Step |
Hyp |
Ref |
Expression |
1 |
|
6re |
|- 6 e. RR |
2 |
1
|
a1i |
|- ( N e. RR -> 6 e. RR ) |
3 |
|
id |
|- ( N e. RR -> N e. RR ) |
4 |
2 3 3
|
leadd2d |
|- ( N e. RR -> ( 6 <_ N <-> ( N + 6 ) <_ ( N + N ) ) ) |
5 |
4
|
biimpa |
|- ( ( N e. RR /\ 6 <_ N ) -> ( N + 6 ) <_ ( N + N ) ) |
6 |
|
recn |
|- ( N e. RR -> N e. CC ) |
7 |
6
|
times2d |
|- ( N e. RR -> ( N x. 2 ) = ( N + N ) ) |
8 |
7
|
adantr |
|- ( ( N e. RR /\ 6 <_ N ) -> ( N x. 2 ) = ( N + N ) ) |
9 |
5 8
|
breqtrrd |
|- ( ( N e. RR /\ 6 <_ N ) -> ( N + 6 ) <_ ( N x. 2 ) ) |
10 |
|
4cn |
|- 4 e. CC |
11 |
10
|
a1i |
|- ( N e. RR -> 4 e. CC ) |
12 |
|
2cn |
|- 2 e. CC |
13 |
12
|
a1i |
|- ( N e. RR -> 2 e. CC ) |
14 |
6 11 13
|
addassd |
|- ( N e. RR -> ( ( N + 4 ) + 2 ) = ( N + ( 4 + 2 ) ) ) |
15 |
|
4p2e6 |
|- ( 4 + 2 ) = 6 |
16 |
15
|
oveq2i |
|- ( N + ( 4 + 2 ) ) = ( N + 6 ) |
17 |
14 16
|
eqtrdi |
|- ( N e. RR -> ( ( N + 4 ) + 2 ) = ( N + 6 ) ) |
18 |
17
|
breq1d |
|- ( N e. RR -> ( ( ( N + 4 ) + 2 ) <_ ( N x. 2 ) <-> ( N + 6 ) <_ ( N x. 2 ) ) ) |
19 |
18
|
adantr |
|- ( ( N e. RR /\ 6 <_ N ) -> ( ( ( N + 4 ) + 2 ) <_ ( N x. 2 ) <-> ( N + 6 ) <_ ( N x. 2 ) ) ) |
20 |
9 19
|
mpbird |
|- ( ( N e. RR /\ 6 <_ N ) -> ( ( N + 4 ) + 2 ) <_ ( N x. 2 ) ) |
21 |
|
4re |
|- 4 e. RR |
22 |
21
|
a1i |
|- ( N e. RR -> 4 e. RR ) |
23 |
|
4ne0 |
|- 4 =/= 0 |
24 |
23
|
a1i |
|- ( N e. RR -> 4 =/= 0 ) |
25 |
3 22 24
|
redivcld |
|- ( N e. RR -> ( N / 4 ) e. RR ) |
26 |
|
peano2re |
|- ( ( N / 4 ) e. RR -> ( ( N / 4 ) + 1 ) e. RR ) |
27 |
25 26
|
syl |
|- ( N e. RR -> ( ( N / 4 ) + 1 ) e. RR ) |
28 |
|
peano2rem |
|- ( N e. RR -> ( N - 1 ) e. RR ) |
29 |
28
|
rehalfcld |
|- ( N e. RR -> ( ( N - 1 ) / 2 ) e. RR ) |
30 |
|
4pos |
|- 0 < 4 |
31 |
21 30
|
pm3.2i |
|- ( 4 e. RR /\ 0 < 4 ) |
32 |
31
|
a1i |
|- ( N e. RR -> ( 4 e. RR /\ 0 < 4 ) ) |
33 |
|
lemul1 |
|- ( ( ( ( N / 4 ) + 1 ) e. RR /\ ( ( N - 1 ) / 2 ) e. RR /\ ( 4 e. RR /\ 0 < 4 ) ) -> ( ( ( N / 4 ) + 1 ) <_ ( ( N - 1 ) / 2 ) <-> ( ( ( N / 4 ) + 1 ) x. 4 ) <_ ( ( ( N - 1 ) / 2 ) x. 4 ) ) ) |
34 |
27 29 32 33
|
syl3anc |
|- ( N e. RR -> ( ( ( N / 4 ) + 1 ) <_ ( ( N - 1 ) / 2 ) <-> ( ( ( N / 4 ) + 1 ) x. 4 ) <_ ( ( ( N - 1 ) / 2 ) x. 4 ) ) ) |
35 |
25
|
recnd |
|- ( N e. RR -> ( N / 4 ) e. CC ) |
36 |
|
1cnd |
|- ( N e. RR -> 1 e. CC ) |
37 |
6 11 24
|
divcan1d |
|- ( N e. RR -> ( ( N / 4 ) x. 4 ) = N ) |
38 |
10
|
mulid2i |
|- ( 1 x. 4 ) = 4 |
39 |
38
|
a1i |
|- ( N e. RR -> ( 1 x. 4 ) = 4 ) |
40 |
37 39
|
oveq12d |
|- ( N e. RR -> ( ( ( N / 4 ) x. 4 ) + ( 1 x. 4 ) ) = ( N + 4 ) ) |
41 |
35 11 36 40
|
joinlmuladdmuld |
|- ( N e. RR -> ( ( ( N / 4 ) + 1 ) x. 4 ) = ( N + 4 ) ) |
42 |
|
2t2e4 |
|- ( 2 x. 2 ) = 4 |
43 |
42
|
eqcomi |
|- 4 = ( 2 x. 2 ) |
44 |
43
|
a1i |
|- ( N e. RR -> 4 = ( 2 x. 2 ) ) |
45 |
44
|
oveq2d |
|- ( N e. RR -> ( ( ( N - 1 ) / 2 ) x. 4 ) = ( ( ( N - 1 ) / 2 ) x. ( 2 x. 2 ) ) ) |
46 |
29
|
recnd |
|- ( N e. RR -> ( ( N - 1 ) / 2 ) e. CC ) |
47 |
|
mulass |
|- ( ( ( ( N - 1 ) / 2 ) e. CC /\ 2 e. CC /\ 2 e. CC ) -> ( ( ( ( N - 1 ) / 2 ) x. 2 ) x. 2 ) = ( ( ( N - 1 ) / 2 ) x. ( 2 x. 2 ) ) ) |
48 |
47
|
eqcomd |
|- ( ( ( ( N - 1 ) / 2 ) e. CC /\ 2 e. CC /\ 2 e. CC ) -> ( ( ( N - 1 ) / 2 ) x. ( 2 x. 2 ) ) = ( ( ( ( N - 1 ) / 2 ) x. 2 ) x. 2 ) ) |
49 |
46 13 13 48
|
syl3anc |
|- ( N e. RR -> ( ( ( N - 1 ) / 2 ) x. ( 2 x. 2 ) ) = ( ( ( ( N - 1 ) / 2 ) x. 2 ) x. 2 ) ) |
50 |
28
|
recnd |
|- ( N e. RR -> ( N - 1 ) e. CC ) |
51 |
|
2ne0 |
|- 2 =/= 0 |
52 |
51
|
a1i |
|- ( N e. RR -> 2 =/= 0 ) |
53 |
50 13 52
|
divcan1d |
|- ( N e. RR -> ( ( ( N - 1 ) / 2 ) x. 2 ) = ( N - 1 ) ) |
54 |
53
|
oveq1d |
|- ( N e. RR -> ( ( ( ( N - 1 ) / 2 ) x. 2 ) x. 2 ) = ( ( N - 1 ) x. 2 ) ) |
55 |
6 36 13
|
subdird |
|- ( N e. RR -> ( ( N - 1 ) x. 2 ) = ( ( N x. 2 ) - ( 1 x. 2 ) ) ) |
56 |
12
|
mulid2i |
|- ( 1 x. 2 ) = 2 |
57 |
56
|
a1i |
|- ( N e. RR -> ( 1 x. 2 ) = 2 ) |
58 |
57
|
oveq2d |
|- ( N e. RR -> ( ( N x. 2 ) - ( 1 x. 2 ) ) = ( ( N x. 2 ) - 2 ) ) |
59 |
54 55 58
|
3eqtrd |
|- ( N e. RR -> ( ( ( ( N - 1 ) / 2 ) x. 2 ) x. 2 ) = ( ( N x. 2 ) - 2 ) ) |
60 |
45 49 59
|
3eqtrd |
|- ( N e. RR -> ( ( ( N - 1 ) / 2 ) x. 4 ) = ( ( N x. 2 ) - 2 ) ) |
61 |
41 60
|
breq12d |
|- ( N e. RR -> ( ( ( ( N / 4 ) + 1 ) x. 4 ) <_ ( ( ( N - 1 ) / 2 ) x. 4 ) <-> ( N + 4 ) <_ ( ( N x. 2 ) - 2 ) ) ) |
62 |
3 22
|
readdcld |
|- ( N e. RR -> ( N + 4 ) e. RR ) |
63 |
|
2re |
|- 2 e. RR |
64 |
63
|
a1i |
|- ( N e. RR -> 2 e. RR ) |
65 |
3 64
|
remulcld |
|- ( N e. RR -> ( N x. 2 ) e. RR ) |
66 |
|
leaddsub |
|- ( ( ( N + 4 ) e. RR /\ 2 e. RR /\ ( N x. 2 ) e. RR ) -> ( ( ( N + 4 ) + 2 ) <_ ( N x. 2 ) <-> ( N + 4 ) <_ ( ( N x. 2 ) - 2 ) ) ) |
67 |
66
|
bicomd |
|- ( ( ( N + 4 ) e. RR /\ 2 e. RR /\ ( N x. 2 ) e. RR ) -> ( ( N + 4 ) <_ ( ( N x. 2 ) - 2 ) <-> ( ( N + 4 ) + 2 ) <_ ( N x. 2 ) ) ) |
68 |
62 64 65 67
|
syl3anc |
|- ( N e. RR -> ( ( N + 4 ) <_ ( ( N x. 2 ) - 2 ) <-> ( ( N + 4 ) + 2 ) <_ ( N x. 2 ) ) ) |
69 |
34 61 68
|
3bitrd |
|- ( N e. RR -> ( ( ( N / 4 ) + 1 ) <_ ( ( N - 1 ) / 2 ) <-> ( ( N + 4 ) + 2 ) <_ ( N x. 2 ) ) ) |
70 |
69
|
adantr |
|- ( ( N e. RR /\ 6 <_ N ) -> ( ( ( N / 4 ) + 1 ) <_ ( ( N - 1 ) / 2 ) <-> ( ( N + 4 ) + 2 ) <_ ( N x. 2 ) ) ) |
71 |
20 70
|
mpbird |
|- ( ( N e. RR /\ 6 <_ N ) -> ( ( N / 4 ) + 1 ) <_ ( ( N - 1 ) / 2 ) ) |