Step | Hyp | Ref
| Expression |
1 | | fzfid 13934 |
. . . . . 6
β’ (π β (π...π) β Fin) |
2 | | simpl 483 |
. . . . . . 7
β’ ((π β§ π β (π...π)) β π) |
3 | | elfzuz 13493 |
. . . . . . . . 9
β’ (π β (π...π) β π β (β€β₯βπ)) |
4 | | fsumsermpt.z |
. . . . . . . . 9
β’ π =
(β€β₯βπ) |
5 | 3, 4 | eleqtrrdi 2844 |
. . . . . . . 8
β’ (π β (π...π) β π β π) |
6 | 5 | adantl 482 |
. . . . . . 7
β’ ((π β§ π β (π...π)) β π β π) |
7 | | fsumsermpt.a |
. . . . . . 7
β’ ((π β§ π β π) β π΄ β β) |
8 | 2, 6, 7 | syl2anc 584 |
. . . . . 6
β’ ((π β§ π β (π...π)) β π΄ β β) |
9 | 1, 8 | fsumcl 15675 |
. . . . 5
β’ (π β Ξ£π β (π...π)π΄ β β) |
10 | 9 | adantr 481 |
. . . 4
β’ ((π β§ π β π) β Ξ£π β (π...π)π΄ β β) |
11 | 10 | ralrimiva 3146 |
. . 3
β’ (π β βπ β π Ξ£π β (π...π)π΄ β β) |
12 | | fsumsermpt.f |
. . . . 5
β’ πΉ = (π β π β¦ Ξ£π β (π...π)π΄) |
13 | | oveq2 7413 |
. . . . . . 7
β’ (π = π β (π...π) = (π...π)) |
14 | 13 | sumeq1d 15643 |
. . . . . 6
β’ (π = π β Ξ£π β (π...π)π΄ = Ξ£π β (π...π)π΄) |
15 | 14 | cbvmptv 5260 |
. . . . 5
β’ (π β π β¦ Ξ£π β (π...π)π΄) = (π β π β¦ Ξ£π β (π...π)π΄) |
16 | 12, 15 | eqtri 2760 |
. . . 4
β’ πΉ = (π β π β¦ Ξ£π β (π...π)π΄) |
17 | 16 | fnmpt 6687 |
. . 3
β’
(βπ β
π Ξ£π β (π...π)π΄ β β β πΉ Fn π) |
18 | 11, 17 | syl 17 |
. 2
β’ (π β πΉ Fn π) |
19 | | fsumsermpt.m |
. . . . 5
β’ (π β π β β€) |
20 | | simpr 485 |
. . . . . . 7
β’ ((π β§ π β π) β π β π) |
21 | | nfv 1917 |
. . . . . . . . 9
β’
β²π(π β§ π β π) |
22 | | nfcv 2903 |
. . . . . . . . . . 11
β’
β²ππ |
23 | 22 | nfcsb1 3916 |
. . . . . . . . . 10
β’
β²πβ¦π / πβ¦π΄ |
24 | 23 | nfel1 2919 |
. . . . . . . . 9
β’
β²πβ¦π / πβ¦π΄ β β |
25 | 21, 24 | nfim 1899 |
. . . . . . . 8
β’
β²π((π β§ π β π) β β¦π / πβ¦π΄ β β) |
26 | | eleq1w 2816 |
. . . . . . . . . 10
β’ (π = π β (π β π β π β π)) |
27 | 26 | anbi2d 629 |
. . . . . . . . 9
β’ (π = π β ((π β§ π β π) β (π β§ π β π))) |
28 | | csbeq1a 3906 |
. . . . . . . . . 10
β’ (π = π β π΄ = β¦π / πβ¦π΄) |
29 | 28 | eleq1d 2818 |
. . . . . . . . 9
β’ (π = π β (π΄ β β β β¦π / πβ¦π΄ β β)) |
30 | 27, 29 | imbi12d 344 |
. . . . . . . 8
β’ (π = π β (((π β§ π β π) β π΄ β β) β ((π β§ π β π) β β¦π / πβ¦π΄ β β))) |
31 | 25, 30, 7 | chvarfv 2233 |
. . . . . . 7
β’ ((π β§ π β π) β β¦π / πβ¦π΄ β β) |
32 | | eqid 2732 |
. . . . . . . 8
β’ (π β π β¦ π΄) = (π β π β¦ π΄) |
33 | 22, 23, 28, 32 | fvmptf 7016 |
. . . . . . 7
β’ ((π β π β§ β¦π / πβ¦π΄ β β) β ((π β π β¦ π΄)βπ) = β¦π / πβ¦π΄) |
34 | 20, 31, 33 | syl2anc 584 |
. . . . . 6
β’ ((π β§ π β π) β ((π β π β¦ π΄)βπ) = β¦π / πβ¦π΄) |
35 | 34, 31 | eqeltrd 2833 |
. . . . 5
β’ ((π β§ π β π) β ((π β π β¦ π΄)βπ) β β) |
36 | | addcl 11188 |
. . . . . 6
β’ ((π β β β§ π₯ β β) β (π + π₯) β β) |
37 | 36 | adantl 482 |
. . . . 5
β’ ((π β§ (π β β β§ π₯ β β)) β (π + π₯) β β) |
38 | 4, 19, 35, 37 | seqf 13985 |
. . . 4
β’ (π β seqπ( + , (π β π β¦ π΄)):πβΆβ) |
39 | 38 | ffnd 6715 |
. . 3
β’ (π β seqπ( + , (π β π β¦ π΄)) Fn π) |
40 | | fsumsermpt.g |
. . . . 5
β’ πΊ = seqπ( + , (π β π β¦ π΄)) |
41 | 40 | a1i 11 |
. . . 4
β’ (π β πΊ = seqπ( + , (π β π β¦ π΄))) |
42 | 41 | fneq1d 6639 |
. . 3
β’ (π β (πΊ Fn π β seqπ( + , (π β π β¦ π΄)) Fn π)) |
43 | 39, 42 | mpbird 256 |
. 2
β’ (π β πΊ Fn π) |
44 | | simpr 485 |
. . . . 5
β’ ((π β§ π β π) β π β π) |
45 | 16 | fvmpt2 7006 |
. . . . 5
β’ ((π β π β§ Ξ£π β (π...π)π΄ β β) β (πΉβπ) = Ξ£π β (π...π)π΄) |
46 | 44, 10, 45 | syl2anc 584 |
. . . 4
β’ ((π β§ π β π) β (πΉβπ) = Ξ£π β (π...π)π΄) |
47 | | nfcv 2903 |
. . . . . 6
β’
β²π(π...π) |
48 | | nfcv 2903 |
. . . . . 6
β’
β²π(π...π) |
49 | | nfcv 2903 |
. . . . . 6
β’
β²ππ΄ |
50 | 28, 47, 48, 49, 23 | cbvsum 15637 |
. . . . 5
β’
Ξ£π β
(π...π)π΄ = Ξ£π β (π...π)β¦π / πβ¦π΄ |
51 | 50 | a1i 11 |
. . . 4
β’ ((π β§ π β π) β Ξ£π β (π...π)π΄ = Ξ£π β (π...π)β¦π / πβ¦π΄) |
52 | 46, 51 | eqtrd 2772 |
. . 3
β’ ((π β§ π β π) β (πΉβπ) = Ξ£π β (π...π)β¦π / πβ¦π΄) |
53 | | simpl 483 |
. . . . . 6
β’ ((π β§ π β (π...π)) β π) |
54 | | elfzuz 13493 |
. . . . . . . 8
β’ (π β (π...π) β π β (β€β₯βπ)) |
55 | 54, 4 | eleqtrrdi 2844 |
. . . . . . 7
β’ (π β (π...π) β π β π) |
56 | 55 | adantl 482 |
. . . . . 6
β’ ((π β§ π β (π...π)) β π β π) |
57 | 53, 56, 34 | syl2anc 584 |
. . . . 5
β’ ((π β§ π β (π...π)) β ((π β π β¦ π΄)βπ) = β¦π / πβ¦π΄) |
58 | 57 | adantlr 713 |
. . . 4
β’ (((π β§ π β π) β§ π β (π...π)) β ((π β π β¦ π΄)βπ) = β¦π / πβ¦π΄) |
59 | | id 22 |
. . . . . 6
β’ (π β π β π β π) |
60 | 59, 4 | eleqtrdi 2843 |
. . . . 5
β’ (π β π β π β (β€β₯βπ)) |
61 | 60 | adantl 482 |
. . . 4
β’ ((π β§ π β π) β π β (β€β₯βπ)) |
62 | 53, 56, 31 | syl2anc 584 |
. . . . 5
β’ ((π β§ π β (π...π)) β β¦π / πβ¦π΄ β β) |
63 | 62 | adantlr 713 |
. . . 4
β’ (((π β§ π β π) β§ π β (π...π)) β β¦π / πβ¦π΄ β β) |
64 | 58, 61, 63 | fsumser 15672 |
. . 3
β’ ((π β§ π β π) β Ξ£π β (π...π)β¦π / πβ¦π΄ = (seqπ( + , (π β π β¦ π΄))βπ)) |
65 | 40 | fveq1i 6889 |
. . . . 5
β’ (πΊβπ) = (seqπ( + , (π β π β¦ π΄))βπ) |
66 | 65 | eqcomi 2741 |
. . . 4
β’ (seqπ( + , (π β π β¦ π΄))βπ) = (πΊβπ) |
67 | 66 | a1i 11 |
. . 3
β’ ((π β§ π β π) β (seqπ( + , (π β π β¦ π΄))βπ) = (πΊβπ)) |
68 | 52, 64, 67 | 3eqtrd 2776 |
. 2
β’ ((π β§ π β π) β (πΉβπ) = (πΊβπ)) |
69 | 18, 43, 68 | eqfnfvd 7032 |
1
β’ (π β πΉ = πΊ) |