Metamath Proof Explorer


Theorem monotuz

Description: A function defined on an upper set of integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014)

Ref Expression
Hypotheses monotuz.1 φ y H F < G
monotuz.2 φ x H C
monotuz.3 H = I
monotuz.4 x = y + 1 C = G
monotuz.5 x = y C = F
monotuz.6 x = A C = D
monotuz.7 x = B C = E
Assertion monotuz φ A H B H A < B D < E

Proof

Step Hyp Ref Expression
1 monotuz.1 φ y H F < G
2 monotuz.2 φ x H C
3 monotuz.3 H = I
4 monotuz.4 x = y + 1 C = G
5 monotuz.5 x = y C = F
6 monotuz.6 x = A C = D
7 monotuz.7 x = B C = E
8 csbeq1 a = b a / x C = b / x C
9 csbeq1 a = A a / x C = A / x C
10 csbeq1 a = B a / x C = B / x C
11 uzssz I
12 zssre
13 11 12 sstri I
14 3 13 eqsstri H
15 nfv x φ a H
16 nfcsb1v _ x a / x C
17 16 nfel1 x a / x C
18 15 17 nfim x φ a H a / x C
19 eleq1 x = a x H a H
20 19 anbi2d x = a φ x H φ a H
21 csbeq1a x = a C = a / x C
22 21 eleq1d x = a C a / x C
23 20 22 imbi12d x = a φ x H C φ a H a / x C
24 18 23 2 chvarfv φ a H a / x C
25 simpl φ a H a < b φ a H
26 25 adantlrr φ a H b H a < b φ a H
27 3 11 eqsstri H
28 simplrl φ a H b H a < b a H
29 27 28 sseldi φ a H b H a < b a
30 simplrr φ a H b H a < b b H
31 27 30 sseldi φ a H b H a < b b
32 simpr φ a H b H a < b a < b
33 csbeq1 c = a + 1 c / x C = a + 1 / x C
34 33 breq2d c = a + 1 a / x C < c / x C a / x C < a + 1 / x C
35 34 imbi2d c = a + 1 φ a H a / x C < c / x C φ a H a / x C < a + 1 / x C
36 csbeq1 c = d c / x C = d / x C
37 36 breq2d c = d a / x C < c / x C a / x C < d / x C
38 37 imbi2d c = d φ a H a / x C < c / x C φ a H a / x C < d / x C
39 csbeq1 c = d + 1 c / x C = d + 1 / x C
40 39 breq2d c = d + 1 a / x C < c / x C a / x C < d + 1 / x C
41 40 imbi2d c = d + 1 φ a H a / x C < c / x C φ a H a / x C < d + 1 / x C
42 csbeq1 c = b c / x C = b / x C
43 42 breq2d c = b a / x C < c / x C a / x C < b / x C
44 43 imbi2d c = b φ a H a / x C < c / x C φ a H a / x C < b / x C
45 eleq1 y = a y H a H
46 45 anbi2d y = a φ y H φ a H
47 vex y V
48 47 5 csbie y / x C = F
49 csbeq1 y = a y / x C = a / x C
50 48 49 eqtr3id y = a F = a / x C
51 ovex y + 1 V
52 51 4 csbie y + 1 / x C = G
53 oveq1 y = a y + 1 = a + 1
54 53 csbeq1d y = a y + 1 / x C = a + 1 / x C
55 52 54 eqtr3id y = a G = a + 1 / x C
56 50 55 breq12d y = a F < G a / x C < a + 1 / x C
57 46 56 imbi12d y = a φ y H F < G φ a H a / x C < a + 1 / x C
58 57 1 vtoclg a φ a H a / x C < a + 1 / x C
59 24 3ad2ant2 a d a < d φ a H a / x C < d / x C a / x C
60 simp2l a d a < d φ a H a / x C < d / x C φ
61 zre a a
62 61 3ad2ant1 a d a < d a
63 zre d d
64 63 3ad2ant2 a d a < d d
65 simp3 a d a < d a < d
66 62 64 65 ltled a d a < d a d
67 66 3ad2ant1 a d a < d φ a H a / x C < d / x C a d
68 simp11 a d a < d φ a H a / x C < d / x C a
69 simp12 a d a < d φ a H a / x C < d / x C d
70 eluz a d d a a d
71 68 69 70 syl2anc a d a < d φ a H a / x C < d / x C d a a d
72 67 71 mpbird a d a < d φ a H a / x C < d / x C d a
73 simp2r a d a < d φ a H a / x C < d / x C a H
74 73 3 eleqtrdi a d a < d φ a H a / x C < d / x C a I
75 uztrn d a a I d I
76 72 74 75 syl2anc a d a < d φ a H a / x C < d / x C d I
77 76 3 eleqtrrdi a d a < d φ a H a / x C < d / x C d H
78 nfv x φ d H
79 nfcsb1v _ x d / x C
80 79 nfel1 x d / x C
81 78 80 nfim x φ d H d / x C
82 eleq1 x = d x H d H
83 82 anbi2d x = d φ x H φ d H
84 csbeq1a x = d C = d / x C
85 84 eleq1d x = d C d / x C
86 83 85 imbi12d x = d φ x H C φ d H d / x C
87 81 86 2 chvarfv φ d H d / x C
88 60 77 87 syl2anc a d a < d φ a H a / x C < d / x C d / x C
89 peano2uz d I d + 1 I
90 76 89 syl a d a < d φ a H a / x C < d / x C d + 1 I
91 90 3 eleqtrrdi a d a < d φ a H a / x C < d / x C d + 1 H
92 nfv x φ d + 1 H
93 nfcsb1v _ x d + 1 / x C
94 93 nfel1 x d + 1 / x C
95 92 94 nfim x φ d + 1 H d + 1 / x C
96 ovex d + 1 V
97 eleq1 x = d + 1 x H d + 1 H
98 97 anbi2d x = d + 1 φ x H φ d + 1 H
99 csbeq1a x = d + 1 C = d + 1 / x C
100 99 eleq1d x = d + 1 C d + 1 / x C
101 98 100 imbi12d x = d + 1 φ x H C φ d + 1 H d + 1 / x C
102 95 96 101 2 vtoclf φ d + 1 H d + 1 / x C
103 60 91 102 syl2anc a d a < d φ a H a / x C < d / x C d + 1 / x C
104 simp3 a d a < d φ a H a / x C < d / x C a / x C < d / x C
105 nfv y φ d H d / x C < d + 1 / x C
106 eleq1 y = d y H d H
107 106 anbi2d y = d φ y H φ d H
108 csbeq1 y = d y / x C = d / x C
109 48 108 eqtr3id y = d F = d / x C
110 oveq1 y = d y + 1 = d + 1
111 110 csbeq1d y = d y + 1 / x C = d + 1 / x C
112 52 111 eqtr3id y = d G = d + 1 / x C
113 109 112 breq12d y = d F < G d / x C < d + 1 / x C
114 107 113 imbi12d y = d φ y H F < G φ d H d / x C < d + 1 / x C
115 105 114 1 chvarfv φ d H d / x C < d + 1 / x C
116 60 77 115 syl2anc a d a < d φ a H a / x C < d / x C d / x C < d + 1 / x C
117 59 88 103 104 116 lttrd a d a < d φ a H a / x C < d / x C a / x C < d + 1 / x C
118 117 3exp a d a < d φ a H a / x C < d / x C a / x C < d + 1 / x C
119 118 a2d a d a < d φ a H a / x C < d / x C φ a H a / x C < d + 1 / x C
120 35 38 41 44 58 119 uzind2 a b a < b φ a H a / x C < b / x C
121 29 31 32 120 syl3anc φ a H b H a < b φ a H a / x C < b / x C
122 26 121 mpd φ a H b H a < b a / x C < b / x C
123 122 ex φ a H b H a < b a / x C < b / x C
124 8 9 10 14 24 123 ltord1 φ A H B H A < B A / x C < B / x C
125 nfcvd A H _ x D
126 125 6 csbiegf A H A / x C = D
127 nfcvd B H _ x E
128 127 7 csbiegf B H B / x C = E
129 126 128 breqan12d A H B H A / x C < B / x C D < E
130 129 adantl φ A H B H A / x C < B / x C D < E
131 124 130 bitrd φ A H B H A < B D < E