Step | Hyp | Ref
| Expression |
1 | | fvex 6859 |
. . . 4
β’
(seqΟ((π β V, π§ β V β¦ (((π΄ βo (ββπ)) Β·o (πβ(ββπ))) +o π§)), β
)βdom β) β V |
2 | 1 | csbex 5272 |
. . 3
β’
β¦OrdIso( E , (π supp β
)) / ββ¦(seqΟ((π β V, π§ β V β¦ (((π΄ βo (ββπ)) Β·o (πβ(ββπ))) +o π§)), β
)βdom β) β V |
3 | 2 | a1i 11 |
. 2
β’ ((π β§ π β π) β β¦OrdIso( E , (π supp β
)) / ββ¦(seqΟ((π β V, π§ β V β¦ (((π΄ βo (ββπ)) Β·o (πβ(ββπ))) +o π§)), β
)βdom β) β V) |
4 | | eqid 2733 |
. . . 4
β’ {π β (π΄ βm π΅) β£ π finSupp β
} = {π β (π΄ βm π΅) β£ π finSupp β
} |
5 | | cantnfs.a |
. . . 4
β’ (π β π΄ β On) |
6 | | cantnfs.b |
. . . 4
β’ (π β π΅ β On) |
7 | 4, 5, 6 | cantnffval 9607 |
. . 3
β’ (π β (π΄ CNF π΅) = (π β {π β (π΄ βm π΅) β£ π finSupp β
} β¦
β¦OrdIso( E , (π supp β
)) / ββ¦(seqΟ((π β V, π§ β V β¦ (((π΄ βo (ββπ)) Β·o (πβ(ββπ))) +o π§)), β
)βdom β))) |
8 | | cantnfs.s |
. . . . 5
β’ π = dom (π΄ CNF π΅) |
9 | 4, 5, 6 | cantnfdm 9608 |
. . . . 5
β’ (π β dom (π΄ CNF π΅) = {π β (π΄ βm π΅) β£ π finSupp β
}) |
10 | 8, 9 | eqtrid 2785 |
. . . 4
β’ (π β π = {π β (π΄ βm π΅) β£ π finSupp β
}) |
11 | 10 | mpteq1d 5204 |
. . 3
β’ (π β (π β π β¦ β¦OrdIso( E , (π supp β
)) / ββ¦(seqΟ((π β V, π§ β V β¦ (((π΄ βo (ββπ)) Β·o (πβ(ββπ))) +o π§)), β
)βdom β)) = (π β {π β (π΄ βm π΅) β£ π finSupp β
} β¦
β¦OrdIso( E , (π supp β
)) / ββ¦(seqΟ((π β V, π§ β V β¦ (((π΄ βo (ββπ)) Β·o (πβ(ββπ))) +o π§)), β
)βdom β))) |
12 | 7, 11 | eqtr4d 2776 |
. 2
β’ (π β (π΄ CNF π΅) = (π β π β¦ β¦OrdIso( E , (π supp β
)) / ββ¦(seqΟ((π β V, π§ β V β¦ (((π΄ βo (ββπ)) Β·o (πβ(ββπ))) +o π§)), β
)βdom β))) |
13 | 5 | adantr 482 |
. . . . . . . 8
β’ ((π β§ π₯ β π) β π΄ β On) |
14 | 6 | adantr 482 |
. . . . . . . 8
β’ ((π β§ π₯ β π) β π΅ β On) |
15 | | eqid 2733 |
. . . . . . . 8
β’ OrdIso( E
, (π₯ supp β
)) =
OrdIso( E , (π₯ supp
β
)) |
16 | | simpr 486 |
. . . . . . . 8
β’ ((π β§ π₯ β π) β π₯ β π) |
17 | | eqid 2733 |
. . . . . . . 8
β’
seqΟ((π β V, π§ β V β¦ (((π΄ βo (OrdIso( E , (π₯ supp β
))βπ)) Β·o (π₯β(OrdIso( E , (π₯ supp β
))βπ))) +o π§)), β
) =
seqΟ((π
β V, π§ β V
β¦ (((π΄
βo (OrdIso( E , (π₯ supp β
))βπ)) Β·o (π₯β(OrdIso( E , (π₯ supp β
))βπ))) +o π§)), β
) |
18 | 8, 13, 14, 15, 16, 17 | cantnfval 9612 |
. . . . . . 7
β’ ((π β§ π₯ β π) β ((π΄ CNF π΅)βπ₯) = (seqΟ((π β V, π§ β V β¦ (((π΄ βo (OrdIso( E , (π₯ supp β
))βπ)) Β·o (π₯β(OrdIso( E , (π₯ supp β
))βπ))) +o π§)), β
)βdom OrdIso( E
, (π₯ supp
β
)))) |
19 | 18 | adantr 482 |
. . . . . 6
β’ (((π β§ π₯ β π) β§ π΄ = β
) β ((π΄ CNF π΅)βπ₯) = (seqΟ((π β V, π§ β V β¦ (((π΄ βo (OrdIso( E , (π₯ supp β
))βπ)) Β·o (π₯β(OrdIso( E , (π₯ supp β
))βπ))) +o π§)), β
)βdom OrdIso( E
, (π₯ supp
β
)))) |
20 | | ovex 7394 |
. . . . . . . . . . 11
β’ (π₯ supp β
) β
V |
21 | 8, 13, 14, 15, 16 | cantnfcl 9611 |
. . . . . . . . . . . 12
β’ ((π β§ π₯ β π) β ( E We (π₯ supp β
) β§ dom OrdIso( E , (π₯ supp β
)) β
Ο)) |
22 | 21 | simpld 496 |
. . . . . . . . . . 11
β’ ((π β§ π₯ β π) β E We (π₯ supp β
)) |
23 | 15 | oien 9482 |
. . . . . . . . . . 11
β’ (((π₯ supp β
) β V β§ E
We (π₯ supp β
)) β
dom OrdIso( E , (π₯ supp
β
)) β (π₯ supp
β
)) |
24 | 20, 22, 23 | sylancr 588 |
. . . . . . . . . 10
β’ ((π β§ π₯ β π) β dom OrdIso( E , (π₯ supp β
)) β (π₯ supp β
)) |
25 | 24 | adantr 482 |
. . . . . . . . 9
β’ (((π β§ π₯ β π) β§ π΄ = β
) β dom OrdIso( E , (π₯ supp β
)) β (π₯ supp β
)) |
26 | | suppssdm 8112 |
. . . . . . . . . . 11
β’ (π₯ supp β
) β dom π₯ |
27 | 8, 5, 6 | cantnfs 9610 |
. . . . . . . . . . . 12
β’ (π β (π₯ β π β (π₯:π΅βΆπ΄ β§ π₯ finSupp β
))) |
28 | 27 | simprbda 500 |
. . . . . . . . . . 11
β’ ((π β§ π₯ β π) β π₯:π΅βΆπ΄) |
29 | 26, 28 | fssdm 6692 |
. . . . . . . . . 10
β’ ((π β§ π₯ β π) β (π₯ supp β
) β π΅) |
30 | | feq3 6655 |
. . . . . . . . . . . . . 14
β’ (π΄ = β
β (π₯:π΅βΆπ΄ β π₯:π΅βΆβ
)) |
31 | 28, 30 | syl5ibcom 244 |
. . . . . . . . . . . . 13
β’ ((π β§ π₯ β π) β (π΄ = β
β π₯:π΅βΆβ
)) |
32 | 31 | imp 408 |
. . . . . . . . . . . 12
β’ (((π β§ π₯ β π) β§ π΄ = β
) β π₯:π΅βΆβ
) |
33 | | f00 6728 |
. . . . . . . . . . . 12
β’ (π₯:π΅βΆβ
β (π₯ = β
β§ π΅ = β
)) |
34 | 32, 33 | sylib 217 |
. . . . . . . . . . 11
β’ (((π β§ π₯ β π) β§ π΄ = β
) β (π₯ = β
β§ π΅ = β
)) |
35 | 34 | simprd 497 |
. . . . . . . . . 10
β’ (((π β§ π₯ β π) β§ π΄ = β
) β π΅ = β
) |
36 | | sseq0 4363 |
. . . . . . . . . 10
β’ (((π₯ supp β
) β π΅ β§ π΅ = β
) β (π₯ supp β
) = β
) |
37 | 29, 35, 36 | syl2an2r 684 |
. . . . . . . . 9
β’ (((π β§ π₯ β π) β§ π΄ = β
) β (π₯ supp β
) = β
) |
38 | 25, 37 | breqtrd 5135 |
. . . . . . . 8
β’ (((π β§ π₯ β π) β§ π΄ = β
) β dom OrdIso( E , (π₯ supp β
)) β
β
) |
39 | | en0 8963 |
. . . . . . . 8
β’ (dom
OrdIso( E , (π₯ supp
β
)) β β
β dom OrdIso( E , (π₯ supp β
)) = β
) |
40 | 38, 39 | sylib 217 |
. . . . . . 7
β’ (((π β§ π₯ β π) β§ π΄ = β
) β dom OrdIso( E , (π₯ supp β
)) =
β
) |
41 | 40 | fveq2d 6850 |
. . . . . 6
β’ (((π β§ π₯ β π) β§ π΄ = β
) β
(seqΟ((π
β V, π§ β V
β¦ (((π΄
βo (OrdIso( E , (π₯ supp β
))βπ)) Β·o (π₯β(OrdIso( E , (π₯ supp β
))βπ))) +o π§)), β
)βdom OrdIso( E , (π₯ supp β
))) =
(seqΟ((π
β V, π§ β V
β¦ (((π΄
βo (OrdIso( E , (π₯ supp β
))βπ)) Β·o (π₯β(OrdIso( E , (π₯ supp β
))βπ))) +o π§)), β
)ββ
)) |
42 | | 0ex 5268 |
. . . . . . 7
β’ β
β V |
43 | 17 | seqom0g 8406 |
. . . . . . 7
β’ (β
β V β (seqΟ((π β V, π§ β V β¦ (((π΄ βo (OrdIso( E , (π₯ supp β
))βπ)) Β·o (π₯β(OrdIso( E , (π₯ supp β
))βπ))) +o π§)), β
)ββ
) =
β
) |
44 | 42, 43 | mp1i 13 |
. . . . . 6
β’ (((π β§ π₯ β π) β§ π΄ = β
) β
(seqΟ((π
β V, π§ β V
β¦ (((π΄
βo (OrdIso( E , (π₯ supp β
))βπ)) Β·o (π₯β(OrdIso( E , (π₯ supp β
))βπ))) +o π§)), β
)ββ
) =
β
) |
45 | 19, 41, 44 | 3eqtrd 2777 |
. . . . 5
β’ (((π β§ π₯ β π) β§ π΄ = β
) β ((π΄ CNF π΅)βπ₯) = β
) |
46 | | el1o 8445 |
. . . . 5
β’ (((π΄ CNF π΅)βπ₯) β 1o β ((π΄ CNF π΅)βπ₯) = β
) |
47 | 45, 46 | sylibr 233 |
. . . 4
β’ (((π β§ π₯ β π) β§ π΄ = β
) β ((π΄ CNF π΅)βπ₯) β 1o) |
48 | 35 | oveq2d 7377 |
. . . . 5
β’ (((π β§ π₯ β π) β§ π΄ = β
) β (π΄ βo π΅) = (π΄ βo
β
)) |
49 | 13 | adantr 482 |
. . . . . 6
β’ (((π β§ π₯ β π) β§ π΄ = β
) β π΄ β On) |
50 | | oe0 8472 |
. . . . . 6
β’ (π΄ β On β (π΄ βo β
) =
1o) |
51 | 49, 50 | syl 17 |
. . . . 5
β’ (((π β§ π₯ β π) β§ π΄ = β
) β (π΄ βo β
) =
1o) |
52 | 48, 51 | eqtrd 2773 |
. . . 4
β’ (((π β§ π₯ β π) β§ π΄ = β
) β (π΄ βo π΅) = 1o) |
53 | 47, 52 | eleqtrrd 2837 |
. . 3
β’ (((π β§ π₯ β π) β§ π΄ = β
) β ((π΄ CNF π΅)βπ₯) β (π΄ βo π΅)) |
54 | 13 | adantr 482 |
. . . 4
β’ (((π β§ π₯ β π) β§ π΄ β β
) β π΄ β On) |
55 | 14 | adantr 482 |
. . . 4
β’ (((π β§ π₯ β π) β§ π΄ β β
) β π΅ β On) |
56 | 16 | adantr 482 |
. . . 4
β’ (((π β§ π₯ β π) β§ π΄ β β
) β π₯ β π) |
57 | | on0eln0 6377 |
. . . . . 6
β’ (π΄ β On β (β
β π΄ β π΄ β β
)) |
58 | 13, 57 | syl 17 |
. . . . 5
β’ ((π β§ π₯ β π) β (β
β π΄ β π΄ β β
)) |
59 | 58 | biimpar 479 |
. . . 4
β’ (((π β§ π₯ β π) β§ π΄ β β
) β β
β π΄) |
60 | 29 | adantr 482 |
. . . 4
β’ (((π β§ π₯ β π) β§ π΄ β β
) β (π₯ supp β
) β π΅) |
61 | 8, 54, 55, 56, 59, 55, 60 | cantnflt2 9617 |
. . 3
β’ (((π β§ π₯ β π) β§ π΄ β β
) β ((π΄ CNF π΅)βπ₯) β (π΄ βo π΅)) |
62 | 53, 61 | pm2.61dane 3029 |
. 2
β’ ((π β§ π₯ β π) β ((π΄ CNF π΅)βπ₯) β (π΄ βo π΅)) |
63 | 3, 12, 62 | fmpt2d 7075 |
1
β’ (π β (π΄ CNF π΅):πβΆ(π΄ βo π΅)) |