Step |
Hyp |
Ref |
Expression |
1 |
|
iccpartgtprec.m |
|- ( ph -> M e. NN ) |
2 |
|
iccpartgtprec.p |
|- ( ph -> P e. ( RePart ` M ) ) |
3 |
1
|
nnnn0d |
|- ( ph -> M e. NN0 ) |
4 |
|
elnn0uz |
|- ( M e. NN0 <-> M e. ( ZZ>= ` 0 ) ) |
5 |
3 4
|
sylib |
|- ( ph -> M e. ( ZZ>= ` 0 ) ) |
6 |
|
fzpred |
|- ( M e. ( ZZ>= ` 0 ) -> ( 0 ... M ) = ( { 0 } u. ( ( 0 + 1 ) ... M ) ) ) |
7 |
5 6
|
syl |
|- ( ph -> ( 0 ... M ) = ( { 0 } u. ( ( 0 + 1 ) ... M ) ) ) |
8 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
9 |
8
|
oveq1i |
|- ( ( 0 + 1 ) ... M ) = ( 1 ... M ) |
10 |
9
|
a1i |
|- ( ph -> ( ( 0 + 1 ) ... M ) = ( 1 ... M ) ) |
11 |
10
|
uneq2d |
|- ( ph -> ( { 0 } u. ( ( 0 + 1 ) ... M ) ) = ( { 0 } u. ( 1 ... M ) ) ) |
12 |
7 11
|
eqtrd |
|- ( ph -> ( 0 ... M ) = ( { 0 } u. ( 1 ... M ) ) ) |
13 |
12
|
eleq2d |
|- ( ph -> ( i e. ( 0 ... M ) <-> i e. ( { 0 } u. ( 1 ... M ) ) ) ) |
14 |
|
elun |
|- ( i e. ( { 0 } u. ( 1 ... M ) ) <-> ( i e. { 0 } \/ i e. ( 1 ... M ) ) ) |
15 |
|
velsn |
|- ( i e. { 0 } <-> i = 0 ) |
16 |
15
|
orbi1i |
|- ( ( i e. { 0 } \/ i e. ( 1 ... M ) ) <-> ( i = 0 \/ i e. ( 1 ... M ) ) ) |
17 |
14 16
|
bitri |
|- ( i e. ( { 0 } u. ( 1 ... M ) ) <-> ( i = 0 \/ i e. ( 1 ... M ) ) ) |
18 |
|
fzisfzounsn |
|- ( M e. ( ZZ>= ` 0 ) -> ( 0 ... M ) = ( ( 0 ..^ M ) u. { M } ) ) |
19 |
5 18
|
syl |
|- ( ph -> ( 0 ... M ) = ( ( 0 ..^ M ) u. { M } ) ) |
20 |
19
|
eleq2d |
|- ( ph -> ( j e. ( 0 ... M ) <-> j e. ( ( 0 ..^ M ) u. { M } ) ) ) |
21 |
|
elun |
|- ( j e. ( ( 0 ..^ M ) u. { M } ) <-> ( j e. ( 0 ..^ M ) \/ j e. { M } ) ) |
22 |
|
velsn |
|- ( j e. { M } <-> j = M ) |
23 |
22
|
orbi2i |
|- ( ( j e. ( 0 ..^ M ) \/ j e. { M } ) <-> ( j e. ( 0 ..^ M ) \/ j = M ) ) |
24 |
21 23
|
bitri |
|- ( j e. ( ( 0 ..^ M ) u. { M } ) <-> ( j e. ( 0 ..^ M ) \/ j = M ) ) |
25 |
20 24
|
bitrdi |
|- ( ph -> ( j e. ( 0 ... M ) <-> ( j e. ( 0 ..^ M ) \/ j = M ) ) ) |
26 |
|
simpl |
|- ( ( j e. ( 0 ..^ M ) /\ 0 < j ) -> j e. ( 0 ..^ M ) ) |
27 |
|
simpr |
|- ( ( j e. ( 0 ..^ M ) /\ 0 < j ) -> 0 < j ) |
28 |
27
|
gt0ne0d |
|- ( ( j e. ( 0 ..^ M ) /\ 0 < j ) -> j =/= 0 ) |
29 |
|
fzo1fzo0n0 |
|- ( j e. ( 1 ..^ M ) <-> ( j e. ( 0 ..^ M ) /\ j =/= 0 ) ) |
30 |
26 28 29
|
sylanbrc |
|- ( ( j e. ( 0 ..^ M ) /\ 0 < j ) -> j e. ( 1 ..^ M ) ) |
31 |
1 2
|
iccpartigtl |
|- ( ph -> A. k e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` k ) ) |
32 |
|
fveq2 |
|- ( k = j -> ( P ` k ) = ( P ` j ) ) |
33 |
32
|
breq2d |
|- ( k = j -> ( ( P ` 0 ) < ( P ` k ) <-> ( P ` 0 ) < ( P ` j ) ) ) |
34 |
33
|
rspcv |
|- ( j e. ( 1 ..^ M ) -> ( A. k e. ( 1 ..^ M ) ( P ` 0 ) < ( P ` k ) -> ( P ` 0 ) < ( P ` j ) ) ) |
35 |
30 31 34
|
syl2imc |
|- ( ph -> ( ( j e. ( 0 ..^ M ) /\ 0 < j ) -> ( P ` 0 ) < ( P ` j ) ) ) |
36 |
35
|
expd |
|- ( ph -> ( j e. ( 0 ..^ M ) -> ( 0 < j -> ( P ` 0 ) < ( P ` j ) ) ) ) |
37 |
36
|
impcom |
|- ( ( j e. ( 0 ..^ M ) /\ ph ) -> ( 0 < j -> ( P ` 0 ) < ( P ` j ) ) ) |
38 |
|
breq1 |
|- ( i = 0 -> ( i < j <-> 0 < j ) ) |
39 |
|
fveq2 |
|- ( i = 0 -> ( P ` i ) = ( P ` 0 ) ) |
40 |
39
|
breq1d |
|- ( i = 0 -> ( ( P ` i ) < ( P ` j ) <-> ( P ` 0 ) < ( P ` j ) ) ) |
41 |
38 40
|
imbi12d |
|- ( i = 0 -> ( ( i < j -> ( P ` i ) < ( P ` j ) ) <-> ( 0 < j -> ( P ` 0 ) < ( P ` j ) ) ) ) |
42 |
37 41
|
syl5ibr |
|- ( i = 0 -> ( ( j e. ( 0 ..^ M ) /\ ph ) -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) |
43 |
42
|
expd |
|- ( i = 0 -> ( j e. ( 0 ..^ M ) -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
44 |
43
|
com12 |
|- ( j e. ( 0 ..^ M ) -> ( i = 0 -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
45 |
1 2
|
iccpartlt |
|- ( ph -> ( P ` 0 ) < ( P ` M ) ) |
46 |
|
fveq2 |
|- ( j = M -> ( P ` j ) = ( P ` M ) ) |
47 |
39 46
|
breqan12rd |
|- ( ( j = M /\ i = 0 ) -> ( ( P ` i ) < ( P ` j ) <-> ( P ` 0 ) < ( P ` M ) ) ) |
48 |
45 47
|
syl5ibr |
|- ( ( j = M /\ i = 0 ) -> ( ph -> ( P ` i ) < ( P ` j ) ) ) |
49 |
48
|
a1dd |
|- ( ( j = M /\ i = 0 ) -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) |
50 |
49
|
ex |
|- ( j = M -> ( i = 0 -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
51 |
44 50
|
jaoi |
|- ( ( j e. ( 0 ..^ M ) \/ j = M ) -> ( i = 0 -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
52 |
51
|
com12 |
|- ( i = 0 -> ( ( j e. ( 0 ..^ M ) \/ j = M ) -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
53 |
|
elfzelz |
|- ( i e. ( 1 ... M ) -> i e. ZZ ) |
54 |
53
|
ad3antlr |
|- ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) -> i e. ZZ ) |
55 |
53
|
peano2zd |
|- ( i e. ( 1 ... M ) -> ( i + 1 ) e. ZZ ) |
56 |
55
|
ad2antlr |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> ( i + 1 ) e. ZZ ) |
57 |
|
elfzoelz |
|- ( j e. ( 0 ..^ M ) -> j e. ZZ ) |
58 |
57
|
ad2antrr |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> j e. ZZ ) |
59 |
|
simpr |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> i < j ) |
60 |
57 53
|
anim12ci |
|- ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) -> ( i e. ZZ /\ j e. ZZ ) ) |
61 |
60
|
adantr |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> ( i e. ZZ /\ j e. ZZ ) ) |
62 |
|
zltp1le |
|- ( ( i e. ZZ /\ j e. ZZ ) -> ( i < j <-> ( i + 1 ) <_ j ) ) |
63 |
61 62
|
syl |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> ( i < j <-> ( i + 1 ) <_ j ) ) |
64 |
59 63
|
mpbid |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> ( i + 1 ) <_ j ) |
65 |
56 58 64
|
3jca |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> ( ( i + 1 ) e. ZZ /\ j e. ZZ /\ ( i + 1 ) <_ j ) ) |
66 |
65
|
adantr |
|- ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) -> ( ( i + 1 ) e. ZZ /\ j e. ZZ /\ ( i + 1 ) <_ j ) ) |
67 |
|
eluz2 |
|- ( j e. ( ZZ>= ` ( i + 1 ) ) <-> ( ( i + 1 ) e. ZZ /\ j e. ZZ /\ ( i + 1 ) <_ j ) ) |
68 |
66 67
|
sylibr |
|- ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) -> j e. ( ZZ>= ` ( i + 1 ) ) ) |
69 |
1
|
ad2antlr |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> M e. NN ) |
70 |
2
|
ad2antlr |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> P e. ( RePart ` M ) ) |
71 |
|
1zzd |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> 1 e. ZZ ) |
72 |
|
elfzelz |
|- ( k e. ( i ... j ) -> k e. ZZ ) |
73 |
72
|
adantl |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> k e. ZZ ) |
74 |
|
elfzle1 |
|- ( i e. ( 1 ... M ) -> 1 <_ i ) |
75 |
|
elfzle1 |
|- ( k e. ( i ... j ) -> i <_ k ) |
76 |
|
1red |
|- ( k e. ( i ... j ) -> 1 e. RR ) |
77 |
|
elfzel1 |
|- ( k e. ( i ... j ) -> i e. ZZ ) |
78 |
77
|
zred |
|- ( k e. ( i ... j ) -> i e. RR ) |
79 |
72
|
zred |
|- ( k e. ( i ... j ) -> k e. RR ) |
80 |
|
letr |
|- ( ( 1 e. RR /\ i e. RR /\ k e. RR ) -> ( ( 1 <_ i /\ i <_ k ) -> 1 <_ k ) ) |
81 |
76 78 79 80
|
syl3anc |
|- ( k e. ( i ... j ) -> ( ( 1 <_ i /\ i <_ k ) -> 1 <_ k ) ) |
82 |
75 81
|
mpan2d |
|- ( k e. ( i ... j ) -> ( 1 <_ i -> 1 <_ k ) ) |
83 |
74 82
|
syl5com |
|- ( i e. ( 1 ... M ) -> ( k e. ( i ... j ) -> 1 <_ k ) ) |
84 |
83
|
ad3antlr |
|- ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) -> ( k e. ( i ... j ) -> 1 <_ k ) ) |
85 |
84
|
imp |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> 1 <_ k ) |
86 |
|
eluz2 |
|- ( k e. ( ZZ>= ` 1 ) <-> ( 1 e. ZZ /\ k e. ZZ /\ 1 <_ k ) ) |
87 |
71 73 85 86
|
syl3anbrc |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> k e. ( ZZ>= ` 1 ) ) |
88 |
|
elfzel2 |
|- ( i e. ( 1 ... M ) -> M e. ZZ ) |
89 |
88
|
ad2antlr |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> M e. ZZ ) |
90 |
89
|
ad2antrr |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> M e. ZZ ) |
91 |
79
|
adantl |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> k e. RR ) |
92 |
57
|
zred |
|- ( j e. ( 0 ..^ M ) -> j e. RR ) |
93 |
92
|
ad4antr |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> j e. RR ) |
94 |
69
|
nnred |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> M e. RR ) |
95 |
|
elfzle2 |
|- ( k e. ( i ... j ) -> k <_ j ) |
96 |
95
|
adantl |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> k <_ j ) |
97 |
|
elfzolt2 |
|- ( j e. ( 0 ..^ M ) -> j < M ) |
98 |
97
|
ad4antr |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> j < M ) |
99 |
91 93 94 96 98
|
lelttrd |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> k < M ) |
100 |
|
elfzo2 |
|- ( k e. ( 1 ..^ M ) <-> ( k e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ k < M ) ) |
101 |
87 90 99 100
|
syl3anbrc |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> k e. ( 1 ..^ M ) ) |
102 |
69 70 101
|
iccpartipre |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... j ) ) -> ( P ` k ) e. RR ) |
103 |
1
|
ad2antlr |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... ( j - 1 ) ) ) -> M e. NN ) |
104 |
2
|
ad2antlr |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... ( j - 1 ) ) ) -> P e. ( RePart ` M ) ) |
105 |
57
|
ad3antrrr |
|- ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) -> j e. ZZ ) |
106 |
|
fzoval |
|- ( j e. ZZ -> ( i ..^ j ) = ( i ... ( j - 1 ) ) ) |
107 |
105 106
|
syl |
|- ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) -> ( i ..^ j ) = ( i ... ( j - 1 ) ) ) |
108 |
|
elfzo0le |
|- ( j e. ( 0 ..^ M ) -> j <_ M ) |
109 |
|
0le1 |
|- 0 <_ 1 |
110 |
|
0red |
|- ( i e. ( 1 ... M ) -> 0 e. RR ) |
111 |
|
1red |
|- ( i e. ( 1 ... M ) -> 1 e. RR ) |
112 |
53
|
zred |
|- ( i e. ( 1 ... M ) -> i e. RR ) |
113 |
|
letr |
|- ( ( 0 e. RR /\ 1 e. RR /\ i e. RR ) -> ( ( 0 <_ 1 /\ 1 <_ i ) -> 0 <_ i ) ) |
114 |
110 111 112 113
|
syl3anc |
|- ( i e. ( 1 ... M ) -> ( ( 0 <_ 1 /\ 1 <_ i ) -> 0 <_ i ) ) |
115 |
109 114
|
mpani |
|- ( i e. ( 1 ... M ) -> ( 1 <_ i -> 0 <_ i ) ) |
116 |
74 115
|
mpd |
|- ( i e. ( 1 ... M ) -> 0 <_ i ) |
117 |
108 116
|
anim12ci |
|- ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) -> ( 0 <_ i /\ j <_ M ) ) |
118 |
117
|
adantr |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> ( 0 <_ i /\ j <_ M ) ) |
119 |
|
0zd |
|- ( j e. ( 0 ..^ M ) -> 0 e. ZZ ) |
120 |
|
elfzoel2 |
|- ( j e. ( 0 ..^ M ) -> M e. ZZ ) |
121 |
119 120
|
jca |
|- ( j e. ( 0 ..^ M ) -> ( 0 e. ZZ /\ M e. ZZ ) ) |
122 |
121
|
ad2antrr |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> ( 0 e. ZZ /\ M e. ZZ ) ) |
123 |
|
ssfzo12bi |
|- ( ( ( i e. ZZ /\ j e. ZZ ) /\ ( 0 e. ZZ /\ M e. ZZ ) /\ i < j ) -> ( ( i ..^ j ) C_ ( 0 ..^ M ) <-> ( 0 <_ i /\ j <_ M ) ) ) |
124 |
61 122 59 123
|
syl3anc |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> ( ( i ..^ j ) C_ ( 0 ..^ M ) <-> ( 0 <_ i /\ j <_ M ) ) ) |
125 |
118 124
|
mpbird |
|- ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) -> ( i ..^ j ) C_ ( 0 ..^ M ) ) |
126 |
125
|
adantr |
|- ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) -> ( i ..^ j ) C_ ( 0 ..^ M ) ) |
127 |
107 126
|
eqsstrrd |
|- ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) -> ( i ... ( j - 1 ) ) C_ ( 0 ..^ M ) ) |
128 |
127
|
sselda |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... ( j - 1 ) ) ) -> k e. ( 0 ..^ M ) ) |
129 |
|
iccpartimp |
|- ( ( M e. NN /\ P e. ( RePart ` M ) /\ k e. ( 0 ..^ M ) ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` k ) < ( P ` ( k + 1 ) ) ) ) |
130 |
103 104 128 129
|
syl3anc |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... ( j - 1 ) ) ) -> ( P e. ( RR* ^m ( 0 ... M ) ) /\ ( P ` k ) < ( P ` ( k + 1 ) ) ) ) |
131 |
130
|
simprd |
|- ( ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) /\ k e. ( i ... ( j - 1 ) ) ) -> ( P ` k ) < ( P ` ( k + 1 ) ) ) |
132 |
54 68 102 131
|
smonoord |
|- ( ( ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) /\ i < j ) /\ ph ) -> ( P ` i ) < ( P ` j ) ) |
133 |
132
|
exp31 |
|- ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) -> ( i < j -> ( ph -> ( P ` i ) < ( P ` j ) ) ) ) |
134 |
133
|
com23 |
|- ( ( j e. ( 0 ..^ M ) /\ i e. ( 1 ... M ) ) -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) |
135 |
134
|
ex |
|- ( j e. ( 0 ..^ M ) -> ( i e. ( 1 ... M ) -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
136 |
|
elfzuz |
|- ( i e. ( 1 ... M ) -> i e. ( ZZ>= ` 1 ) ) |
137 |
136
|
adantr |
|- ( ( i e. ( 1 ... M ) /\ i < M ) -> i e. ( ZZ>= ` 1 ) ) |
138 |
88
|
adantr |
|- ( ( i e. ( 1 ... M ) /\ i < M ) -> M e. ZZ ) |
139 |
|
simpr |
|- ( ( i e. ( 1 ... M ) /\ i < M ) -> i < M ) |
140 |
|
elfzo2 |
|- ( i e. ( 1 ..^ M ) <-> ( i e. ( ZZ>= ` 1 ) /\ M e. ZZ /\ i < M ) ) |
141 |
137 138 139 140
|
syl3anbrc |
|- ( ( i e. ( 1 ... M ) /\ i < M ) -> i e. ( 1 ..^ M ) ) |
142 |
1 2
|
iccpartiltu |
|- ( ph -> A. k e. ( 1 ..^ M ) ( P ` k ) < ( P ` M ) ) |
143 |
|
fveq2 |
|- ( k = i -> ( P ` k ) = ( P ` i ) ) |
144 |
143
|
breq1d |
|- ( k = i -> ( ( P ` k ) < ( P ` M ) <-> ( P ` i ) < ( P ` M ) ) ) |
145 |
144
|
rspcv |
|- ( i e. ( 1 ..^ M ) -> ( A. k e. ( 1 ..^ M ) ( P ` k ) < ( P ` M ) -> ( P ` i ) < ( P ` M ) ) ) |
146 |
141 142 145
|
syl2imc |
|- ( ph -> ( ( i e. ( 1 ... M ) /\ i < M ) -> ( P ` i ) < ( P ` M ) ) ) |
147 |
146
|
expd |
|- ( ph -> ( i e. ( 1 ... M ) -> ( i < M -> ( P ` i ) < ( P ` M ) ) ) ) |
148 |
147
|
impcom |
|- ( ( i e. ( 1 ... M ) /\ ph ) -> ( i < M -> ( P ` i ) < ( P ` M ) ) ) |
149 |
148
|
imp |
|- ( ( ( i e. ( 1 ... M ) /\ ph ) /\ i < M ) -> ( P ` i ) < ( P ` M ) ) |
150 |
149
|
a1i |
|- ( j = M -> ( ( ( i e. ( 1 ... M ) /\ ph ) /\ i < M ) -> ( P ` i ) < ( P ` M ) ) ) |
151 |
|
breq2 |
|- ( j = M -> ( i < j <-> i < M ) ) |
152 |
151
|
anbi2d |
|- ( j = M -> ( ( ( i e. ( 1 ... M ) /\ ph ) /\ i < j ) <-> ( ( i e. ( 1 ... M ) /\ ph ) /\ i < M ) ) ) |
153 |
46
|
breq2d |
|- ( j = M -> ( ( P ` i ) < ( P ` j ) <-> ( P ` i ) < ( P ` M ) ) ) |
154 |
150 152 153
|
3imtr4d |
|- ( j = M -> ( ( ( i e. ( 1 ... M ) /\ ph ) /\ i < j ) -> ( P ` i ) < ( P ` j ) ) ) |
155 |
154
|
exp4c |
|- ( j = M -> ( i e. ( 1 ... M ) -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
156 |
135 155
|
jaoi |
|- ( ( j e. ( 0 ..^ M ) \/ j = M ) -> ( i e. ( 1 ... M ) -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
157 |
156
|
com12 |
|- ( i e. ( 1 ... M ) -> ( ( j e. ( 0 ..^ M ) \/ j = M ) -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
158 |
52 157
|
jaoi |
|- ( ( i = 0 \/ i e. ( 1 ... M ) ) -> ( ( j e. ( 0 ..^ M ) \/ j = M ) -> ( ph -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
159 |
158
|
com13 |
|- ( ph -> ( ( j e. ( 0 ..^ M ) \/ j = M ) -> ( ( i = 0 \/ i e. ( 1 ... M ) ) -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
160 |
25 159
|
sylbid |
|- ( ph -> ( j e. ( 0 ... M ) -> ( ( i = 0 \/ i e. ( 1 ... M ) ) -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
161 |
160
|
com3r |
|- ( ( i = 0 \/ i e. ( 1 ... M ) ) -> ( ph -> ( j e. ( 0 ... M ) -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
162 |
17 161
|
sylbi |
|- ( i e. ( { 0 } u. ( 1 ... M ) ) -> ( ph -> ( j e. ( 0 ... M ) -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
163 |
162
|
com12 |
|- ( ph -> ( i e. ( { 0 } u. ( 1 ... M ) ) -> ( j e. ( 0 ... M ) -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
164 |
13 163
|
sylbid |
|- ( ph -> ( i e. ( 0 ... M ) -> ( j e. ( 0 ... M ) -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) ) ) |
165 |
164
|
imp32 |
|- ( ( ph /\ ( i e. ( 0 ... M ) /\ j e. ( 0 ... M ) ) ) -> ( i < j -> ( P ` i ) < ( P ` j ) ) ) |
166 |
165
|
ralrimivva |
|- ( ph -> A. i e. ( 0 ... M ) A. j e. ( 0 ... M ) ( i < j -> ( P ` i ) < ( P ` j ) ) ) |