Step | Hyp | Ref
| Expression |
1 | | nnre 12220 |
. . . . . . . . . . . 12
β’ (π β β β π β
β) |
2 | 1 | ad2antlr 724 |
. . . . . . . . . . 11
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ (β«2βπΉ) = +β) β π β β) |
3 | 2 | ltpnfd 13104 |
. . . . . . . . . 10
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ (β«2βπΉ) = +β) β π < +β) |
4 | | iftrue 4529 |
. . . . . . . . . . 11
β’
((β«2βπΉ) = +β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) = π) |
5 | 4 | adantl 481 |
. . . . . . . . . 10
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ (β«2βπΉ) = +β) β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) = π) |
6 | | simpr 484 |
. . . . . . . . . 10
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ (β«2βπΉ) = +β) β
(β«2βπΉ)
= +β) |
7 | 3, 5, 6 | 3brtr4d 5173 |
. . . . . . . . 9
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ (β«2βπΉ) = +β) β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«2βπΉ)) |
8 | | iffalse 4532 |
. . . . . . . . . . 11
β’ (Β¬
(β«2βπΉ)
= +β β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) = ((β«2βπΉ) β (1 / π))) |
9 | 8 | adantl 481 |
. . . . . . . . . 10
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ Β¬ (β«2βπΉ) = +β) β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) = ((β«2βπΉ) β (1 / π))) |
10 | | itg2cl 25612 |
. . . . . . . . . . . . . . 15
β’ (πΉ:ββΆ(0[,]+β)
β (β«2βπΉ) β
β*) |
11 | | xrrebnd 13150 |
. . . . . . . . . . . . . . 15
β’
((β«2βπΉ) β β* β
((β«2βπΉ) β β β (-β <
(β«2βπΉ)
β§ (β«2βπΉ) < +β))) |
12 | 10, 11 | syl 17 |
. . . . . . . . . . . . . 14
β’ (πΉ:ββΆ(0[,]+β)
β ((β«2βπΉ) β β β (-β <
(β«2βπΉ)
β§ (β«2βπΉ) < +β))) |
13 | | itg2ge0 25615 |
. . . . . . . . . . . . . . . 16
β’ (πΉ:ββΆ(0[,]+β)
β 0 β€ (β«2βπΉ)) |
14 | | mnflt0 13108 |
. . . . . . . . . . . . . . . . 17
β’ -β
< 0 |
15 | | mnfxr 11272 |
. . . . . . . . . . . . . . . . . 18
β’ -β
β β* |
16 | | 0xr 11262 |
. . . . . . . . . . . . . . . . . 18
β’ 0 β
β* |
17 | | xrltletr 13139 |
. . . . . . . . . . . . . . . . . 18
β’
((-β β β* β§ 0 β β*
β§ (β«2βπΉ) β β*) β
((-β < 0 β§ 0 β€ (β«2βπΉ)) β -β <
(β«2βπΉ))) |
18 | 15, 16, 10, 17 | mp3an12i 1461 |
. . . . . . . . . . . . . . . . 17
β’ (πΉ:ββΆ(0[,]+β)
β ((-β < 0 β§ 0 β€ (β«2βπΉ)) β -β <
(β«2βπΉ))) |
19 | 14, 18 | mpani 693 |
. . . . . . . . . . . . . . . 16
β’ (πΉ:ββΆ(0[,]+β)
β (0 β€ (β«2βπΉ) β -β <
(β«2βπΉ))) |
20 | 13, 19 | mpd 15 |
. . . . . . . . . . . . . . 15
β’ (πΉ:ββΆ(0[,]+β)
β -β < (β«2βπΉ)) |
21 | 20 | biantrurd 532 |
. . . . . . . . . . . . . 14
β’ (πΉ:ββΆ(0[,]+β)
β ((β«2βπΉ) < +β β (-β <
(β«2βπΉ)
β§ (β«2βπΉ) < +β))) |
22 | | nltpnft 13146 |
. . . . . . . . . . . . . . . 16
β’
((β«2βπΉ) β β* β
((β«2βπΉ) = +β β Β¬
(β«2βπΉ)
< +β)) |
23 | 10, 22 | syl 17 |
. . . . . . . . . . . . . . 15
β’ (πΉ:ββΆ(0[,]+β)
β ((β«2βπΉ) = +β β Β¬
(β«2βπΉ)
< +β)) |
24 | 23 | con2bid 354 |
. . . . . . . . . . . . . 14
β’ (πΉ:ββΆ(0[,]+β)
β ((β«2βπΉ) < +β β Β¬
(β«2βπΉ)
= +β)) |
25 | 12, 21, 24 | 3bitr2rd 308 |
. . . . . . . . . . . . 13
β’ (πΉ:ββΆ(0[,]+β)
β (Β¬ (β«2βπΉ) = +β β
(β«2βπΉ)
β β)) |
26 | 25 | biimpa 476 |
. . . . . . . . . . . 12
β’ ((πΉ:ββΆ(0[,]+β)
β§ Β¬ (β«2βπΉ) = +β) β
(β«2βπΉ)
β β) |
27 | 26 | adantlr 712 |
. . . . . . . . . . 11
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ Β¬ (β«2βπΉ) = +β) β
(β«2βπΉ)
β β) |
28 | | nnrp 12988 |
. . . . . . . . . . . . 13
β’ (π β β β π β
β+) |
29 | 28 | rpreccld 13029 |
. . . . . . . . . . . 12
β’ (π β β β (1 /
π) β
β+) |
30 | 29 | ad2antlr 724 |
. . . . . . . . . . 11
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ Β¬ (β«2βπΉ) = +β) β (1 / π) β
β+) |
31 | 27, 30 | ltsubrpd 13051 |
. . . . . . . . . 10
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ Β¬ (β«2βπΉ) = +β) β
((β«2βπΉ) β (1 / π)) < (β«2βπΉ)) |
32 | 9, 31 | eqbrtrd 5163 |
. . . . . . . . 9
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ Β¬ (β«2βπΉ) = +β) β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«2βπΉ)) |
33 | 7, 32 | pm2.61dan 810 |
. . . . . . . 8
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«2βπΉ)) |
34 | | nnrecre 12255 |
. . . . . . . . . . . . 13
β’ (π β β β (1 /
π) β
β) |
35 | 34 | ad2antlr 724 |
. . . . . . . . . . . 12
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ Β¬ (β«2βπΉ) = +β) β (1 / π) β β) |
36 | 27, 35 | resubcld 11643 |
. . . . . . . . . . 11
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ Β¬ (β«2βπΉ) = +β) β
((β«2βπΉ) β (1 / π)) β β) |
37 | 2, 36 | ifclda 4558 |
. . . . . . . . . 10
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β) |
38 | 37 | rexrd 11265 |
. . . . . . . . 9
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β
β*) |
39 | 10 | adantr 480 |
. . . . . . . . 9
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β (β«2βπΉ) β
β*) |
40 | | xrltnle 11282 |
. . . . . . . . 9
β’
((if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β* β§
(β«2βπΉ)
β β*) β (if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«2βπΉ) β Β¬
(β«2βπΉ)
β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
41 | 38, 39, 40 | syl2anc 583 |
. . . . . . . 8
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β (if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«2βπΉ) β Β¬
(β«2βπΉ)
β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
42 | 33, 41 | mpbid 231 |
. . . . . . 7
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β Β¬ (β«2βπΉ) β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))) |
43 | | itg2leub 25614 |
. . . . . . . 8
β’ ((πΉ:ββΆ(0[,]+β)
β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β*) β
((β«2βπΉ) β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β βπ β dom β«1(π βr β€ πΉ β
(β«1βπ)
β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))))) |
44 | 38, 43 | syldan 590 |
. . . . . . 7
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β ((β«2βπΉ) β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β βπ β dom β«1(π βr β€ πΉ β
(β«1βπ)
β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))))) |
45 | 42, 44 | mtbid 324 |
. . . . . 6
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β Β¬ βπ
β dom β«1(π βr β€ πΉ β (β«1βπ) β€
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
46 | | rexanali 3096 |
. . . . . 6
β’
(βπ β dom
β«1(π
βr β€ πΉ
β§ Β¬ (β«1βπ) β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))) β Β¬ βπ β dom β«1(π βr β€ πΉ β
(β«1βπ)
β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
47 | 45, 46 | sylibr 233 |
. . . . 5
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β βπ β dom
β«1(π
βr β€ πΉ
β§ Β¬ (β«1βπ) β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
48 | | itg1cl 25564 |
. . . . . . . 8
β’ (π β dom β«1
β (β«1βπ) β β) |
49 | | ltnle 11294 |
. . . . . . . 8
β’
((if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β β§
(β«1βπ)
β β) β (if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1βπ) β Β¬
(β«1βπ)
β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
50 | 37, 48, 49 | syl2an 595 |
. . . . . . 7
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ π β dom
β«1) β (if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1βπ) β Β¬
(β«1βπ)
β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
51 | 50 | anbi2d 628 |
. . . . . 6
β’ (((πΉ:ββΆ(0[,]+β)
β§ π β β)
β§ π β dom
β«1) β ((π βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1βπ)) β (π βr β€ πΉ β§ Β¬ (β«1βπ) β€
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))))) |
52 | 51 | rexbidva 3170 |
. . . . 5
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β (βπ β dom
β«1(π
βr β€ πΉ
β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1βπ)) β βπ β dom
β«1(π
βr β€ πΉ
β§ Β¬ (β«1βπ) β€ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))))) |
53 | 47, 52 | mpbird 257 |
. . . 4
β’ ((πΉ:ββΆ(0[,]+β)
β§ π β β)
β βπ β dom
β«1(π
βr β€ πΉ
β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1βπ))) |
54 | 53 | ralrimiva 3140 |
. . 3
β’ (πΉ:ββΆ(0[,]+β)
β βπ β
β βπ β dom
β«1(π
βr β€ πΉ
β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1βπ))) |
55 | | ovex 7437 |
. . . . 5
β’ (β
βm β) β V |
56 | | i1ff 25555 |
. . . . . . 7
β’ (π₯ β dom β«1
β π₯:ββΆβ) |
57 | | reex 11200 |
. . . . . . . 8
β’ β
β V |
58 | 57, 57 | elmap 8864 |
. . . . . . 7
β’ (π₯ β (β
βm β) β π₯:ββΆβ) |
59 | 56, 58 | sylibr 233 |
. . . . . 6
β’ (π₯ β dom β«1
β π₯ β (β
βm β)) |
60 | 59 | ssriv 3981 |
. . . . 5
β’ dom
β«1 β (β βm β) |
61 | 55, 60 | ssexi 5315 |
. . . 4
β’ dom
β«1 β V |
62 | | nnenom 13948 |
. . . 4
β’ β
β Ο |
63 | | breq1 5144 |
. . . . 5
β’ (π = (πβπ) β (π βr β€ πΉ β (πβπ) βr β€ πΉ)) |
64 | | fveq2 6884 |
. . . . . 6
β’ (π = (πβπ) β (β«1βπ) =
(β«1β(πβπ))) |
65 | 64 | breq2d 5153 |
. . . . 5
β’ (π = (πβπ) β (if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1βπ) β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ)))) |
66 | 63, 65 | anbi12d 630 |
. . . 4
β’ (π = (πβπ) β ((π βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1βπ)) β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) |
67 | 61, 62, 66 | axcc4 10433 |
. . 3
β’
(βπ β
β βπ β dom
β«1(π
βr β€ πΉ
β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1βπ)) β βπ(π:ββΆdom β«1 β§
βπ β β
((πβπ) βr β€ πΉ β§
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) |
68 | 54, 67 | syl 17 |
. 2
β’ (πΉ:ββΆ(0[,]+β)
β βπ(π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) |
69 | | simprl 768 |
. . . . 5
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β π:ββΆdom
β«1) |
70 | | simpl 482 |
. . . . . . 7
β’ (((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))) β (πβπ) βr β€ πΉ) |
71 | 70 | ralimi 3077 |
. . . . . 6
β’
(βπ β
β ((πβπ) βr β€ πΉ β§
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))) β βπ β β (πβπ) βr β€ πΉ) |
72 | 71 | ad2antll 726 |
. . . . 5
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β βπ β β (πβπ) βr β€ πΉ) |
73 | 10 | adantr 480 |
. . . . . 6
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β (β«2βπΉ) β
β*) |
74 | | ffvelcdm 7076 |
. . . . . . . . . . . 12
β’ ((π:ββΆdom
β«1 β§ π
β β) β (πβπ) β dom
β«1) |
75 | | itg1cl 25564 |
. . . . . . . . . . . 12
β’ ((πβπ) β dom β«1 β
(β«1β(πβπ)) β β) |
76 | 74, 75 | syl 17 |
. . . . . . . . . . 11
β’ ((π:ββΆdom
β«1 β§ π
β β) β (β«1β(πβπ)) β β) |
77 | 76 | fmpttd 7109 |
. . . . . . . . . 10
β’ (π:ββΆdom
β«1 β (π
β β β¦ (β«1β(πβπ))):ββΆβ) |
78 | 77 | ad2antrl 725 |
. . . . . . . . 9
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β (π β β β¦
(β«1β(πβπ))):ββΆβ) |
79 | 78 | frnd 6718 |
. . . . . . . 8
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β ran (π β β β¦
(β«1β(πβπ))) β β) |
80 | | ressxr 11259 |
. . . . . . . 8
β’ β
β β* |
81 | 79, 80 | sstrdi 3989 |
. . . . . . 7
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β ran (π β β β¦
(β«1β(πβπ))) β
β*) |
82 | | supxrcl 13297 |
. . . . . . 7
β’ (ran
(π β β β¦
(β«1β(πβπ))) β β* β
sup(ran (π β β
β¦ (β«1β(πβπ))), β*, < ) β
β*) |
83 | 81, 82 | syl 17 |
. . . . . 6
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
β*) |
84 | 38 | adantlr 712 |
. . . . . . . . . . . . 13
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β
β*) |
85 | 76 | adantll 711 |
. . . . . . . . . . . . . 14
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β (β«1β(πβπ)) β β) |
86 | 85 | rexrd 11265 |
. . . . . . . . . . . . 13
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β (β«1β(πβπ)) β
β*) |
87 | | xrltle 13131 |
. . . . . . . . . . . . 13
β’
((if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β* β§
(β«1β(πβπ)) β β*) β
(if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ)) β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ (β«1β(πβπ)))) |
88 | 84, 86, 87 | syl2anc 583 |
. . . . . . . . . . . 12
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β (if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ)) β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ (β«1β(πβπ)))) |
89 | | 2fveq3 6889 |
. . . . . . . . . . . . . . . . 17
β’ (π = π β (β«1β(πβπ)) = (β«1β(πβπ))) |
90 | 89 | cbvmptv 5254 |
. . . . . . . . . . . . . . . 16
β’ (π β β β¦
(β«1β(πβπ))) = (π β β β¦
(β«1β(πβπ))) |
91 | 90 | rneqi 5929 |
. . . . . . . . . . . . . . 15
β’ ran
(π β β β¦
(β«1β(πβπ))) = ran (π β β β¦
(β«1β(πβπ))) |
92 | 77 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
β’ ((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β (π β β β¦
(β«1β(πβπ))):ββΆβ) |
93 | 92 | frnd 6718 |
. . . . . . . . . . . . . . . . 17
β’ ((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β ran (π β β β¦
(β«1β(πβπ))) β β) |
94 | 93, 80 | sstrdi 3989 |
. . . . . . . . . . . . . . . 16
β’ ((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β ran (π β β β¦
(β«1β(πβπ))) β
β*) |
95 | 94 | adantr 480 |
. . . . . . . . . . . . . . 15
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β ran (π β β β¦
(β«1β(πβπ))) β
β*) |
96 | 91, 95 | eqsstrrid 4026 |
. . . . . . . . . . . . . 14
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β ran (π β β β¦
(β«1β(πβπ))) β
β*) |
97 | | 2fveq3 6889 |
. . . . . . . . . . . . . . . . 17
β’ (π = π β (β«1β(πβπ)) = (β«1β(πβπ))) |
98 | | eqid 2726 |
. . . . . . . . . . . . . . . . 17
β’ (π β β β¦
(β«1β(πβπ))) = (π β β β¦
(β«1β(πβπ))) |
99 | | fvex 6897 |
. . . . . . . . . . . . . . . . 17
β’
(β«1β(πβπ)) β V |
100 | 97, 98, 99 | fvmpt 6991 |
. . . . . . . . . . . . . . . 16
β’ (π β β β ((π β β β¦
(β«1β(πβπ)))βπ) = (β«1β(πβπ))) |
101 | | fvex 6897 |
. . . . . . . . . . . . . . . . . 18
β’
(β«1β(πβπ)) β V |
102 | 101, 98 | fnmpti 6686 |
. . . . . . . . . . . . . . . . 17
β’ (π β β β¦
(β«1β(πβπ))) Fn β |
103 | | fnfvelrn 7075 |
. . . . . . . . . . . . . . . . 17
β’ (((π β β β¦
(β«1β(πβπ))) Fn β β§ π β β) β ((π β β β¦
(β«1β(πβπ)))βπ) β ran (π β β β¦
(β«1β(πβπ)))) |
104 | 102, 103 | mpan 687 |
. . . . . . . . . . . . . . . 16
β’ (π β β β ((π β β β¦
(β«1β(πβπ)))βπ) β ran (π β β β¦
(β«1β(πβπ)))) |
105 | 100, 104 | eqeltrrd 2828 |
. . . . . . . . . . . . . . 15
β’ (π β β β
(β«1β(πβπ)) β ran (π β β β¦
(β«1β(πβπ)))) |
106 | 105 | adantl 481 |
. . . . . . . . . . . . . 14
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β (β«1β(πβπ)) β ran (π β β β¦
(β«1β(πβπ)))) |
107 | | supxrub 13306 |
. . . . . . . . . . . . . 14
β’ ((ran
(π β β β¦
(β«1β(πβπ))) β β* β§
(β«1β(πβπ)) β ran (π β β β¦
(β«1β(πβπ)))) β (β«1β(πβπ)) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
)) |
108 | 96, 106, 107 | syl2anc 583 |
. . . . . . . . . . . . 13
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β (β«1β(πβπ)) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
)) |
109 | 91 | supeq1i 9441 |
. . . . . . . . . . . . . . 15
β’ sup(ran
(π β β β¦
(β«1β(πβπ))), β*, < ) = sup(ran
(π β β β¦
(β«1β(πβπ))), β*, <
) |
110 | 95, 82 | syl 17 |
. . . . . . . . . . . . . . 15
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
β*) |
111 | 109, 110 | eqeltrrid 2832 |
. . . . . . . . . . . . . 14
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
β*) |
112 | | xrletr 13140 |
. . . . . . . . . . . . . 14
β’
((if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β* β§
(β«1β(πβπ)) β β* β§ sup(ran
(π β β β¦
(β«1β(πβπ))), β*, < ) β
β*) β ((if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ (β«1β(πβπ)) β§ (β«1β(πβπ)) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, < )) β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
113 | 84, 86, 111, 112 | syl3anc 1368 |
. . . . . . . . . . . . 13
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β ((if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ (β«1β(πβπ)) β§ (β«1β(πβπ)) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, < )) β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
114 | 108, 113 | mpan2d 691 |
. . . . . . . . . . . 12
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β (if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ (β«1β(πβπ)) β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
115 | 88, 114 | syld 47 |
. . . . . . . . . . 11
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β (if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ)) β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
116 | 115 | adantld 490 |
. . . . . . . . . 10
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β (((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))) β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
117 | 116 | ralimdva 3161 |
. . . . . . . . 9
β’ ((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β (βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))) β βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
118 | 117 | impr 454 |
. . . . . . . 8
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
)) |
119 | | breq2 5145 |
. . . . . . . . . . 11
β’ (π₯ = sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
(if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
120 | 119 | ralbidv 3171 |
. . . . . . . . . 10
β’ (π₯ = sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
(βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
121 | | breq2 5145 |
. . . . . . . . . 10
β’ (π₯ = sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
((β«2βπΉ) β€ π₯ β (β«2βπΉ) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
122 | 120, 121 | imbi12d 344 |
. . . . . . . . 9
β’ (π₯ = sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
((βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β (β«2βπΉ) β€ π₯) β (βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
(β«2βπΉ)
β€ sup(ran (π β
β β¦ (β«1β(πβπ))), β*, <
)))) |
123 | | elxr 13099 |
. . . . . . . . . . . 12
β’ (π₯ β β*
β (π₯ β β
β¨ π₯ = +β β¨
π₯ =
-β)) |
124 | | simplrl 774 |
. . . . . . . . . . . . . . . . . . 19
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ (β«2βπΉ) = +β) β π₯ β
β) |
125 | | arch 12470 |
. . . . . . . . . . . . . . . . . . 19
β’ (π₯ β β β
βπ β β
π₯ < π) |
126 | 124, 125 | syl 17 |
. . . . . . . . . . . . . . . . . 18
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ (β«2βπΉ) = +β) β
βπ β β
π₯ < π) |
127 | 4 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ (β«2βπΉ) = +β) β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) = π) |
128 | 127 | breq2d 5153 |
. . . . . . . . . . . . . . . . . . 19
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ (β«2βπΉ) = +β) β (π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β π₯ < π)) |
129 | 128 | rexbidv 3172 |
. . . . . . . . . . . . . . . . . 18
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ (β«2βπΉ) = +β) β
(βπ β β
π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β βπ β β π₯ < π)) |
130 | 126, 129 | mpbird 257 |
. . . . . . . . . . . . . . . . 17
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ (β«2βπΉ) = +β) β
βπ β β
π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))) |
131 | 26 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β (β«2βπΉ) β β) |
132 | | simplrl 774 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β π₯ β
β) |
133 | 131, 132 | resubcld 11643 |
. . . . . . . . . . . . . . . . . . 19
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β ((β«2βπΉ) β π₯) β β) |
134 | | simplrr 775 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β π₯ <
(β«2βπΉ)) |
135 | 132, 131 | posdifd 11802 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β (π₯ <
(β«2βπΉ)
β 0 < ((β«2βπΉ) β π₯))) |
136 | 134, 135 | mpbid 231 |
. . . . . . . . . . . . . . . . . . 19
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β 0 < ((β«2βπΉ) β π₯)) |
137 | | nnrecl 12471 |
. . . . . . . . . . . . . . . . . . 19
β’
((((β«2βπΉ) β π₯) β β β§ 0 <
((β«2βπΉ) β π₯)) β βπ β β (1 / π) < ((β«2βπΉ) β π₯)) |
138 | 133, 136,
137 | syl2anc 583 |
. . . . . . . . . . . . . . . . . 18
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β βπ
β β (1 / π) <
((β«2βπΉ) β π₯)) |
139 | 34 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β§ π β
β) β (1 / π)
β β) |
140 | 131 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β§ π β
β) β (β«2βπΉ) β β) |
141 | 132 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β§ π β
β) β π₯ β
β) |
142 | | ltsub13 11696 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((1 /
π) β β β§
(β«2βπΉ)
β β β§ π₯
β β) β ((1 / π) < ((β«2βπΉ) β π₯) β π₯ < ((β«2βπΉ) β (1 / π)))) |
143 | 139, 140,
141, 142 | syl3anc 1368 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β§ π β
β) β ((1 / π)
< ((β«2βπΉ) β π₯) β π₯ < ((β«2βπΉ) β (1 / π)))) |
144 | 8 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β§ π β
β) β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) = ((β«2βπΉ) β (1 / π))) |
145 | 144 | breq2d 5153 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β§ π β
β) β (π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β π₯ < ((β«2βπΉ) β (1 / π)))) |
146 | 143, 145 | bitr4d 282 |
. . . . . . . . . . . . . . . . . . 19
β’ ((((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β§ π β
β) β ((1 / π)
< ((β«2βπΉ) β π₯) β π₯ < if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
147 | 146 | rexbidva 3170 |
. . . . . . . . . . . . . . . . . 18
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β (βπ β β (1 / π) < ((β«2βπΉ) β π₯) β βπ β β π₯ < if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
148 | 138, 147 | mpbid 231 |
. . . . . . . . . . . . . . . . 17
β’ (((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β§ Β¬
(β«2βπΉ)
= +β) β βπ
β β π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))) |
149 | 130, 148 | pm2.61dan 810 |
. . . . . . . . . . . . . . . 16
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β§ π₯ <
(β«2βπΉ))) β βπ β β π₯ < if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))) |
150 | 149 | expr 456 |
. . . . . . . . . . . . . . 15
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ β β)
β (π₯ <
(β«2βπΉ)
β βπ β
β π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))))) |
151 | | rexr 11261 |
. . . . . . . . . . . . . . . 16
β’ (π₯ β β β π₯ β
β*) |
152 | | xrltnle 11282 |
. . . . . . . . . . . . . . . 16
β’ ((π₯ β β*
β§ (β«2βπΉ) β β*) β (π₯ <
(β«2βπΉ)
β Β¬ (β«2βπΉ) β€ π₯)) |
153 | 151, 10, 152 | syl2anr 596 |
. . . . . . . . . . . . . . 15
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ β β)
β (π₯ <
(β«2βπΉ)
β Β¬ (β«2βπΉ) β€ π₯)) |
154 | 151 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . 18
β’ (((πΉ:ββΆ(0[,]+β)
β§ π₯ β β)
β§ π β β)
β π₯ β
β*) |
155 | 38 | adantlr 712 |
. . . . . . . . . . . . . . . . . 18
β’ (((πΉ:ββΆ(0[,]+β)
β§ π₯ β β)
β§ π β β)
β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β
β*) |
156 | | xrltnle 11282 |
. . . . . . . . . . . . . . . . . 18
β’ ((π₯ β β*
β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β*) β (π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β Β¬
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯)) |
157 | 154, 155,
156 | syl2anc 583 |
. . . . . . . . . . . . . . . . 17
β’ (((πΉ:ββΆ(0[,]+β)
β§ π₯ β β)
β§ π β β)
β (π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β Β¬
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯)) |
158 | 157 | rexbidva 3170 |
. . . . . . . . . . . . . . . 16
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ β β)
β (βπ β
β π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β βπ β β Β¬
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯)) |
159 | | rexnal 3094 |
. . . . . . . . . . . . . . . 16
β’
(βπ β
β Β¬ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β Β¬ βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯) |
160 | 158, 159 | bitrdi 287 |
. . . . . . . . . . . . . . 15
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ β β)
β (βπ β
β π₯ <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β Β¬ βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯)) |
161 | 150, 153,
160 | 3imtr3d 293 |
. . . . . . . . . . . . . 14
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ β β)
β (Β¬ (β«2βπΉ) β€ π₯ β Β¬ βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯)) |
162 | 161 | con4d 115 |
. . . . . . . . . . . . 13
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ β β)
β (βπ β
β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β (β«2βπΉ) β€ π₯)) |
163 | 10 | adantr 480 |
. . . . . . . . . . . . . . . 16
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ = +β) β
(β«2βπΉ)
β β*) |
164 | | pnfge 13113 |
. . . . . . . . . . . . . . . 16
β’
((β«2βπΉ) β β* β
(β«2βπΉ)
β€ +β) |
165 | 163, 164 | syl 17 |
. . . . . . . . . . . . . . 15
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ = +β) β
(β«2βπΉ)
β€ +β) |
166 | | simpr 484 |
. . . . . . . . . . . . . . 15
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ = +β) β
π₯ =
+β) |
167 | 165, 166 | breqtrrd 5169 |
. . . . . . . . . . . . . 14
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ = +β) β
(β«2βπΉ)
β€ π₯) |
168 | 167 | a1d 25 |
. . . . . . . . . . . . 13
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ = +β) β
(βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β (β«2βπΉ) β€ π₯)) |
169 | | 1nn 12224 |
. . . . . . . . . . . . . . . 16
β’ 1 β
β |
170 | 169 | ne0ii 4332 |
. . . . . . . . . . . . . . 15
β’ β
β β
|
171 | | r19.2z 4489 |
. . . . . . . . . . . . . . 15
β’ ((β
β β
β§ βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯) β βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯) |
172 | 170, 171 | mpan 687 |
. . . . . . . . . . . . . 14
β’
(βπ β
β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯) |
173 | 37 | adantlr 712 |
. . . . . . . . . . . . . . . . . 18
β’ (((πΉ:ββΆ(0[,]+β)
β§ π₯ = -β) β§
π β β) β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β) |
174 | | mnflt 13106 |
. . . . . . . . . . . . . . . . . . 19
β’
(if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β β -β <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π)))) |
175 | | rexr 11261 |
. . . . . . . . . . . . . . . . . . . 20
β’
(if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β
β*) |
176 | | xrltnle 11282 |
. . . . . . . . . . . . . . . . . . . 20
β’
((-β β β* β§
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β*) β
(-β < if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β Β¬
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ -β)) |
177 | 15, 175, 176 | sylancr 586 |
. . . . . . . . . . . . . . . . . . 19
β’
(if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β β (-β <
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β Β¬
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ -β)) |
178 | 174, 177 | mpbid 231 |
. . . . . . . . . . . . . . . . . 18
β’
(if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β β β Β¬
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ -β) |
179 | 173, 178 | syl 17 |
. . . . . . . . . . . . . . . . 17
β’ (((πΉ:ββΆ(0[,]+β)
β§ π₯ = -β) β§
π β β) β
Β¬ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ -β) |
180 | | simplr 766 |
. . . . . . . . . . . . . . . . . 18
β’ (((πΉ:ββΆ(0[,]+β)
β§ π₯ = -β) β§
π β β) β
π₯ =
-β) |
181 | 180 | breq2d 5153 |
. . . . . . . . . . . . . . . . 17
β’ (((πΉ:ββΆ(0[,]+β)
β§ π₯ = -β) β§
π β β) β
(if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ -β)) |
182 | 179, 181 | mtbird 325 |
. . . . . . . . . . . . . . . 16
β’ (((πΉ:ββΆ(0[,]+β)
β§ π₯ = -β) β§
π β β) β
Β¬ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯) |
183 | 182 | nrexdv 3143 |
. . . . . . . . . . . . . . 15
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ = -β) β
Β¬ βπ β
β if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯) |
184 | 183 | pm2.21d 121 |
. . . . . . . . . . . . . 14
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ = -β) β
(βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β (β«2βπΉ) β€ π₯)) |
185 | 172, 184 | syl5 34 |
. . . . . . . . . . . . 13
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ = -β) β
(βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β (β«2βπΉ) β€ π₯)) |
186 | 162, 168,
185 | 3jaodan 1427 |
. . . . . . . . . . . 12
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π₯ β β
β¨ π₯ = +β β¨
π₯ = -β)) β
(βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β (β«2βπΉ) β€ π₯)) |
187 | 123, 186 | sylan2b 593 |
. . . . . . . . . . 11
β’ ((πΉ:ββΆ(0[,]+β)
β§ π₯ β
β*) β (βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β (β«2βπΉ) β€ π₯)) |
188 | 187 | ralrimiva 3140 |
. . . . . . . . . 10
β’ (πΉ:ββΆ(0[,]+β)
β βπ₯ β
β* (βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β (β«2βπΉ) β€ π₯)) |
189 | 188 | adantr 480 |
. . . . . . . . 9
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β βπ₯ β β* (βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ π₯ β (β«2βπΉ) β€ π₯)) |
190 | 109, 83 | eqeltrrid 2832 |
. . . . . . . . 9
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
β*) |
191 | 122, 189,
190 | rspcdva 3607 |
. . . . . . . 8
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β (βπ β β
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β
(β«2βπΉ)
β€ sup(ran (π β
β β¦ (β«1β(πβπ))), β*, <
))) |
192 | 118, 191 | mpd 15 |
. . . . . . 7
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β (β«2βπΉ) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
)) |
193 | 192, 109 | breqtrrdi 5183 |
. . . . . 6
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β (β«2βπΉ) β€ sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
)) |
194 | | itg2ub 25613 |
. . . . . . . . . . . . . . 15
β’ ((πΉ:ββΆ(0[,]+β)
β§ (πβπ) β dom β«1
β§ (πβπ) βr β€ πΉ) β
(β«1β(πβπ)) β€ (β«2βπΉ)) |
195 | 194 | 3expia 1118 |
. . . . . . . . . . . . . 14
β’ ((πΉ:ββΆ(0[,]+β)
β§ (πβπ) β dom β«1)
β ((πβπ) βr β€ πΉ β
(β«1β(πβπ)) β€ (β«2βπΉ))) |
196 | 74, 195 | sylan2 592 |
. . . . . . . . . . . . 13
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ π
β β)) β ((πβπ) βr β€ πΉ β (β«1β(πβπ)) β€ (β«2βπΉ))) |
197 | 196 | anassrs 467 |
. . . . . . . . . . . 12
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β ((πβπ) βr β€ πΉ β (β«1β(πβπ)) β€ (β«2βπΉ))) |
198 | 197 | adantrd 491 |
. . . . . . . . . . 11
β’ (((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β§ π
β β) β (((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))) β (β«1β(πβπ)) β€ (β«2βπΉ))) |
199 | 198 | ralimdva 3161 |
. . . . . . . . . 10
β’ ((πΉ:ββΆ(0[,]+β)
β§ π:ββΆdom
β«1) β (βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))) β βπ β β
(β«1β(πβπ)) β€ (β«2βπΉ))) |
200 | 199 | impr 454 |
. . . . . . . . 9
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β βπ β β
(β«1β(πβπ)) β€ (β«2βπΉ)) |
201 | | eqid 2726 |
. . . . . . . . . . . . 13
β’ (π β β β¦
(β«1β(πβπ))) = (π β β β¦
(β«1β(πβπ))) |
202 | 89, 201, 101 | fvmpt 6991 |
. . . . . . . . . . . 12
β’ (π β β β ((π β β β¦
(β«1β(πβπ)))βπ) = (β«1β(πβπ))) |
203 | 202 | breq1d 5151 |
. . . . . . . . . . 11
β’ (π β β β (((π β β β¦
(β«1β(πβπ)))βπ) β€ (β«2βπΉ) β
(β«1β(πβπ)) β€ (β«2βπΉ))) |
204 | 203 | ralbiia 3085 |
. . . . . . . . . 10
β’
(βπ β
β ((π β β
β¦ (β«1β(πβπ)))βπ) β€ (β«2βπΉ) β βπ β β
(β«1β(πβπ)) β€ (β«2βπΉ)) |
205 | 89 | breq1d 5151 |
. . . . . . . . . . 11
β’ (π = π β ((β«1β(πβπ)) β€ (β«2βπΉ) β
(β«1β(πβπ)) β€ (β«2βπΉ))) |
206 | 205 | cbvralvw 3228 |
. . . . . . . . . 10
β’
(βπ β
β (β«1β(πβπ)) β€ (β«2βπΉ) β βπ β β
(β«1β(πβπ)) β€ (β«2βπΉ)) |
207 | 204, 206 | bitr4i 278 |
. . . . . . . . 9
β’
(βπ β
β ((π β β
β¦ (β«1β(πβπ)))βπ) β€ (β«2βπΉ) β βπ β β
(β«1β(πβπ)) β€ (β«2βπΉ)) |
208 | 200, 207 | sylibr 233 |
. . . . . . . 8
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β βπ β β ((π β β β¦
(β«1β(πβπ)))βπ) β€ (β«2βπΉ)) |
209 | | ffn 6710 |
. . . . . . . . 9
β’ ((π β β β¦
(β«1β(πβπ))):ββΆβ β (π β β β¦
(β«1β(πβπ))) Fn β) |
210 | | breq1 5144 |
. . . . . . . . . 10
β’ (π§ = ((π β β β¦
(β«1β(πβπ)))βπ) β (π§ β€ (β«2βπΉ) β ((π β β β¦
(β«1β(πβπ)))βπ) β€ (β«2βπΉ))) |
211 | 210 | ralrn 7082 |
. . . . . . . . 9
β’ ((π β β β¦
(β«1β(πβπ))) Fn β β (βπ§ β ran (π β β β¦
(β«1β(πβπ)))π§ β€ (β«2βπΉ) β βπ β β ((π β β β¦
(β«1β(πβπ)))βπ) β€ (β«2βπΉ))) |
212 | 78, 209, 211 | 3syl 18 |
. . . . . . . 8
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β (βπ§ β ran (π β β β¦
(β«1β(πβπ)))π§ β€ (β«2βπΉ) β βπ β β ((π β β β¦
(β«1β(πβπ)))βπ) β€ (β«2βπΉ))) |
213 | 208, 212 | mpbird 257 |
. . . . . . 7
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β βπ§ β ran (π β β β¦
(β«1β(πβπ)))π§ β€ (β«2βπΉ)) |
214 | | supxrleub 13308 |
. . . . . . . 8
β’ ((ran
(π β β β¦
(β«1β(πβπ))) β β* β§
(β«2βπΉ)
β β*) β (sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β€
(β«2βπΉ)
β βπ§ β ran
(π β β β¦
(β«1β(πβπ)))π§ β€ (β«2βπΉ))) |
215 | 81, 73, 214 | syl2anc 583 |
. . . . . . 7
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β (sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β€
(β«2βπΉ)
β βπ§ β ran
(π β β β¦
(β«1β(πβπ)))π§ β€ (β«2βπΉ))) |
216 | 213, 215 | mpbird 257 |
. . . . . 6
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β sup(ran (π β β β¦
(β«1β(πβπ))), β*, < ) β€
(β«2βπΉ)) |
217 | 73, 83, 193, 216 | xrletrid 13137 |
. . . . 5
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β (β«2βπΉ) = sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
)) |
218 | 69, 72, 217 | 3jca 1125 |
. . . 4
β’ ((πΉ:ββΆ(0[,]+β)
β§ (π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ))))) β (π:ββΆdom β«1 β§
βπ β β
(πβπ) βr β€ πΉ β§ (β«2βπΉ) = sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |
219 | 218 | ex 412 |
. . 3
β’ (πΉ:ββΆ(0[,]+β)
β ((π:ββΆdom β«1 β§
βπ β β
((πβπ) βr β€ πΉ β§
if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ)))) β (π:ββΆdom β«1 β§
βπ β β
(πβπ) βr β€ πΉ β§ (β«2βπΉ) = sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
)))) |
220 | 219 | eximdv 1912 |
. 2
β’ (πΉ:ββΆ(0[,]+β)
β (βπ(π:ββΆdom
β«1 β§ βπ β β ((πβπ) βr β€ πΉ β§ if((β«2βπΉ) = +β, π, ((β«2βπΉ) β (1 / π))) < (β«1β(πβπ)))) β βπ(π:ββΆdom β«1 β§
βπ β β
(πβπ) βr β€ πΉ β§ (β«2βπΉ) = sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
)))) |
221 | 68, 220 | mpd 15 |
1
β’ (πΉ:ββΆ(0[,]+β)
β βπ(π:ββΆdom
β«1 β§ βπ β β (πβπ) βr β€ πΉ β§ (β«2βπΉ) = sup(ran (π β β β¦
(β«1β(πβπ))), β*, <
))) |