Step | Hyp | Ref
| Expression |
1 | | caubl.5 |
. . 3
β’ (π β βπ β β+ βπ β β (2nd
β(πΉβπ)) < π) |
2 | | 2fveq3 6887 |
. . . . . . . . . . . . 13
β’ (π = π β ((ballβπ·)β(πΉβπ)) = ((ballβπ·)β(πΉβπ))) |
3 | 2 | sseq1d 4006 |
. . . . . . . . . . . 12
β’ (π = π β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)))) |
4 | 3 | imbi2d 340 |
. . . . . . . . . . 11
β’ (π = π β (((π β§ π β β) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ))) β ((π β§ π β β) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ))))) |
5 | | 2fveq3 6887 |
. . . . . . . . . . . . 13
β’ (π = π β ((ballβπ·)β(πΉβπ)) = ((ballβπ·)β(πΉβπ))) |
6 | 5 | sseq1d 4006 |
. . . . . . . . . . . 12
β’ (π = π β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)))) |
7 | 6 | imbi2d 340 |
. . . . . . . . . . 11
β’ (π = π β (((π β§ π β β) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ))) β ((π β§ π β β) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ))))) |
8 | | 2fveq3 6887 |
. . . . . . . . . . . . 13
β’ (π = (π + 1) β ((ballβπ·)β(πΉβπ)) = ((ballβπ·)β(πΉβ(π + 1)))) |
9 | 8 | sseq1d 4006 |
. . . . . . . . . . . 12
β’ (π = (π + 1) β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ)))) |
10 | 9 | imbi2d 340 |
. . . . . . . . . . 11
β’ (π = (π + 1) β (((π β§ π β β) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ))) β ((π β§ π β β) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ))))) |
11 | | ssid 3997 |
. . . . . . . . . . . 12
β’
((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) |
12 | 11 | 2a1i 12 |
. . . . . . . . . . 11
β’ (π β β€ β ((π β§ π β β) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)))) |
13 | | caubl.4 |
. . . . . . . . . . . . . . . 16
β’ (π β βπ β β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ))) |
14 | | eluznn 12901 |
. . . . . . . . . . . . . . . 16
β’ ((π β β β§ π β
(β€β₯βπ)) β π β β) |
15 | | fvoveq1 7425 |
. . . . . . . . . . . . . . . . . . 19
β’ (π = π β (πΉβ(π + 1)) = (πΉβ(π + 1))) |
16 | 15 | fveq2d 6886 |
. . . . . . . . . . . . . . . . . 18
β’ (π = π β ((ballβπ·)β(πΉβ(π + 1))) = ((ballβπ·)β(πΉβ(π + 1)))) |
17 | | 2fveq3 6887 |
. . . . . . . . . . . . . . . . . 18
β’ (π = π β ((ballβπ·)β(πΉβπ)) = ((ballβπ·)β(πΉβπ))) |
18 | 16, 17 | sseq12d 4008 |
. . . . . . . . . . . . . . . . 17
β’ (π = π β (((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ)))) |
19 | 18 | rspccva 3603 |
. . . . . . . . . . . . . . . 16
β’
((βπ β
β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ)) β§ π β β) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ))) |
20 | 13, 14, 19 | syl2an 595 |
. . . . . . . . . . . . . . 15
β’ ((π β§ (π β β β§ π β (β€β₯βπ))) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ))) |
21 | 20 | anassrs 467 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β β) β§ π β (β€β₯βπ)) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ))) |
22 | | sstr2 3982 |
. . . . . . . . . . . . . 14
β’
(((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ)) β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ)))) |
23 | 21, 22 | syl 17 |
. . . . . . . . . . . . 13
β’ (((π β§ π β β) β§ π β (β€β₯βπ)) β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ)))) |
24 | 23 | expcom 413 |
. . . . . . . . . . . 12
β’ (π β
(β€β₯βπ) β ((π β§ π β β) β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ))))) |
25 | 24 | a2d 29 |
. . . . . . . . . . 11
β’ (π β
(β€β₯βπ) β (((π β§ π β β) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ))) β ((π β§ π β β) β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ))))) |
26 | 4, 7, 10, 7, 12, 25 | uzind4 12889 |
. . . . . . . . . 10
β’ (π β
(β€β₯βπ) β ((π β§ π β β) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)))) |
27 | 26 | com12 32 |
. . . . . . . . 9
β’ ((π β§ π β β) β (π β (β€β₯βπ) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)))) |
28 | 27 | ad2ant2r 744 |
. . . . . . . 8
β’ (((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β (π β (β€β₯βπ) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)))) |
29 | | relxp 5685 |
. . . . . . . . . . . . . . . 16
β’ Rel
(π Γ
β+) |
30 | | caubl.3 |
. . . . . . . . . . . . . . . . . 18
β’ (π β πΉ:ββΆ(π Γ
β+)) |
31 | 30 | ad3antrrr 727 |
. . . . . . . . . . . . . . . . 17
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β πΉ:ββΆ(π Γ
β+)) |
32 | | simplrl 774 |
. . . . . . . . . . . . . . . . 17
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β π β β) |
33 | 31, 32 | ffvelcdmd 7078 |
. . . . . . . . . . . . . . . 16
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (πΉβπ) β (π Γ
β+)) |
34 | | 1st2nd 8019 |
. . . . . . . . . . . . . . . 16
β’ ((Rel
(π Γ
β+) β§ (πΉβπ) β (π Γ β+)) β (πΉβπ) = β¨(1st β(πΉβπ)), (2nd β(πΉβπ))β©) |
35 | 29, 33, 34 | sylancr 586 |
. . . . . . . . . . . . . . 15
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (πΉβπ) = β¨(1st β(πΉβπ)), (2nd β(πΉβπ))β©) |
36 | 35 | fveq2d 6886 |
. . . . . . . . . . . . . 14
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β ((ballβπ·)β(πΉβπ)) = ((ballβπ·)ββ¨(1st β(πΉβπ)), (2nd β(πΉβπ))β©)) |
37 | | df-ov 7405 |
. . . . . . . . . . . . . 14
β’
((1st β(πΉβπ))(ballβπ·)(2nd β(πΉβπ))) = ((ballβπ·)ββ¨(1st β(πΉβπ)), (2nd β(πΉβπ))β©) |
38 | 36, 37 | eqtr4di 2782 |
. . . . . . . . . . . . 13
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β ((ballβπ·)β(πΉβπ)) = ((1st β(πΉβπ))(ballβπ·)(2nd β(πΉβπ)))) |
39 | | caubl.2 |
. . . . . . . . . . . . . . 15
β’ (π β π· β (βMetβπ)) |
40 | 39 | ad3antrrr 727 |
. . . . . . . . . . . . . 14
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β π· β (βMetβπ)) |
41 | | xp1st 8001 |
. . . . . . . . . . . . . . 15
β’ ((πΉβπ) β (π Γ β+) β
(1st β(πΉβπ)) β π) |
42 | 33, 41 | syl 17 |
. . . . . . . . . . . . . 14
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (1st
β(πΉβπ)) β π) |
43 | | xp2nd 8002 |
. . . . . . . . . . . . . . . 16
β’ ((πΉβπ) β (π Γ β+) β
(2nd β(πΉβπ)) β
β+) |
44 | 33, 43 | syl 17 |
. . . . . . . . . . . . . . 15
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (2nd
β(πΉβπ)) β
β+) |
45 | 44 | rpxrd 13018 |
. . . . . . . . . . . . . 14
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (2nd
β(πΉβπ)) β
β*) |
46 | | simpllr 773 |
. . . . . . . . . . . . . . 15
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β π β β+) |
47 | 46 | rpxrd 13018 |
. . . . . . . . . . . . . 14
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β π β β*) |
48 | | simplrr 775 |
. . . . . . . . . . . . . . 15
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (2nd
β(πΉβπ)) < π) |
49 | | rpre 12983 |
. . . . . . . . . . . . . . . . 17
β’
((2nd β(πΉβπ)) β β+ β
(2nd β(πΉβπ)) β β) |
50 | | rpre 12983 |
. . . . . . . . . . . . . . . . 17
β’ (π β β+
β π β
β) |
51 | | ltle 11301 |
. . . . . . . . . . . . . . . . 17
β’
(((2nd β(πΉβπ)) β β β§ π β β) β ((2nd
β(πΉβπ)) < π β (2nd β(πΉβπ)) β€ π)) |
52 | 49, 50, 51 | syl2an 595 |
. . . . . . . . . . . . . . . 16
β’
(((2nd β(πΉβπ)) β β+ β§ π β β+)
β ((2nd β(πΉβπ)) < π β (2nd β(πΉβπ)) β€ π)) |
53 | 44, 46, 52 | syl2anc 583 |
. . . . . . . . . . . . . . 15
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β ((2nd
β(πΉβπ)) < π β (2nd β(πΉβπ)) β€ π)) |
54 | 48, 53 | mpd 15 |
. . . . . . . . . . . . . 14
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (2nd
β(πΉβπ)) β€ π) |
55 | | ssbl 24273 |
. . . . . . . . . . . . . 14
β’ (((π· β (βMetβπ) β§ (1st
β(πΉβπ)) β π) β§ ((2nd β(πΉβπ)) β β* β§ π β β*)
β§ (2nd β(πΉβπ)) β€ π) β ((1st β(πΉβπ))(ballβπ·)(2nd β(πΉβπ))) β ((1st β(πΉβπ))(ballβπ·)π)) |
56 | 40, 42, 45, 47, 54, 55 | syl221anc 1378 |
. . . . . . . . . . . . 13
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β ((1st
β(πΉβπ))(ballβπ·)(2nd β(πΉβπ))) β ((1st β(πΉβπ))(ballβπ·)π)) |
57 | 38, 56 | eqsstrd 4013 |
. . . . . . . . . . . 12
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β ((ballβπ·)β(πΉβπ)) β ((1st β(πΉβπ))(ballβπ·)π)) |
58 | | sstr2 3982 |
. . . . . . . . . . . 12
β’
(((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β (((ballβπ·)β(πΉβπ)) β ((1st β(πΉβπ))(ballβπ·)π) β ((ballβπ·)β(πΉβπ)) β ((1st β(πΉβπ))(ballβπ·)π))) |
59 | 57, 58 | syl5com 31 |
. . . . . . . . . . 11
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((1st β(πΉβπ))(ballβπ·)π))) |
60 | | simprl 768 |
. . . . . . . . . . . . . . . 16
β’ (((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β π β β) |
61 | 60, 14 | sylan 579 |
. . . . . . . . . . . . . . 15
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β π β β) |
62 | 31, 61 | ffvelcdmd 7078 |
. . . . . . . . . . . . . 14
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (πΉβπ) β (π Γ
β+)) |
63 | | xp1st 8001 |
. . . . . . . . . . . . . 14
β’ ((πΉβπ) β (π Γ β+) β
(1st β(πΉβπ)) β π) |
64 | 62, 63 | syl 17 |
. . . . . . . . . . . . 13
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (1st
β(πΉβπ)) β π) |
65 | | xp2nd 8002 |
. . . . . . . . . . . . . 14
β’ ((πΉβπ) β (π Γ β+) β
(2nd β(πΉβπ)) β
β+) |
66 | 62, 65 | syl 17 |
. . . . . . . . . . . . 13
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (2nd
β(πΉβπ)) β
β+) |
67 | | blcntr 24263 |
. . . . . . . . . . . . 13
β’ ((π· β (βMetβπ) β§ (1st
β(πΉβπ)) β π β§ (2nd β(πΉβπ)) β β+) β
(1st β(πΉβπ)) β ((1st β(πΉβπ))(ballβπ·)(2nd β(πΉβπ)))) |
68 | 40, 64, 66, 67 | syl3anc 1368 |
. . . . . . . . . . . 12
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (1st
β(πΉβπ)) β ((1st
β(πΉβπ))(ballβπ·)(2nd β(πΉβπ)))) |
69 | | 1st2nd 8019 |
. . . . . . . . . . . . . . 15
β’ ((Rel
(π Γ
β+) β§ (πΉβπ) β (π Γ β+)) β (πΉβπ) = β¨(1st β(πΉβπ)), (2nd β(πΉβπ))β©) |
70 | 29, 62, 69 | sylancr 586 |
. . . . . . . . . . . . . 14
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (πΉβπ) = β¨(1st β(πΉβπ)), (2nd β(πΉβπ))β©) |
71 | 70 | fveq2d 6886 |
. . . . . . . . . . . . 13
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β ((ballβπ·)β(πΉβπ)) = ((ballβπ·)ββ¨(1st β(πΉβπ)), (2nd β(πΉβπ))β©)) |
72 | | df-ov 7405 |
. . . . . . . . . . . . 13
β’
((1st β(πΉβπ))(ballβπ·)(2nd β(πΉβπ))) = ((ballβπ·)ββ¨(1st β(πΉβπ)), (2nd β(πΉβπ))β©) |
73 | 71, 72 | eqtr4di 2782 |
. . . . . . . . . . . 12
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β ((ballβπ·)β(πΉβπ)) = ((1st β(πΉβπ))(ballβπ·)(2nd β(πΉβπ)))) |
74 | 68, 73 | eleqtrrd 2828 |
. . . . . . . . . . 11
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (1st
β(πΉβπ)) β ((ballβπ·)β(πΉβπ))) |
75 | | ssel 3968 |
. . . . . . . . . . 11
β’
(((ballβπ·)β(πΉβπ)) β ((1st β(πΉβπ))(ballβπ·)π) β ((1st β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β (1st β(πΉβπ)) β ((1st β(πΉβπ))(ballβπ·)π))) |
76 | 59, 74, 75 | syl6ci 71 |
. . . . . . . . . 10
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β (1st β(πΉβπ)) β ((1st β(πΉβπ))(ballβπ·)π))) |
77 | | elbl2 24240 |
. . . . . . . . . . 11
β’ (((π· β (βMetβπ) β§ π β β*) β§
((1st β(πΉβπ)) β π β§ (1st β(πΉβπ)) β π)) β ((1st β(πΉβπ)) β ((1st β(πΉβπ))(ballβπ·)π) β ((1st β(πΉβπ))π·(1st β(πΉβπ))) < π)) |
78 | 40, 47, 42, 64, 77 | syl22anc 836 |
. . . . . . . . . 10
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β ((1st
β(πΉβπ)) β ((1st
β(πΉβπ))(ballβπ·)π) β ((1st β(πΉβπ))π·(1st β(πΉβπ))) < π)) |
79 | 76, 78 | sylibd 238 |
. . . . . . . . 9
β’ ((((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β§ π β (β€β₯βπ)) β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((1st β(πΉβπ))π·(1st β(πΉβπ))) < π)) |
80 | 79 | ex 412 |
. . . . . . . 8
β’ (((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β (π β (β€β₯βπ) β (((ballβπ·)β(πΉβπ)) β ((ballβπ·)β(πΉβπ)) β ((1st β(πΉβπ))π·(1st β(πΉβπ))) < π))) |
81 | 28, 80 | mpdd 43 |
. . . . . . 7
β’ (((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β (π β (β€β₯βπ) β ((1st
β(πΉβπ))π·(1st β(πΉβπ))) < π)) |
82 | 81 | ralrimiv 3137 |
. . . . . 6
β’ (((π β§ π β β+) β§ (π β β β§
(2nd β(πΉβπ)) < π)) β βπ β (β€β₯βπ)((1st β(πΉβπ))π·(1st β(πΉβπ))) < π) |
83 | 82 | expr 456 |
. . . . 5
β’ (((π β§ π β β+) β§ π β β) β
((2nd β(πΉβπ)) < π β βπ β (β€β₯βπ)((1st β(πΉβπ))π·(1st β(πΉβπ))) < π)) |
84 | 83 | reximdva 3160 |
. . . 4
β’ ((π β§ π β β+) β
(βπ β β
(2nd β(πΉβπ)) < π β βπ β β βπ β (β€β₯βπ)((1st β(πΉβπ))π·(1st β(πΉβπ))) < π)) |
85 | 84 | ralimdva 3159 |
. . 3
β’ (π β (βπ β β+
βπ β β
(2nd β(πΉβπ)) < π β βπ β β+ βπ β β βπ β
(β€β₯βπ)((1st β(πΉβπ))π·(1st β(πΉβπ))) < π)) |
86 | 1, 85 | mpd 15 |
. 2
β’ (π β βπ β β+ βπ β β βπ β
(β€β₯βπ)((1st β(πΉβπ))π·(1st β(πΉβπ))) < π) |
87 | | nnuz 12864 |
. . 3
β’ β =
(β€β₯β1) |
88 | | 1zzd 12592 |
. . 3
β’ (π β 1 β
β€) |
89 | | fvco3 6981 |
. . . 4
β’ ((πΉ:ββΆ(π Γ β+)
β§ π β β)
β ((1st β πΉ)βπ) = (1st β(πΉβπ))) |
90 | 30, 89 | sylan 579 |
. . 3
β’ ((π β§ π β β) β ((1st
β πΉ)βπ) = (1st
β(πΉβπ))) |
91 | | fvco3 6981 |
. . . 4
β’ ((πΉ:ββΆ(π Γ β+)
β§ π β β)
β ((1st β πΉ)βπ) = (1st β(πΉβπ))) |
92 | 30, 91 | sylan 579 |
. . 3
β’ ((π β§ π β β) β ((1st
β πΉ)βπ) = (1st
β(πΉβπ))) |
93 | | 1stcof 7999 |
. . . 4
β’ (πΉ:ββΆ(π Γ β+)
β (1st β πΉ):ββΆπ) |
94 | 30, 93 | syl 17 |
. . 3
β’ (π β (1st β
πΉ):ββΆπ) |
95 | 87, 39, 88, 90, 92, 94 | iscauf 25152 |
. 2
β’ (π β ((1st β
πΉ) β (Cauβπ·) β βπ β β+
βπ β β
βπ β
(β€β₯βπ)((1st β(πΉβπ))π·(1st β(πΉβπ))) < π)) |
96 | 86, 95 | mpbird 257 |
1
β’ (π β (1st β
πΉ) β (Cauβπ·)) |