Step |
Hyp |
Ref |
Expression |
1 |
|
iccpartgtprec.m |
|- ( ph -> M e. NN ) |
2 |
|
iccpartgtprec.p |
|- ( ph -> P e. ( RePart ` M ) ) |
3 |
|
ral0 |
|- A. i e. (/) ( P ` 0 ) < ( P ` i ) |
4 |
|
oveq2 |
|- ( M = 1 -> ( 1 ..^ M ) = ( 1 ..^ 1 ) ) |
5 |
|
fzo0 |
|- ( 1 ..^ 1 ) = (/) |
6 |
4 5
|
eqtrdi |
|- ( M = 1 -> ( 1 ..^ M ) = (/) ) |
7 |
6
|
raleqdv |
|- ( M = 1 -> ( A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) <-> A. i e. (/) ( P ` 0 ) < ( P ` i ) ) ) |
8 |
3 7
|
mpbiri |
|- ( M = 1 -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) |
9 |
8
|
a1d |
|- ( M = 1 -> ( ph -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) ) |
10 |
1
|
nnnn0d |
|- ( ph -> M e. NN0 ) |
11 |
|
0elfz |
|- ( M e. NN0 -> 0 e. ( 0 ... M ) ) |
12 |
10 11
|
syl |
|- ( ph -> 0 e. ( 0 ... M ) ) |
13 |
1 2 12
|
iccpartxr |
|- ( ph -> ( P ` 0 ) e. RR* ) |
14 |
13
|
adantr |
|- ( ( ph /\ -. M = 1 ) -> ( P ` 0 ) e. RR* ) |
15 |
|
elxr |
|- ( ( P ` 0 ) e. RR* <-> ( ( P ` 0 ) e. RR \/ ( P ` 0 ) = +oo \/ ( P ` 0 ) = -oo ) ) |
16 |
|
0zd |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> 0 e. ZZ ) |
17 |
|
elfzouz |
|- ( i e. ( 1 ..^ M ) -> i e. ( ZZ>= ` 1 ) ) |
18 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
19 |
18
|
fveq2i |
|- ( ZZ>= ` ( 0 + 1 ) ) = ( ZZ>= ` 1 ) |
20 |
17 19
|
eleqtrrdi |
|- ( i e. ( 1 ..^ M ) -> i e. ( ZZ>= ` ( 0 + 1 ) ) ) |
21 |
20
|
adantl |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> i e. ( ZZ>= ` ( 0 + 1 ) ) ) |
22 |
|
fveq2 |
|- ( k = 0 -> ( P ` k ) = ( P ` 0 ) ) |
23 |
22
|
eqcomd |
|- ( k = 0 -> ( P ` 0 ) = ( P ` k ) ) |
24 |
23
|
eleq1d |
|- ( k = 0 -> ( ( P ` 0 ) e. RR <-> ( P ` k ) e. RR ) ) |
25 |
24
|
biimpcd |
|- ( ( P ` 0 ) e. RR -> ( k = 0 -> ( P ` k ) e. RR ) ) |
26 |
25
|
ad3antrrr |
|- ( ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) /\ k e. ( 0 ... i ) ) -> ( k = 0 -> ( P ` k ) e. RR ) ) |
27 |
1
|
adantr |
|- ( ( ph /\ ( i e. ( 1 ..^ M ) /\ ( k e. ( 0 ... i ) /\ k =/= 0 ) ) ) -> M e. NN ) |
28 |
2
|
adantr |
|- ( ( ph /\ ( i e. ( 1 ..^ M ) /\ ( k e. ( 0 ... i ) /\ k =/= 0 ) ) ) -> P e. ( RePart ` M ) ) |
29 |
|
elfz2nn0 |
|- ( k e. ( 0 ... i ) <-> ( k e. NN0 /\ i e. NN0 /\ k <_ i ) ) |
30 |
|
elfzo2 |
|- ( i e. ( 1 ..^ M ) <-> ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) ) |
31 |
|
simpl1 |
|- ( ( ( k e. NN0 /\ i e. NN0 /\ k <_ i ) /\ ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) ) -> k e. NN0 ) |
32 |
|
simpr2 |
|- ( ( ( k e. NN0 /\ i e. NN0 /\ k <_ i ) /\ ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) ) -> M e. ZZ ) |
33 |
|
nn0ge0 |
|- ( i e. NN0 -> 0 <_ i ) |
34 |
|
0red |
|- ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ ) -> 0 e. RR ) |
35 |
|
eluzelre |
|- ( i e. ( ZZ>= ` 1 ) -> i e. RR ) |
36 |
35
|
adantr |
|- ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ ) -> i e. RR ) |
37 |
|
zre |
|- ( M e. ZZ -> M e. RR ) |
38 |
37
|
adantl |
|- ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ ) -> M e. RR ) |
39 |
|
lelttr |
|- ( ( 0 e. RR /\ i e. RR /\ M e. RR ) -> ( ( 0 <_ i /\ i < M ) -> 0 < M ) ) |
40 |
34 36 38 39
|
syl3anc |
|- ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ ) -> ( ( 0 <_ i /\ i < M ) -> 0 < M ) ) |
41 |
40
|
expcomd |
|- ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ ) -> ( i < M -> ( 0 <_ i -> 0 < M ) ) ) |
42 |
41
|
3impia |
|- ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) -> ( 0 <_ i -> 0 < M ) ) |
43 |
33 42
|
syl5com |
|- ( i e. NN0 -> ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) -> 0 < M ) ) |
44 |
43
|
3ad2ant2 |
|- ( ( k e. NN0 /\ i e. NN0 /\ k <_ i ) -> ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) -> 0 < M ) ) |
45 |
44
|
imp |
|- ( ( ( k e. NN0 /\ i e. NN0 /\ k <_ i ) /\ ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) ) -> 0 < M ) |
46 |
|
elnnz |
|- ( M e. NN <-> ( M e. ZZ /\ 0 < M ) ) |
47 |
32 45 46
|
sylanbrc |
|- ( ( ( k e. NN0 /\ i e. NN0 /\ k <_ i ) /\ ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) ) -> M e. NN ) |
48 |
|
nn0re |
|- ( k e. NN0 -> k e. RR ) |
49 |
48
|
ad2antrl |
|- ( ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ ) /\ ( k e. NN0 /\ i e. NN0 ) ) -> k e. RR ) |
50 |
|
nn0re |
|- ( i e. NN0 -> i e. RR ) |
51 |
50
|
adantl |
|- ( ( k e. NN0 /\ i e. NN0 ) -> i e. RR ) |
52 |
51
|
adantl |
|- ( ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ ) /\ ( k e. NN0 /\ i e. NN0 ) ) -> i e. RR ) |
53 |
38
|
adantr |
|- ( ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ ) /\ ( k e. NN0 /\ i e. NN0 ) ) -> M e. RR ) |
54 |
|
lelttr |
|- ( ( k e. RR /\ i e. RR /\ M e. RR ) -> ( ( k <_ i /\ i < M ) -> k < M ) ) |
55 |
54
|
expd |
|- ( ( k e. RR /\ i e. RR /\ M e. RR ) -> ( k <_ i -> ( i < M -> k < M ) ) ) |
56 |
49 52 53 55
|
syl3anc |
|- ( ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ ) /\ ( k e. NN0 /\ i e. NN0 ) ) -> ( k <_ i -> ( i < M -> k < M ) ) ) |
57 |
56
|
exp31 |
|- ( i e. ( ZZ>= ` 1 ) -> ( M e. ZZ -> ( ( k e. NN0 /\ i e. NN0 ) -> ( k <_ i -> ( i < M -> k < M ) ) ) ) ) |
58 |
57
|
com34 |
|- ( i e. ( ZZ>= ` 1 ) -> ( M e. ZZ -> ( k <_ i -> ( ( k e. NN0 /\ i e. NN0 ) -> ( i < M -> k < M ) ) ) ) ) |
59 |
58
|
com35 |
|- ( i e. ( ZZ>= ` 1 ) -> ( M e. ZZ -> ( i < M -> ( ( k e. NN0 /\ i e. NN0 ) -> ( k <_ i -> k < M ) ) ) ) ) |
60 |
59
|
3imp |
|- ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) -> ( ( k e. NN0 /\ i e. NN0 ) -> ( k <_ i -> k < M ) ) ) |
61 |
60
|
expdcom |
|- ( k e. NN0 -> ( i e. NN0 -> ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) -> ( k <_ i -> k < M ) ) ) ) |
62 |
61
|
com34 |
|- ( k e. NN0 -> ( i e. NN0 -> ( k <_ i -> ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) -> k < M ) ) ) ) |
63 |
62
|
3imp1 |
|- ( ( ( k e. NN0 /\ i e. NN0 /\ k <_ i ) /\ ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) ) -> k < M ) |
64 |
|
elfzo0 |
|- ( k e. ( 0 ..^ M ) <-> ( k e. NN0 /\ M e. NN /\ k < M ) ) |
65 |
31 47 63 64
|
syl3anbrc |
|- ( ( ( k e. NN0 /\ i e. NN0 /\ k <_ i ) /\ ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) ) -> k e. ( 0 ..^ M ) ) |
66 |
65
|
ex |
|- ( ( k e. NN0 /\ i e. NN0 /\ k <_ i ) -> ( ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) -> k e. ( 0 ..^ M ) ) ) |
67 |
30 66
|
syl5bi |
|- ( ( k e. NN0 /\ i e. NN0 /\ k <_ i ) -> ( i e. ( 1 ..^ M ) -> k e. ( 0 ..^ M ) ) ) |
68 |
29 67
|
sylbi |
|- ( k e. ( 0 ... i ) -> ( i e. ( 1 ..^ M ) -> k e. ( 0 ..^ M ) ) ) |
69 |
68
|
adantr |
|- ( ( k e. ( 0 ... i ) /\ k =/= 0 ) -> ( i e. ( 1 ..^ M ) -> k e. ( 0 ..^ M ) ) ) |
70 |
69
|
impcom |
|- ( ( i e. ( 1 ..^ M ) /\ ( k e. ( 0 ... i ) /\ k =/= 0 ) ) -> k e. ( 0 ..^ M ) ) |
71 |
|
simpr |
|- ( ( k e. ( 0 ... i ) /\ k =/= 0 ) -> k =/= 0 ) |
72 |
71
|
adantl |
|- ( ( i e. ( 1 ..^ M ) /\ ( k e. ( 0 ... i ) /\ k =/= 0 ) ) -> k =/= 0 ) |
73 |
|
fzo1fzo0n0 |
|- ( k e. ( 1 ..^ M ) <-> ( k e. ( 0 ..^ M ) /\ k =/= 0 ) ) |
74 |
70 72 73
|
sylanbrc |
|- ( ( i e. ( 1 ..^ M ) /\ ( k e. ( 0 ... i ) /\ k =/= 0 ) ) -> k e. ( 1 ..^ M ) ) |
75 |
74
|
adantl |
|- ( ( ph /\ ( i e. ( 1 ..^ M ) /\ ( k e. ( 0 ... i ) /\ k =/= 0 ) ) ) -> k e. ( 1 ..^ M ) ) |
76 |
27 28 75
|
iccpartipre |
|- ( ( ph /\ ( i e. ( 1 ..^ M ) /\ ( k e. ( 0 ... i ) /\ k =/= 0 ) ) ) -> ( P ` k ) e. RR ) |
77 |
76
|
exp32 |
|- ( ph -> ( i e. ( 1 ..^ M ) -> ( ( k e. ( 0 ... i ) /\ k =/= 0 ) -> ( P ` k ) e. RR ) ) ) |
78 |
77
|
ad2antrl |
|- ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) -> ( i e. ( 1 ..^ M ) -> ( ( k e. ( 0 ... i ) /\ k =/= 0 ) -> ( P ` k ) e. RR ) ) ) |
79 |
78
|
imp |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( ( k e. ( 0 ... i ) /\ k =/= 0 ) -> ( P ` k ) e. RR ) ) |
80 |
79
|
expdimp |
|- ( ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) /\ k e. ( 0 ... i ) ) -> ( k =/= 0 -> ( P ` k ) e. RR ) ) |
81 |
26 80
|
pm2.61dne |
|- ( ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) /\ k e. ( 0 ... i ) ) -> ( P ` k ) e. RR ) |
82 |
1
|
adantr |
|- ( ( ph /\ -. M = 1 ) -> M e. NN ) |
83 |
82
|
ad3antlr |
|- ( ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) /\ k e. ( 0 ... ( i - 1 ) ) ) -> M e. NN ) |
84 |
2
|
adantr |
|- ( ( ph /\ -. M = 1 ) -> P e. ( RePart ` M ) ) |
85 |
84
|
ad3antlr |
|- ( ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) /\ k e. ( 0 ... ( i - 1 ) ) ) -> P e. ( RePart ` M ) ) |
86 |
|
elfzoelz |
|- ( i e. ( 1 ..^ M ) -> i e. ZZ ) |
87 |
86
|
adantl |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> i e. ZZ ) |
88 |
|
fzoval |
|- ( i e. ZZ -> ( 0 ..^ i ) = ( 0 ... ( i - 1 ) ) ) |
89 |
88
|
eqcomd |
|- ( i e. ZZ -> ( 0 ... ( i - 1 ) ) = ( 0 ..^ i ) ) |
90 |
87 89
|
syl |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( 0 ... ( i - 1 ) ) = ( 0 ..^ i ) ) |
91 |
90
|
eleq2d |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( k e. ( 0 ... ( i - 1 ) ) <-> k e. ( 0 ..^ i ) ) ) |
92 |
|
elfzouz2 |
|- ( i e. ( 1 ..^ M ) -> M e. ( ZZ>= ` i ) ) |
93 |
92
|
adantl |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> M e. ( ZZ>= ` i ) ) |
94 |
|
fzoss2 |
|- ( M e. ( ZZ>= ` i ) -> ( 0 ..^ i ) C_ ( 0 ..^ M ) ) |
95 |
93 94
|
syl |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( 0 ..^ i ) C_ ( 0 ..^ M ) ) |
96 |
95
|
sseld |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( k e. ( 0 ..^ i ) -> k e. ( 0 ..^ M ) ) ) |
97 |
91 96
|
sylbid |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( k e. ( 0 ... ( i - 1 ) ) -> k e. ( 0 ..^ M ) ) ) |
98 |
97
|
imp |
|- ( ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) /\ k e. ( 0 ... ( i - 1 ) ) ) -> k e. ( 0 ..^ M ) ) |
99 |
|
iccpartimp |
|- ( ( M e. NN /\ P e. ( RePart ` M ) /\ k e. ( 0 ..^ M ) ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` k ) < ( P ` ( k + 1 ) ) ) ) |
100 |
83 85 98 99
|
syl3anc |
|- ( ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) /\ k e. ( 0 ... ( i - 1 ) ) ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` k ) < ( P ` ( k + 1 ) ) ) ) |
101 |
100
|
simprd |
|- ( ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) /\ k e. ( 0 ... ( i - 1 ) ) ) -> ( P ` k ) < ( P ` ( k + 1 ) ) ) |
102 |
16 21 81 101
|
smonoord |
|- ( ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( P ` 0 ) < ( P ` i ) ) |
103 |
102
|
ralrimiva |
|- ( ( ( P ` 0 ) e. RR /\ ( ph /\ -. M = 1 ) ) -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) |
104 |
103
|
ex |
|- ( ( P ` 0 ) e. RR -> ( ( ph /\ -. M = 1 ) -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) ) |
105 |
|
lbfzo0 |
|- ( 0 e. ( 0 ..^ M ) <-> M e. NN ) |
106 |
1 105
|
sylibr |
|- ( ph -> 0 e. ( 0 ..^ M ) ) |
107 |
1 2 106
|
3jca |
|- ( ph -> ( M e. NN /\ P e. ( RePart ` M ) /\ 0 e. ( 0 ..^ M ) ) ) |
108 |
107
|
ad2antrl |
|- ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) -> ( M e. NN /\ P e. ( RePart ` M ) /\ 0 e. ( 0 ..^ M ) ) ) |
109 |
108
|
adantr |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( M e. NN /\ P e. ( RePart ` M ) /\ 0 e. ( 0 ..^ M ) ) ) |
110 |
|
iccpartimp |
|- ( ( M e. NN /\ P e. ( RePart ` M ) /\ 0 e. ( 0 ..^ M ) ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` 0 ) < ( P ` ( 0 + 1 ) ) ) ) |
111 |
109 110
|
syl |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` 0 ) < ( P ` ( 0 + 1 ) ) ) ) |
112 |
111
|
simprd |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( P ` 0 ) < ( P ` ( 0 + 1 ) ) ) |
113 |
|
breq1 |
|- ( ( P ` 0 ) = +oo -> ( ( P ` 0 ) < ( P ` ( 0 + 1 ) ) <-> +oo < ( P ` ( 0 + 1 ) ) ) ) |
114 |
113
|
adantr |
|- ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) -> ( ( P ` 0 ) < ( P ` ( 0 + 1 ) ) <-> +oo < ( P ` ( 0 + 1 ) ) ) ) |
115 |
114
|
adantr |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( ( P ` 0 ) < ( P ` ( 0 + 1 ) ) <-> +oo < ( P ` ( 0 + 1 ) ) ) ) |
116 |
112 115
|
mpbid |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> +oo < ( P ` ( 0 + 1 ) ) ) |
117 |
1
|
ad2antrl |
|- ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) -> M e. NN ) |
118 |
117
|
adantr |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> M e. NN ) |
119 |
2
|
ad2antrl |
|- ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) -> P e. ( RePart ` M ) ) |
120 |
119
|
adantr |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> P e. ( RePart ` M ) ) |
121 |
|
1nn0 |
|- 1 e. NN0 |
122 |
121
|
a1i |
|- ( M e. NN -> 1 e. NN0 ) |
123 |
|
nnnn0 |
|- ( M e. NN -> M e. NN0 ) |
124 |
|
nnge1 |
|- ( M e. NN -> 1 <_ M ) |
125 |
122 123 124
|
3jca |
|- ( M e. NN -> ( 1 e. NN0 /\ M e. NN0 /\ 1 <_ M ) ) |
126 |
1 125
|
syl |
|- ( ph -> ( 1 e. NN0 /\ M e. NN0 /\ 1 <_ M ) ) |
127 |
|
elfz2nn0 |
|- ( 1 e. ( 0 ... M ) <-> ( 1 e. NN0 /\ M e. NN0 /\ 1 <_ M ) ) |
128 |
126 127
|
sylibr |
|- ( ph -> 1 e. ( 0 ... M ) ) |
129 |
18 128
|
eqeltrid |
|- ( ph -> ( 0 + 1 ) e. ( 0 ... M ) ) |
130 |
129
|
ad2antrl |
|- ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) -> ( 0 + 1 ) e. ( 0 ... M ) ) |
131 |
130
|
adantr |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( 0 + 1 ) e. ( 0 ... M ) ) |
132 |
118 120 131
|
iccpartxr |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( P ` ( 0 + 1 ) ) e. RR* ) |
133 |
|
pnfnlt |
|- ( ( P ` ( 0 + 1 ) ) e. RR* -> -. +oo < ( P ` ( 0 + 1 ) ) ) |
134 |
132 133
|
syl |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> -. +oo < ( P ` ( 0 + 1 ) ) ) |
135 |
116 134
|
pm2.21dd |
|- ( ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) /\ i e. ( 1 ..^ M ) ) -> ( P ` 0 ) < ( P ` i ) ) |
136 |
135
|
ralrimiva |
|- ( ( ( P ` 0 ) = +oo /\ ( ph /\ -. M = 1 ) ) -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) |
137 |
136
|
ex |
|- ( ( P ` 0 ) = +oo -> ( ( ph /\ -. M = 1 ) -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) ) |
138 |
1
|
adantr |
|- ( ( ph /\ i e. ( 1 ..^ M ) ) -> M e. NN ) |
139 |
2
|
adantr |
|- ( ( ph /\ i e. ( 1 ..^ M ) ) -> P e. ( RePart ` M ) ) |
140 |
|
simpr |
|- ( ( ph /\ i e. ( 1 ..^ M ) ) -> i e. ( 1 ..^ M ) ) |
141 |
138 139 140
|
iccpartipre |
|- ( ( ph /\ i e. ( 1 ..^ M ) ) -> ( P ` i ) e. RR ) |
142 |
|
mnflt |
|- ( ( P ` i ) e. RR -> -oo < ( P ` i ) ) |
143 |
141 142
|
syl |
|- ( ( ph /\ i e. ( 1 ..^ M ) ) -> -oo < ( P ` i ) ) |
144 |
143
|
ralrimiva |
|- ( ph -> A. i e. ( 1 ..^ M ) -oo < ( P ` i ) ) |
145 |
144
|
ad2antrl |
|- ( ( ( P ` 0 ) = -oo /\ ( ph /\ -. M = 1 ) ) -> A. i e. ( 1 ..^ M ) -oo < ( P ` i ) ) |
146 |
|
breq1 |
|- ( ( P ` 0 ) = -oo -> ( ( P ` 0 ) < ( P ` i ) <-> -oo < ( P ` i ) ) ) |
147 |
146
|
adantr |
|- ( ( ( P ` 0 ) = -oo /\ ( ph /\ -. M = 1 ) ) -> ( ( P ` 0 ) < ( P ` i ) <-> -oo < ( P ` i ) ) ) |
148 |
147
|
ralbidv |
|- ( ( ( P ` 0 ) = -oo /\ ( ph /\ -. M = 1 ) ) -> ( A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) <-> A. i e. ( 1 ..^ M ) -oo < ( P ` i ) ) ) |
149 |
145 148
|
mpbird |
|- ( ( ( P ` 0 ) = -oo /\ ( ph /\ -. M = 1 ) ) -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) |
150 |
149
|
ex |
|- ( ( P ` 0 ) = -oo -> ( ( ph /\ -. M = 1 ) -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) ) |
151 |
104 137 150
|
3jaoi |
|- ( ( ( P ` 0 ) e. RR \/ ( P ` 0 ) = +oo \/ ( P ` 0 ) = -oo ) -> ( ( ph /\ -. M = 1 ) -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) ) |
152 |
15 151
|
sylbi |
|- ( ( P ` 0 ) e. RR* -> ( ( ph /\ -. M = 1 ) -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) ) |
153 |
14 152
|
mpcom |
|- ( ( ph /\ -. M = 1 ) -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) |
154 |
153
|
expcom |
|- ( -. M = 1 -> ( ph -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) ) |
155 |
9 154
|
pm2.61i |
|- ( ph -> A. i e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` i ) ) |