Step |
Hyp |
Ref |
Expression |
1 |
|
xblss2.1 |
|- ( ph -> D e. ( *Met ` X ) ) |
2 |
|
xblss2.2 |
|- ( ph -> P e. X ) |
3 |
|
xblss2.3 |
|- ( ph -> Q e. X ) |
4 |
|
xblss2.4 |
|- ( ph -> R e. RR* ) |
5 |
|
xblss2.5 |
|- ( ph -> S e. RR* ) |
6 |
|
xblss2.6 |
|- ( ph -> ( P D Q ) e. RR ) |
7 |
|
xblss2.7 |
|- ( ph -> ( P D Q ) <_ ( S +e -e R ) ) |
8 |
|
elbl |
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( x e. ( P ( ball ` D ) R ) <-> ( x e. X /\ ( P D x ) < R ) ) ) |
9 |
1 2 4 8
|
syl3anc |
|- ( ph -> ( x e. ( P ( ball ` D ) R ) <-> ( x e. X /\ ( P D x ) < R ) ) ) |
10 |
9
|
simprbda |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> x e. X ) |
11 |
1
|
adantr |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> D e. ( *Met ` X ) ) |
12 |
3
|
adantr |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> Q e. X ) |
13 |
|
xmetcl |
|- ( ( D e. ( *Met ` X ) /\ Q e. X /\ x e. X ) -> ( Q D x ) e. RR* ) |
14 |
11 12 10 13
|
syl3anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( Q D x ) e. RR* ) |
15 |
14
|
adantr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R e. RR ) -> ( Q D x ) e. RR* ) |
16 |
6
|
adantr |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( P D Q ) e. RR ) |
17 |
16
|
rexrd |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( P D Q ) e. RR* ) |
18 |
4
|
adantr |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> R e. RR* ) |
19 |
17 18
|
xaddcld |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( ( P D Q ) +e R ) e. RR* ) |
20 |
19
|
adantr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R e. RR ) -> ( ( P D Q ) +e R ) e. RR* ) |
21 |
5
|
ad2antrr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R e. RR ) -> S e. RR* ) |
22 |
2
|
adantr |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> P e. X ) |
23 |
|
xmetcl |
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* ) |
24 |
11 22 10 23
|
syl3anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( P D x ) e. RR* ) |
25 |
17 24
|
xaddcld |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( ( P D Q ) +e ( P D x ) ) e. RR* ) |
26 |
|
xmettri2 |
|- ( ( D e. ( *Met ` X ) /\ ( P e. X /\ Q e. X /\ x e. X ) ) -> ( Q D x ) <_ ( ( P D Q ) +e ( P D x ) ) ) |
27 |
11 22 12 10 26
|
syl13anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( Q D x ) <_ ( ( P D Q ) +e ( P D x ) ) ) |
28 |
9
|
simplbda |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( P D x ) < R ) |
29 |
|
xltadd2 |
|- ( ( ( P D x ) e. RR* /\ R e. RR* /\ ( P D Q ) e. RR ) -> ( ( P D x ) < R <-> ( ( P D Q ) +e ( P D x ) ) < ( ( P D Q ) +e R ) ) ) |
30 |
24 18 16 29
|
syl3anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( ( P D x ) < R <-> ( ( P D Q ) +e ( P D x ) ) < ( ( P D Q ) +e R ) ) ) |
31 |
28 30
|
mpbid |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( ( P D Q ) +e ( P D x ) ) < ( ( P D Q ) +e R ) ) |
32 |
14 25 19 27 31
|
xrlelttrd |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( Q D x ) < ( ( P D Q ) +e R ) ) |
33 |
32
|
adantr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R e. RR ) -> ( Q D x ) < ( ( P D Q ) +e R ) ) |
34 |
5
|
adantr |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> S e. RR* ) |
35 |
18
|
xnegcld |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> -e R e. RR* ) |
36 |
34 35
|
xaddcld |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( S +e -e R ) e. RR* ) |
37 |
7
|
adantr |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( P D Q ) <_ ( S +e -e R ) ) |
38 |
|
xleadd1a |
|- ( ( ( ( P D Q ) e. RR* /\ ( S +e -e R ) e. RR* /\ R e. RR* ) /\ ( P D Q ) <_ ( S +e -e R ) ) -> ( ( P D Q ) +e R ) <_ ( ( S +e -e R ) +e R ) ) |
39 |
17 36 18 37 38
|
syl31anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( ( P D Q ) +e R ) <_ ( ( S +e -e R ) +e R ) ) |
40 |
39
|
adantr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R e. RR ) -> ( ( P D Q ) +e R ) <_ ( ( S +e -e R ) +e R ) ) |
41 |
|
xnpcan |
|- ( ( S e. RR* /\ R e. RR ) -> ( ( S +e -e R ) +e R ) = S ) |
42 |
34 41
|
sylan |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R e. RR ) -> ( ( S +e -e R ) +e R ) = S ) |
43 |
40 42
|
breqtrd |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R e. RR ) -> ( ( P D Q ) +e R ) <_ S ) |
44 |
15 20 21 33 43
|
xrltletrd |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R e. RR ) -> ( Q D x ) < S ) |
45 |
28
|
adantr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( P D x ) < R ) |
46 |
7
|
ad2antrr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( P D Q ) <_ ( S +e -e R ) ) |
47 |
|
0xr |
|- 0 e. RR* |
48 |
47
|
a1i |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> 0 e. RR* ) |
49 |
|
xmetge0 |
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ Q e. X ) -> 0 <_ ( P D Q ) ) |
50 |
11 22 12 49
|
syl3anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> 0 <_ ( P D Q ) ) |
51 |
48 17 36 50 37
|
xrletrd |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> 0 <_ ( S +e -e R ) ) |
52 |
|
ge0nemnf |
|- ( ( ( S +e -e R ) e. RR* /\ 0 <_ ( S +e -e R ) ) -> ( S +e -e R ) =/= -oo ) |
53 |
36 51 52
|
syl2anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( S +e -e R ) =/= -oo ) |
54 |
53
|
adantr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( S +e -e R ) =/= -oo ) |
55 |
5
|
ad2antrr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> S e. RR* ) |
56 |
|
xaddmnf1 |
|- ( ( S e. RR* /\ S =/= +oo ) -> ( S +e -oo ) = -oo ) |
57 |
56
|
ex |
|- ( S e. RR* -> ( S =/= +oo -> ( S +e -oo ) = -oo ) ) |
58 |
55 57
|
syl |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( S =/= +oo -> ( S +e -oo ) = -oo ) ) |
59 |
|
simpr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> R = +oo ) |
60 |
|
xnegeq |
|- ( R = +oo -> -e R = -e +oo ) |
61 |
59 60
|
syl |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> -e R = -e +oo ) |
62 |
|
xnegpnf |
|- -e +oo = -oo |
63 |
61 62
|
eqtrdi |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> -e R = -oo ) |
64 |
63
|
oveq2d |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( S +e -e R ) = ( S +e -oo ) ) |
65 |
64
|
eqeq1d |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( ( S +e -e R ) = -oo <-> ( S +e -oo ) = -oo ) ) |
66 |
58 65
|
sylibrd |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( S =/= +oo -> ( S +e -e R ) = -oo ) ) |
67 |
66
|
necon1d |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( ( S +e -e R ) =/= -oo -> S = +oo ) ) |
68 |
54 67
|
mpd |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> S = +oo ) |
69 |
68 63
|
oveq12d |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( S +e -e R ) = ( +oo +e -oo ) ) |
70 |
|
pnfaddmnf |
|- ( +oo +e -oo ) = 0 |
71 |
69 70
|
eqtrdi |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( S +e -e R ) = 0 ) |
72 |
46 71
|
breqtrd |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( P D Q ) <_ 0 ) |
73 |
50
|
biantrud |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( ( P D Q ) <_ 0 <-> ( ( P D Q ) <_ 0 /\ 0 <_ ( P D Q ) ) ) ) |
74 |
|
xrletri3 |
|- ( ( ( P D Q ) e. RR* /\ 0 e. RR* ) -> ( ( P D Q ) = 0 <-> ( ( P D Q ) <_ 0 /\ 0 <_ ( P D Q ) ) ) ) |
75 |
17 47 74
|
sylancl |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( ( P D Q ) = 0 <-> ( ( P D Q ) <_ 0 /\ 0 <_ ( P D Q ) ) ) ) |
76 |
|
xmeteq0 |
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ Q e. X ) -> ( ( P D Q ) = 0 <-> P = Q ) ) |
77 |
11 22 12 76
|
syl3anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( ( P D Q ) = 0 <-> P = Q ) ) |
78 |
73 75 77
|
3bitr2d |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( ( P D Q ) <_ 0 <-> P = Q ) ) |
79 |
78
|
adantr |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( ( P D Q ) <_ 0 <-> P = Q ) ) |
80 |
72 79
|
mpbid |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> P = Q ) |
81 |
80
|
oveq1d |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( P D x ) = ( Q D x ) ) |
82 |
59 68
|
eqtr4d |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> R = S ) |
83 |
45 81 82
|
3brtr3d |
|- ( ( ( ph /\ x e. ( P ( ball ` D ) R ) ) /\ R = +oo ) -> ( Q D x ) < S ) |
84 |
|
xmetge0 |
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> 0 <_ ( P D x ) ) |
85 |
11 22 10 84
|
syl3anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> 0 <_ ( P D x ) ) |
86 |
48 24 18 85 28
|
xrlelttrd |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> 0 < R ) |
87 |
48 18 86
|
xrltled |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> 0 <_ R ) |
88 |
|
ge0nemnf |
|- ( ( R e. RR* /\ 0 <_ R ) -> R =/= -oo ) |
89 |
18 87 88
|
syl2anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> R =/= -oo ) |
90 |
18 89
|
jca |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( R e. RR* /\ R =/= -oo ) ) |
91 |
|
xrnemnf |
|- ( ( R e. RR* /\ R =/= -oo ) <-> ( R e. RR \/ R = +oo ) ) |
92 |
90 91
|
sylib |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( R e. RR \/ R = +oo ) ) |
93 |
44 83 92
|
mpjaodan |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( Q D x ) < S ) |
94 |
|
elbl |
|- ( ( D e. ( *Met ` X ) /\ Q e. X /\ S e. RR* ) -> ( x e. ( Q ( ball ` D ) S ) <-> ( x e. X /\ ( Q D x ) < S ) ) ) |
95 |
11 12 34 94
|
syl3anc |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> ( x e. ( Q ( ball ` D ) S ) <-> ( x e. X /\ ( Q D x ) < S ) ) ) |
96 |
10 93 95
|
mpbir2and |
|- ( ( ph /\ x e. ( P ( ball ` D ) R ) ) -> x e. ( Q ( ball ` D ) S ) ) |
97 |
96
|
ex |
|- ( ph -> ( x e. ( P ( ball ` D ) R ) -> x e. ( Q ( ball ` D ) S ) ) ) |
98 |
97
|
ssrdv |
|- ( ph -> ( P ( ball ` D ) R ) C_ ( Q ( ball ` D ) S ) ) |