Step | Hyp | Ref
| Expression |
1 | | nnex 12166 |
. . . 4
β’ β
β V |
2 | | inss1 4193 |
. . . . 5
β’ (β
β© π) β
β |
3 | | prmssnn 16559 |
. . . . 5
β’ β
β β |
4 | 2, 3 | sstri 3958 |
. . . 4
β’ (β
β© π) β
β |
5 | | ssdomg 8947 |
. . . 4
β’ (β
β V β ((β β© π) β β β (β β©
π) βΌ
β)) |
6 | 1, 4, 5 | mp2 9 |
. . 3
β’ (β
β© π) βΌ
β |
7 | 6 | a1i 11 |
. 2
β’ (π β (β β© π) βΌ
β) |
8 | | logno1 26007 |
. . . 4
β’ Β¬
(π₯ β
β+ β¦ (logβπ₯)) β π(1) |
9 | | rpvmasum.a |
. . . . . . . . . . 11
β’ (π β π β β) |
10 | 9 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ (β β© π) β Fin) β π β
β) |
11 | 10 | phicld 16651 |
. . . . . . . . 9
β’ ((π β§ (β β© π) β Fin) β
(Οβπ) β
β) |
12 | 11 | nnred 12175 |
. . . . . . . 8
β’ ((π β§ (β β© π) β Fin) β
(Οβπ) β
β) |
13 | 12 | adantr 482 |
. . . . . . 7
β’ (((π β§ (β β© π) β Fin) β§ π₯ β β+)
β (Οβπ)
β β) |
14 | | simpr 486 |
. . . . . . . . . 10
β’ ((π β§ (β β© π) β Fin) β (β
β© π) β
Fin) |
15 | | inss2 4194 |
. . . . . . . . . 10
β’
((1...(ββπ₯)) β© (β β© π)) β (β β© π) |
16 | | ssfi 9124 |
. . . . . . . . . 10
β’
(((β β© π)
β Fin β§ ((1...(ββπ₯)) β© (β β© π)) β (β β© π)) β ((1...(ββπ₯)) β© (β β© π)) β Fin) |
17 | 14, 15, 16 | sylancl 587 |
. . . . . . . . 9
β’ ((π β§ (β β© π) β Fin) β
((1...(ββπ₯))
β© (β β© π))
β Fin) |
18 | | elinel2 4161 |
. . . . . . . . . 10
β’ (π β
((1...(ββπ₯))
β© (β β© π))
β π β (β
β© π)) |
19 | | simpr 486 |
. . . . . . . . . . . . . 14
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β π β (β β© π)) |
20 | 4, 19 | sselid 3947 |
. . . . . . . . . . . . 13
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β π β β) |
21 | 20 | nnrpd 12962 |
. . . . . . . . . . . 12
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β π β β+) |
22 | | relogcl 25947 |
. . . . . . . . . . . 12
β’ (π β β+
β (logβπ) β
β) |
23 | 21, 22 | syl 17 |
. . . . . . . . . . 11
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β (logβπ) β
β) |
24 | 23, 20 | nndivred 12214 |
. . . . . . . . . 10
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β ((logβπ) / π) β β) |
25 | 18, 24 | sylan2 594 |
. . . . . . . . 9
β’ (((π β§ (β β© π) β Fin) β§ π β
((1...(ββπ₯))
β© (β β© π)))
β ((logβπ) /
π) β
β) |
26 | 17, 25 | fsumrecl 15626 |
. . . . . . . 8
β’ ((π β§ (β β© π) β Fin) β
Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π) β β) |
27 | 26 | adantr 482 |
. . . . . . 7
β’ (((π β§ (β β© π) β Fin) β§ π₯ β β+)
β Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π) β β) |
28 | | rpssre 12929 |
. . . . . . . 8
β’
β+ β β |
29 | 12 | recnd 11190 |
. . . . . . . 8
β’ ((π β§ (β β© π) β Fin) β
(Οβπ) β
β) |
30 | | o1const 15509 |
. . . . . . . 8
β’
((β+ β β β§ (Οβπ) β β) β (π₯ β β+
β¦ (Οβπ))
β π(1)) |
31 | 28, 29, 30 | sylancr 588 |
. . . . . . 7
β’ ((π β§ (β β© π) β Fin) β (π₯ β β+
β¦ (Οβπ))
β π(1)) |
32 | 28 | a1i 11 |
. . . . . . . . 9
β’ ((π β§ (β β© π) β Fin) β
β+ β β) |
33 | | 1red 11163 |
. . . . . . . . 9
β’ ((π β§ (β β© π) β Fin) β 1 β
β) |
34 | 14, 24 | fsumrecl 15626 |
. . . . . . . . 9
β’ ((π β§ (β β© π) β Fin) β
Ξ£π β (β
β© π)((logβπ) / π) β β) |
35 | | log1 25957 |
. . . . . . . . . . . . 13
β’
(logβ1) = 0 |
36 | 20 | nnge1d 12208 |
. . . . . . . . . . . . . 14
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β 1 β€ π) |
37 | | 1rp 12926 |
. . . . . . . . . . . . . . 15
β’ 1 β
β+ |
38 | | logleb 25974 |
. . . . . . . . . . . . . . 15
β’ ((1
β β+ β§ π β β+) β (1 β€
π β (logβ1) β€
(logβπ))) |
39 | 37, 21, 38 | sylancr 588 |
. . . . . . . . . . . . . 14
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β (1 β€ π β (logβ1) β€
(logβπ))) |
40 | 36, 39 | mpbid 231 |
. . . . . . . . . . . . 13
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β (logβ1) β€
(logβπ)) |
41 | 35, 40 | eqbrtrrid 5146 |
. . . . . . . . . . . 12
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β 0 β€
(logβπ)) |
42 | 23, 21, 41 | divge0d 13004 |
. . . . . . . . . . 11
β’ (((π β§ (β β© π) β Fin) β§ π β (β β© π)) β 0 β€
((logβπ) / π)) |
43 | 15 | a1i 11 |
. . . . . . . . . . 11
β’ ((π β§ (β β© π) β Fin) β
((1...(ββπ₯))
β© (β β© π))
β (β β© π)) |
44 | 14, 24, 42, 43 | fsumless 15688 |
. . . . . . . . . 10
β’ ((π β§ (β β© π) β Fin) β
Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π) β€ Ξ£π β (β β© π)((logβπ) / π)) |
45 | 44 | adantr 482 |
. . . . . . . . 9
β’ (((π β§ (β β© π) β Fin) β§ (π₯ β β+
β§ 1 β€ π₯)) β
Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π) β€ Ξ£π β (β β© π)((logβπ) / π)) |
46 | 32, 27, 33, 34, 45 | ello1d 15412 |
. . . . . . . 8
β’ ((π β§ (β β© π) β Fin) β (π₯ β β+
β¦ Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) β β€π(1)) |
47 | | 0red 11165 |
. . . . . . . . 9
β’ ((π β§ (β β© π) β Fin) β 0 β
β) |
48 | 18, 42 | sylan2 594 |
. . . . . . . . . . 11
β’ (((π β§ (β β© π) β Fin) β§ π β
((1...(ββπ₯))
β© (β β© π)))
β 0 β€ ((logβπ) / π)) |
49 | 17, 25, 48 | fsumge0 15687 |
. . . . . . . . . 10
β’ ((π β§ (β β© π) β Fin) β 0 β€
Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) |
50 | 49 | adantr 482 |
. . . . . . . . 9
β’ (((π β§ (β β© π) β Fin) β§ π₯ β β+)
β 0 β€ Ξ£π
β ((1...(ββπ₯)) β© (β β© π))((logβπ) / π)) |
51 | 27, 47, 50 | o1lo12 15427 |
. . . . . . . 8
β’ ((π β§ (β β© π) β Fin) β ((π₯ β β+
β¦ Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) β π(1) β (π₯ β β+
β¦ Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) β β€π(1))) |
52 | 46, 51 | mpbird 257 |
. . . . . . 7
β’ ((π β§ (β β© π) β Fin) β (π₯ β β+
β¦ Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) β π(1)) |
53 | 13, 27, 31, 52 | o1mul2 15514 |
. . . . . 6
β’ ((π β§ (β β© π) β Fin) β (π₯ β β+
β¦ ((Οβπ)
Β· Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π))) β π(1)) |
54 | 12, 26 | remulcld 11192 |
. . . . . . . . 9
β’ ((π β§ (β β© π) β Fin) β
((Οβπ) Β·
Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) β β) |
55 | 54 | recnd 11190 |
. . . . . . . 8
β’ ((π β§ (β β© π) β Fin) β
((Οβπ) Β·
Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) β β) |
56 | 55 | adantr 482 |
. . . . . . 7
β’ (((π β§ (β β© π) β Fin) β§ π₯ β β+)
β ((Οβπ)
Β· Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) β β) |
57 | | relogcl 25947 |
. . . . . . . . 9
β’ (π₯ β β+
β (logβπ₯) β
β) |
58 | 57 | adantl 483 |
. . . . . . . 8
β’ (((π β§ (β β© π) β Fin) β§ π₯ β β+)
β (logβπ₯) β
β) |
59 | 58 | recnd 11190 |
. . . . . . 7
β’ (((π β§ (β β© π) β Fin) β§ π₯ β β+)
β (logβπ₯) β
β) |
60 | | rpvmasum.z |
. . . . . . . . 9
β’ π =
(β€/nβ€βπ) |
61 | | rpvmasum.l |
. . . . . . . . 9
β’ πΏ = (β€RHomβπ) |
62 | | rpvmasum.u |
. . . . . . . . 9
β’ π = (Unitβπ) |
63 | | rpvmasum.b |
. . . . . . . . 9
β’ (π β π΄ β π) |
64 | | rpvmasum.t |
. . . . . . . . 9
β’ π = (β‘πΏ β {π΄}) |
65 | 60, 61, 9, 62, 63, 64 | rplogsum 26891 |
. . . . . . . 8
β’ (π β (π₯ β β+ β¦
(((Οβπ) Β·
Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) β (logβπ₯))) β π(1)) |
66 | 65 | adantr 482 |
. . . . . . 7
β’ ((π β§ (β β© π) β Fin) β (π₯ β β+
β¦ (((Οβπ)
Β· Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π)) β (logβπ₯))) β π(1)) |
67 | 56, 59, 66 | o1dif 15519 |
. . . . . 6
β’ ((π β§ (β β© π) β Fin) β ((π₯ β β+
β¦ ((Οβπ)
Β· Ξ£π β
((1...(ββπ₯))
β© (β β© π))((logβπ) / π))) β π(1) β (π₯ β β+
β¦ (logβπ₯))
β π(1))) |
68 | 53, 67 | mpbid 231 |
. . . . 5
β’ ((π β§ (β β© π) β Fin) β (π₯ β β+
β¦ (logβπ₯))
β π(1)) |
69 | 68 | ex 414 |
. . . 4
β’ (π β ((β β© π) β Fin β (π₯ β β+
β¦ (logβπ₯))
β π(1))) |
70 | 8, 69 | mtoi 198 |
. . 3
β’ (π β Β¬ (β β© π) β Fin) |
71 | | nnenom 13892 |
. . . . 5
β’ β
β Ο |
72 | | sdomentr 9062 |
. . . . 5
β’
(((β β© π)
βΊ β β§ β β Ο) β (β β© π) βΊ
Ο) |
73 | 71, 72 | mpan2 690 |
. . . 4
β’ ((β
β© π) βΊ β
β (β β© π)
βΊ Ο) |
74 | | isfinite2 9252 |
. . . 4
β’ ((β
β© π) βΊ Ο
β (β β© π)
β Fin) |
75 | 73, 74 | syl 17 |
. . 3
β’ ((β
β© π) βΊ β
β (β β© π)
β Fin) |
76 | 70, 75 | nsyl 140 |
. 2
β’ (π β Β¬ (β β© π) βΊ
β) |
77 | | bren2 8930 |
. 2
β’ ((β
β© π) β β
β ((β β© π)
βΌ β β§ Β¬ (β β© π) βΊ β)) |
78 | 7, 76, 77 | sylanbrc 584 |
1
β’ (π β (β β© π) β
β) |