Metamath Proof Explorer


Theorem ovolshftlem1

Description: Lemma for ovolshft . (Contributed by Mario Carneiro, 22-Mar-2014)

Ref Expression
Hypotheses ovolshft.1 φ A
ovolshft.2 φ C
ovolshft.3 φ B = x | x C A
ovolshft.4 M = y * | f 2 B ran . f y = sup ran seq 1 + abs f * <
ovolshft.5 S = seq 1 + abs F
ovolshft.6 G = n 1 st F n + C 2 nd F n + C
ovolshft.7 φ F : 2
ovolshft.8 φ A ran . F
Assertion ovolshftlem1 φ sup ran S * < M

Proof

Step Hyp Ref Expression
1 ovolshft.1 φ A
2 ovolshft.2 φ C
3 ovolshft.3 φ B = x | x C A
4 ovolshft.4 M = y * | f 2 B ran . f y = sup ran seq 1 + abs f * <
5 ovolshft.5 S = seq 1 + abs F
6 ovolshft.6 G = n 1 st F n + C 2 nd F n + C
7 ovolshft.7 φ F : 2
8 ovolshft.8 φ A ran . F
9 ovolfcl F : 2 n 1 st F n 2 nd F n 1 st F n 2 nd F n
10 7 9 sylan φ n 1 st F n 2 nd F n 1 st F n 2 nd F n
11 10 simp1d φ n 1 st F n
12 10 simp2d φ n 2 nd F n
13 2 adantr φ n C
14 10 simp3d φ n 1 st F n 2 nd F n
15 11 12 13 14 leadd1dd φ n 1 st F n + C 2 nd F n + C
16 df-br 1 st F n + C 2 nd F n + C 1 st F n + C 2 nd F n + C
17 15 16 sylib φ n 1 st F n + C 2 nd F n + C
18 11 13 readdcld φ n 1 st F n + C
19 12 13 readdcld φ n 2 nd F n + C
20 18 19 opelxpd φ n 1 st F n + C 2 nd F n + C 2
21 17 20 elind φ n 1 st F n + C 2 nd F n + C 2
22 21 6 fmptd φ G : 2
23 eqid abs G = abs G
24 23 ovolfsf G : 2 abs G : 0 +∞
25 ffn abs G : 0 +∞ abs G Fn
26 22 24 25 3syl φ abs G Fn
27 eqid abs F = abs F
28 27 ovolfsf F : 2 abs F : 0 +∞
29 ffn abs F : 0 +∞ abs F Fn
30 7 28 29 3syl φ abs F Fn
31 opex 1 st F n + C 2 nd F n + C V
32 6 fvmpt2 n 1 st F n + C 2 nd F n + C V G n = 1 st F n + C 2 nd F n + C
33 31 32 mpan2 n G n = 1 st F n + C 2 nd F n + C
34 33 fveq2d n 2 nd G n = 2 nd 1 st F n + C 2 nd F n + C
35 ovex 1 st F n + C V
36 ovex 2 nd F n + C V
37 35 36 op2nd 2 nd 1 st F n + C 2 nd F n + C = 2 nd F n + C
38 34 37 eqtrdi n 2 nd G n = 2 nd F n + C
39 33 fveq2d n 1 st G n = 1 st 1 st F n + C 2 nd F n + C
40 35 36 op1st 1 st 1 st F n + C 2 nd F n + C = 1 st F n + C
41 39 40 eqtrdi n 1 st G n = 1 st F n + C
42 38 41 oveq12d n 2 nd G n 1 st G n = 2 nd F n + C - 1 st F n + C
43 42 adantl φ n 2 nd G n 1 st G n = 2 nd F n + C - 1 st F n + C
44 12 recnd φ n 2 nd F n
45 11 recnd φ n 1 st F n
46 13 recnd φ n C
47 44 45 46 pnpcan2d φ n 2 nd F n + C - 1 st F n + C = 2 nd F n 1 st F n
48 43 47 eqtrd φ n 2 nd G n 1 st G n = 2 nd F n 1 st F n
49 23 ovolfsval G : 2 n abs G n = 2 nd G n 1 st G n
50 22 49 sylan φ n abs G n = 2 nd G n 1 st G n
51 27 ovolfsval F : 2 n abs F n = 2 nd F n 1 st F n
52 7 51 sylan φ n abs F n = 2 nd F n 1 st F n
53 48 50 52 3eqtr4d φ n abs G n = abs F n
54 26 30 53 eqfnfvd φ abs G = abs F
55 54 seqeq3d φ seq 1 + abs G = seq 1 + abs F
56 55 5 eqtr4di φ seq 1 + abs G = S
57 56 rneqd φ ran seq 1 + abs G = ran S
58 57 supeq1d φ sup ran seq 1 + abs G * < = sup ran S * <
59 3 eleq2d φ y B y x | x C A
60 oveq1 x = y x C = y C
61 60 eleq1d x = y x C A y C A
62 61 elrab y x | x C A y y C A
63 59 62 bitrdi φ y B y y C A
64 63 biimpa φ y B y y C A
65 breq2 x = y C 1 st F n < x 1 st F n < y C
66 breq1 x = y C x < 2 nd F n y C < 2 nd F n
67 65 66 anbi12d x = y C 1 st F n < x x < 2 nd F n 1 st F n < y C y C < 2 nd F n
68 67 rexbidv x = y C n 1 st F n < x x < 2 nd F n n 1 st F n < y C y C < 2 nd F n
69 ovolfioo A F : 2 A ran . F x A n 1 st F n < x x < 2 nd F n
70 1 7 69 syl2anc φ A ran . F x A n 1 st F n < x x < 2 nd F n
71 8 70 mpbid φ x A n 1 st F n < x x < 2 nd F n
72 71 adantr φ y y C A x A n 1 st F n < x x < 2 nd F n
73 simprr φ y y C A y C A
74 68 72 73 rspcdva φ y y C A n 1 st F n < y C y C < 2 nd F n
75 41 adantl φ y y C A n 1 st G n = 1 st F n + C
76 75 breq1d φ y y C A n 1 st G n < y 1 st F n + C < y
77 11 adantlr φ y y C A n 1 st F n
78 2 ad2antrr φ y y C A n C
79 simplrl φ y y C A n y
80 77 78 79 ltaddsubd φ y y C A n 1 st F n + C < y 1 st F n < y C
81 76 80 bitrd φ y y C A n 1 st G n < y 1 st F n < y C
82 38 adantl φ y y C A n 2 nd G n = 2 nd F n + C
83 82 breq2d φ y y C A n y < 2 nd G n y < 2 nd F n + C
84 12 adantlr φ y y C A n 2 nd F n
85 79 78 84 ltsubaddd φ y y C A n y C < 2 nd F n y < 2 nd F n + C
86 83 85 bitr4d φ y y C A n y < 2 nd G n y C < 2 nd F n
87 81 86 anbi12d φ y y C A n 1 st G n < y y < 2 nd G n 1 st F n < y C y C < 2 nd F n
88 87 rexbidva φ y y C A n 1 st G n < y y < 2 nd G n n 1 st F n < y C y C < 2 nd F n
89 74 88 mpbird φ y y C A n 1 st G n < y y < 2 nd G n
90 64 89 syldan φ y B n 1 st G n < y y < 2 nd G n
91 90 ralrimiva φ y B n 1 st G n < y y < 2 nd G n
92 ssrab2 x | x C A
93 3 92 eqsstrdi φ B
94 ovolfioo B G : 2 B ran . G y B n 1 st G n < y y < 2 nd G n
95 93 22 94 syl2anc φ B ran . G y B n 1 st G n < y y < 2 nd G n
96 91 95 mpbird φ B ran . G
97 eqid seq 1 + abs G = seq 1 + abs G
98 4 97 elovolmr G : 2 B ran . G sup ran seq 1 + abs G * < M
99 22 96 98 syl2anc φ sup ran seq 1 + abs G * < M
100 58 99 eqeltrrd φ sup ran S * < M