
16 81 134
|
syl2anc |
|- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> B <_ sup ( A , RR* , < ) ) |
136 |
15 135
|
pm2.61dan |
|- ( ph -> B <_ sup ( A , RR* , < ) ) |