Metamath Proof Explorer


Theorem knoppcnlem4

Description: Lemma for knoppcn . (Contributed by Asger C. Ipsen, 4-Apr-2021) (Revised by Asger C. Ipsen, 5-Jul-2021)

Ref Expression
Hypotheses knoppcnlem4.t T = x x + 1 2 x
knoppcnlem4.f F = y n 0 C n T 2 N n y
knoppcnlem4.n φ N
knoppcnlem4.1 φ C
knoppcnlem4.2 φ A
knoppcnlem4.3 φ M 0
Assertion knoppcnlem4 φ F A M m 0 C m M

Proof

Step Hyp Ref Expression
1 knoppcnlem4.t T = x x + 1 2 x
2 knoppcnlem4.f F = y n 0 C n T 2 N n y
3 knoppcnlem4.n φ N
4 knoppcnlem4.1 φ C
5 knoppcnlem4.2 φ A
6 knoppcnlem4.3 φ M 0
7 2 5 6 knoppcnlem1 φ F A M = C M T 2 N M A
8 7 fveq2d φ F A M = C M T 2 N M A
9 4 recnd φ C
10 9 6 expcld φ C M
11 2re 2
12 11 a1i φ 2
13 nnre N N
14 3 13 syl φ N
15 12 14 remulcld φ 2 N
16 15 6 reexpcld φ 2 N M
17 16 5 remulcld φ 2 N M A
18 1 17 dnicld2 φ T 2 N M A
19 18 recnd φ T 2 N M A
20 10 19 absmuld φ C M T 2 N M A = C M T 2 N M A
21 9 6 absexpd φ C M = C M
22 21 oveq1d φ C M T 2 N M A = C M T 2 N M A
23 20 22 eqtrd φ C M T 2 N M A = C M T 2 N M A
24 19 abscld φ T 2 N M A
25 1red φ 1
26 9 abscld φ C
27 26 6 reexpcld φ C M
28 9 absge0d φ 0 C
29 26 6 28 expge0d φ 0 C M
30 1 dnival 2 N M A T 2 N M A = 2 N M A + 1 2 2 N M A
31 17 30 syl φ T 2 N M A = 2 N M A + 1 2 2 N M A
32 31 fveq2d φ T 2 N M A = 2 N M A + 1 2 2 N M A
33 halfre 1 2
34 33 a1i φ 1 2
35 17 34 readdcld φ 2 N M A + 1 2
36 reflcl 2 N M A + 1 2 2 N M A + 1 2
37 35 36 syl φ 2 N M A + 1 2
38 37 17 resubcld φ 2 N M A + 1 2 2 N M A
39 38 recnd φ 2 N M A + 1 2 2 N M A
40 absidm 2 N M A + 1 2 2 N M A 2 N M A + 1 2 2 N M A = 2 N M A + 1 2 2 N M A
41 39 40 syl φ 2 N M A + 1 2 2 N M A = 2 N M A + 1 2 2 N M A
42 32 41 eqtrd φ T 2 N M A = 2 N M A + 1 2 2 N M A
43 31 18 eqeltrrd φ 2 N M A + 1 2 2 N M A
44 rddif 2 N M A 2 N M A + 1 2 2 N M A 1 2
45 17 44 syl φ 2 N M A + 1 2 2 N M A 1 2
46 halflt1 1 2 < 1
47 1re 1
48 33 47 ltlei 1 2 < 1 1 2 1
49 46 48 ax-mp 1 2 1
50 49 a1i φ 1 2 1
51 43 34 25 45 50 letrd φ 2 N M A + 1 2 2 N M A 1
52 42 51 eqbrtrd φ T 2 N M A 1
53 24 25 27 29 52 lemul2ad φ C M T 2 N M A C M 1
54 ax-1rid C M C M 1 = C M
55 27 54 syl φ C M 1 = C M
56 53 55 breqtrd φ C M T 2 N M A C M
57 23 56 eqbrtrd φ C M T 2 N M A C M
58 eqidd φ m 0 C m = m 0 C m
59 oveq2 m = M C m = C M
60 59 adantl φ m = M C m = C M
61 58 60 6 27 fvmptd φ m 0 C m M = C M
62 61 eqcomd φ C M = m 0 C m M
63 57 62 breqtrd φ C M T 2 N M A m 0 C m M
64 8 63 eqbrtrd φ F A M m 0 C m M