Step |
Hyp |
Ref |
Expression |
1 |
|
monoords.fk |
|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR ) |
2 |
|
monoords.flt |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> ( F ` k ) < ( F ` ( k + 1 ) ) ) |
3 |
|
monoords.i |
|- ( ph -> I e. ( M ... N ) ) |
4 |
|
monoords.j |
|- ( ph -> J e. ( M ... N ) ) |
5 |
|
monoords.iltj |
|- ( ph -> I < J ) |
6 |
3
|
ancli |
|- ( ph -> ( ph /\ I e. ( M ... N ) ) ) |
7 |
|
eleq1 |
|- ( k = I -> ( k e. ( M ... N ) <-> I e. ( M ... N ) ) ) |
8 |
7
|
anbi2d |
|- ( k = I -> ( ( ph /\ k e. ( M ... N ) ) <-> ( ph /\ I e. ( M ... N ) ) ) ) |
9 |
|
fveq2 |
|- ( k = I -> ( F ` k ) = ( F ` I ) ) |
10 |
9
|
eleq1d |
|- ( k = I -> ( ( F ` k ) e. RR <-> ( F ` I ) e. RR ) ) |
11 |
8 10
|
imbi12d |
|- ( k = I -> ( ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR ) <-> ( ( ph /\ I e. ( M ... N ) ) -> ( F ` I ) e. RR ) ) ) |
12 |
11 1
|
vtoclg |
|- ( I e. ( M ... N ) -> ( ( ph /\ I e. ( M ... N ) ) -> ( F ` I ) e. RR ) ) |
13 |
3 6 12
|
sylc |
|- ( ph -> ( F ` I ) e. RR ) |
14 |
|
elfzel1 |
|- ( I e. ( M ... N ) -> M e. ZZ ) |
15 |
3 14
|
syl |
|- ( ph -> M e. ZZ ) |
16 |
3
|
elfzelzd |
|- ( ph -> I e. ZZ ) |
17 |
|
elfzle1 |
|- ( I e. ( M ... N ) -> M <_ I ) |
18 |
3 17
|
syl |
|- ( ph -> M <_ I ) |
19 |
|
eluz2 |
|- ( I e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ I e. ZZ /\ M <_ I ) ) |
20 |
15 16 18 19
|
syl3anbrc |
|- ( ph -> I e. ( ZZ>= ` M ) ) |
21 |
|
elfzuz2 |
|- ( I e. ( M ... N ) -> N e. ( ZZ>= ` M ) ) |
22 |
3 21
|
syl |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
23 |
|
eluzelz |
|- ( N e. ( ZZ>= ` M ) -> N e. ZZ ) |
24 |
22 23
|
syl |
|- ( ph -> N e. ZZ ) |
25 |
16
|
zred |
|- ( ph -> I e. RR ) |
26 |
4
|
elfzelzd |
|- ( ph -> J e. ZZ ) |
27 |
26
|
zred |
|- ( ph -> J e. RR ) |
28 |
24
|
zred |
|- ( ph -> N e. RR ) |
29 |
|
elfzle2 |
|- ( J e. ( M ... N ) -> J <_ N ) |
30 |
4 29
|
syl |
|- ( ph -> J <_ N ) |
31 |
25 27 28 5 30
|
ltletrd |
|- ( ph -> I < N ) |
32 |
|
elfzo2 |
|- ( I e. ( M ..^ N ) <-> ( I e. ( ZZ>= ` M ) /\ N e. ZZ /\ I < N ) ) |
33 |
20 24 31 32
|
syl3anbrc |
|- ( ph -> I e. ( M ..^ N ) ) |
34 |
|
fzofzp1 |
|- ( I e. ( M ..^ N ) -> ( I + 1 ) e. ( M ... N ) ) |
35 |
33 34
|
syl |
|- ( ph -> ( I + 1 ) e. ( M ... N ) ) |
36 |
35
|
ancli |
|- ( ph -> ( ph /\ ( I + 1 ) e. ( M ... N ) ) ) |
37 |
|
eleq1 |
|- ( k = ( I + 1 ) -> ( k e. ( M ... N ) <-> ( I + 1 ) e. ( M ... N ) ) ) |
38 |
37
|
anbi2d |
|- ( k = ( I + 1 ) -> ( ( ph /\ k e. ( M ... N ) ) <-> ( ph /\ ( I + 1 ) e. ( M ... N ) ) ) ) |
39 |
|
fveq2 |
|- ( k = ( I + 1 ) -> ( F ` k ) = ( F ` ( I + 1 ) ) ) |
40 |
39
|
eleq1d |
|- ( k = ( I + 1 ) -> ( ( F ` k ) e. RR <-> ( F ` ( I + 1 ) ) e. RR ) ) |
41 |
38 40
|
imbi12d |
|- ( k = ( I + 1 ) -> ( ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR ) <-> ( ( ph /\ ( I + 1 ) e. ( M ... N ) ) -> ( F ` ( I + 1 ) ) e. RR ) ) ) |
42 |
41 1
|
vtoclg |
|- ( ( I + 1 ) e. ( M ... N ) -> ( ( ph /\ ( I + 1 ) e. ( M ... N ) ) -> ( F ` ( I + 1 ) ) e. RR ) ) |
43 |
35 36 42
|
sylc |
|- ( ph -> ( F ` ( I + 1 ) ) e. RR ) |
44 |
4
|
ancli |
|- ( ph -> ( ph /\ J e. ( M ... N ) ) ) |
45 |
|
eleq1 |
|- ( k = J -> ( k e. ( M ... N ) <-> J e. ( M ... N ) ) ) |
46 |
45
|
anbi2d |
|- ( k = J -> ( ( ph /\ k e. ( M ... N ) ) <-> ( ph /\ J e. ( M ... N ) ) ) ) |
47 |
|
fveq2 |
|- ( k = J -> ( F ` k ) = ( F ` J ) ) |
48 |
47
|
eleq1d |
|- ( k = J -> ( ( F ` k ) e. RR <-> ( F ` J ) e. RR ) ) |
49 |
46 48
|
imbi12d |
|- ( k = J -> ( ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR ) <-> ( ( ph /\ J e. ( M ... N ) ) -> ( F ` J ) e. RR ) ) ) |
50 |
49 1
|
vtoclg |
|- ( J e. ( M ... N ) -> ( ( ph /\ J e. ( M ... N ) ) -> ( F ` J ) e. RR ) ) |
51 |
4 44 50
|
sylc |
|- ( ph -> ( F ` J ) e. RR ) |
52 |
33
|
ancli |
|- ( ph -> ( ph /\ I e. ( M ..^ N ) ) ) |
53 |
|
eleq1 |
|- ( k = I -> ( k e. ( M ..^ N ) <-> I e. ( M ..^ N ) ) ) |
54 |
53
|
anbi2d |
|- ( k = I -> ( ( ph /\ k e. ( M ..^ N ) ) <-> ( ph /\ I e. ( M ..^ N ) ) ) ) |
55 |
|
fvoveq1 |
|- ( k = I -> ( F ` ( k + 1 ) ) = ( F ` ( I + 1 ) ) ) |
56 |
9 55
|
breq12d |
|- ( k = I -> ( ( F ` k ) < ( F ` ( k + 1 ) ) <-> ( F ` I ) < ( F ` ( I + 1 ) ) ) ) |
57 |
54 56
|
imbi12d |
|- ( k = I -> ( ( ( ph /\ k e. ( M ..^ N ) ) -> ( F ` k ) < ( F ` ( k + 1 ) ) ) <-> ( ( ph /\ I e. ( M ..^ N ) ) -> ( F ` I ) < ( F ` ( I + 1 ) ) ) ) ) |
58 |
57 2
|
vtoclg |
|- ( I e. ( M ..^ N ) -> ( ( ph /\ I e. ( M ..^ N ) ) -> ( F ` I ) < ( F ` ( I + 1 ) ) ) ) |
59 |
33 52 58
|
sylc |
|- ( ph -> ( F ` I ) < ( F ` ( I + 1 ) ) ) |
60 |
16
|
peano2zd |
|- ( ph -> ( I + 1 ) e. ZZ ) |
61 |
|
zltp1le |
|- ( ( I e. ZZ /\ J e. ZZ ) -> ( I < J <-> ( I + 1 ) <_ J ) ) |
62 |
16 26 61
|
syl2anc |
|- ( ph -> ( I < J <-> ( I + 1 ) <_ J ) ) |
63 |
5 62
|
mpbid |
|- ( ph -> ( I + 1 ) <_ J ) |
64 |
|
eluz2 |
|- ( J e. ( ZZ>= ` ( I + 1 ) ) <-> ( ( I + 1 ) e. ZZ /\ J e. ZZ /\ ( I + 1 ) <_ J ) ) |
65 |
60 26 63 64
|
syl3anbrc |
|- ( ph -> J e. ( ZZ>= ` ( I + 1 ) ) ) |
66 |
15
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> M e. ZZ ) |
67 |
24
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> N e. ZZ ) |
68 |
|
elfzelz |
|- ( k e. ( ( I + 1 ) ... J ) -> k e. ZZ ) |
69 |
68
|
adantl |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> k e. ZZ ) |
70 |
66
|
zred |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> M e. RR ) |
71 |
69
|
zred |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> k e. RR ) |
72 |
60
|
zred |
|- ( ph -> ( I + 1 ) e. RR ) |
73 |
72
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> ( I + 1 ) e. RR ) |
74 |
25
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> I e. RR ) |
75 |
18
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> M <_ I ) |
76 |
74
|
ltp1d |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> I < ( I + 1 ) ) |
77 |
70 74 73 75 76
|
lelttrd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> M < ( I + 1 ) ) |
78 |
|
elfzle1 |
|- ( k e. ( ( I + 1 ) ... J ) -> ( I + 1 ) <_ k ) |
79 |
78
|
adantl |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> ( I + 1 ) <_ k ) |
80 |
70 73 71 77 79
|
ltletrd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> M < k ) |
81 |
70 71 80
|
ltled |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> M <_ k ) |
82 |
27
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> J e. RR ) |
83 |
67
|
zred |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> N e. RR ) |
84 |
|
elfzle2 |
|- ( k e. ( ( I + 1 ) ... J ) -> k <_ J ) |
85 |
84
|
adantl |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> k <_ J ) |
86 |
30
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> J <_ N ) |
87 |
71 82 83 85 86
|
letrd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> k <_ N ) |
88 |
66 67 69 81 87
|
elfzd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> k e. ( M ... N ) ) |
89 |
88 1
|
syldan |
|- ( ( ph /\ k e. ( ( I + 1 ) ... J ) ) -> ( F ` k ) e. RR ) |
90 |
15
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> M e. ZZ ) |
91 |
24
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> N e. ZZ ) |
92 |
|
elfzelz |
|- ( k e. ( ( I + 1 ) ... ( J - 1 ) ) -> k e. ZZ ) |
93 |
92
|
adantl |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> k e. ZZ ) |
94 |
90
|
zred |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> M e. RR ) |
95 |
93
|
zred |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> k e. RR ) |
96 |
72
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( I + 1 ) e. RR ) |
97 |
15
|
zred |
|- ( ph -> M e. RR ) |
98 |
25
|
ltp1d |
|- ( ph -> I < ( I + 1 ) ) |
99 |
97 25 72 18 98
|
lelttrd |
|- ( ph -> M < ( I + 1 ) ) |
100 |
99
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> M < ( I + 1 ) ) |
101 |
|
elfzle1 |
|- ( k e. ( ( I + 1 ) ... ( J - 1 ) ) -> ( I + 1 ) <_ k ) |
102 |
101
|
adantl |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( I + 1 ) <_ k ) |
103 |
94 96 95 100 102
|
ltletrd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> M < k ) |
104 |
94 95 103
|
ltled |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> M <_ k ) |
105 |
91
|
zred |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> N e. RR ) |
106 |
|
peano2rem |
|- ( J e. RR -> ( J - 1 ) e. RR ) |
107 |
27 106
|
syl |
|- ( ph -> ( J - 1 ) e. RR ) |
108 |
107
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( J - 1 ) e. RR ) |
109 |
|
elfzle2 |
|- ( k e. ( ( I + 1 ) ... ( J - 1 ) ) -> k <_ ( J - 1 ) ) |
110 |
109
|
adantl |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> k <_ ( J - 1 ) ) |
111 |
27
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> J e. RR ) |
112 |
111
|
ltm1d |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( J - 1 ) < J ) |
113 |
30
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> J <_ N ) |
114 |
108 111 105 112 113
|
ltletrd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( J - 1 ) < N ) |
115 |
95 108 105 110 114
|
lelttrd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> k < N ) |
116 |
95 105 115
|
ltled |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> k <_ N ) |
117 |
90 91 93 104 116
|
elfzd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> k e. ( M ... N ) ) |
118 |
117 1
|
syldan |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( F ` k ) e. RR ) |
119 |
|
peano2zm |
|- ( N e. ZZ -> ( N - 1 ) e. ZZ ) |
120 |
91 119
|
syl |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( N - 1 ) e. ZZ ) |
121 |
120
|
zred |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( N - 1 ) e. RR ) |
122 |
|
1red |
|- ( ph -> 1 e. RR ) |
123 |
27 28 122 30
|
lesub1dd |
|- ( ph -> ( J - 1 ) <_ ( N - 1 ) ) |
124 |
123
|
adantr |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( J - 1 ) <_ ( N - 1 ) ) |
125 |
95 108 121 110 124
|
letrd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> k <_ ( N - 1 ) ) |
126 |
90 120 93 104 125
|
elfzd |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> k e. ( M ... ( N - 1 ) ) ) |
127 |
|
simpr |
|- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> k e. ( M ... ( N - 1 ) ) ) |
128 |
|
fzoval |
|- ( N e. ZZ -> ( M ..^ N ) = ( M ... ( N - 1 ) ) ) |
129 |
24 128
|
syl |
|- ( ph -> ( M ..^ N ) = ( M ... ( N - 1 ) ) ) |
130 |
129
|
eqcomd |
|- ( ph -> ( M ... ( N - 1 ) ) = ( M ..^ N ) ) |
131 |
130
|
adantr |
|- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( M ... ( N - 1 ) ) = ( M ..^ N ) ) |
132 |
127 131
|
eleqtrd |
|- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> k e. ( M ..^ N ) ) |
133 |
|
fzofzp1 |
|- ( k e. ( M ..^ N ) -> ( k + 1 ) e. ( M ... N ) ) |
134 |
132 133
|
syl |
|- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( M ... N ) ) |
135 |
|
simpl |
|- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ph ) |
136 |
135 134
|
jca |
|- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( ph /\ ( k + 1 ) e. ( M ... N ) ) ) |
137 |
|
eleq1 |
|- ( j = ( k + 1 ) -> ( j e. ( M ... N ) <-> ( k + 1 ) e. ( M ... N ) ) ) |
138 |
137
|
anbi2d |
|- ( j = ( k + 1 ) -> ( ( ph /\ j e. ( M ... N ) ) <-> ( ph /\ ( k + 1 ) e. ( M ... N ) ) ) ) |
139 |
|
fveq2 |
|- ( j = ( k + 1 ) -> ( F ` j ) = ( F ` ( k + 1 ) ) ) |
140 |
139
|
eleq1d |
|- ( j = ( k + 1 ) -> ( ( F ` j ) e. RR <-> ( F ` ( k + 1 ) ) e. RR ) ) |
141 |
138 140
|
imbi12d |
|- ( j = ( k + 1 ) -> ( ( ( ph /\ j e. ( M ... N ) ) -> ( F ` j ) e. RR ) <-> ( ( ph /\ ( k + 1 ) e. ( M ... N ) ) -> ( F ` ( k + 1 ) ) e. RR ) ) ) |
142 |
|
eleq1 |
|- ( k = j -> ( k e. ( M ... N ) <-> j e. ( M ... N ) ) ) |
143 |
142
|
anbi2d |
|- ( k = j -> ( ( ph /\ k e. ( M ... N ) ) <-> ( ph /\ j e. ( M ... N ) ) ) ) |
144 |
|
fveq2 |
|- ( k = j -> ( F ` k ) = ( F ` j ) ) |
145 |
144
|
eleq1d |
|- ( k = j -> ( ( F ` k ) e. RR <-> ( F ` j ) e. RR ) ) |
146 |
143 145
|
imbi12d |
|- ( k = j -> ( ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR ) <-> ( ( ph /\ j e. ( M ... N ) ) -> ( F ` j ) e. RR ) ) ) |
147 |
146 1
|
chvarvv |
|- ( ( ph /\ j e. ( M ... N ) ) -> ( F ` j ) e. RR ) |
148 |
141 147
|
vtoclg |
|- ( ( k + 1 ) e. ( M ... N ) -> ( ( ph /\ ( k + 1 ) e. ( M ... N ) ) -> ( F ` ( k + 1 ) ) e. RR ) ) |
149 |
134 136 148
|
sylc |
|- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) e. RR ) |
150 |
126 149
|
syldan |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( F ` ( k + 1 ) ) e. RR ) |
151 |
132 2
|
syldan |
|- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( F ` k ) < ( F ` ( k + 1 ) ) ) |
152 |
126 151
|
syldan |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( F ` k ) < ( F ` ( k + 1 ) ) ) |
153 |
118 150 152
|
ltled |
|- ( ( ph /\ k e. ( ( I + 1 ) ... ( J - 1 ) ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) ) |
154 |
65 89 153
|
monoord |
|- ( ph -> ( F ` ( I + 1 ) ) <_ ( F ` J ) ) |
155 |
13 43 51 59 154
|
ltletrd |
|- ( ph -> ( F ` I ) < ( F ` J ) ) |