Step | Hyp | Ref
| Expression |
1 | | eldiophelnn0 41134 |
. . 3
β’ (π΄ β (Diophβπ) β π β
β0) |
2 | | nnex 12167 |
. . . . . 6
β’ β
β V |
3 | 2 | jctr 526 |
. . . . 5
β’ (π β β0
β (π β
β0 β§ β β V)) |
4 | | 1z 12541 |
. . . . . . 7
β’ 1 β
β€ |
5 | | nnuz 12814 |
. . . . . . . 8
β’ β =
(β€β₯β1) |
6 | 5 | uzinf 13879 |
. . . . . . 7
β’ (1 β
β€ β Β¬ β β Fin) |
7 | 4, 6 | ax-mp 5 |
. . . . . 6
β’ Β¬
β β Fin |
8 | | elfznn 13479 |
. . . . . . 7
β’ (π β (1...π) β π β β) |
9 | 8 | ssriv 3952 |
. . . . . 6
β’
(1...π) β
β |
10 | 7, 9 | pm3.2i 472 |
. . . . 5
β’ (Β¬
β β Fin β§ (1...π) β β) |
11 | | eldioph2b 41133 |
. . . . . 6
β’ (((π β β0
β§ β β V) β§ (Β¬ β β Fin β§ (1...π) β β)) β
(π΄ β
(Diophβπ) β
βπ β
(mzPolyββ)π΄ =
{π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)})) |
12 | | eldioph2b 41133 |
. . . . . 6
β’ (((π β β0
β§ β β V) β§ (Β¬ β β Fin β§ (1...π) β β)) β
(π΅ β
(Diophβπ) β
βπ β
(mzPolyββ)π΅ =
{π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)})) |
13 | 11, 12 | anbi12d 632 |
. . . . 5
β’ (((π β β0
β§ β β V) β§ (Β¬ β β Fin β§ (1...π) β β)) β
((π΄ β
(Diophβπ) β§ π΅ β (Diophβπ)) β (βπ β
(mzPolyββ)π΄ =
{π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} β§ βπ β (mzPolyββ)π΅ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}))) |
14 | 3, 10, 13 | sylancl 587 |
. . . 4
β’ (π β β0
β ((π΄ β
(Diophβπ) β§ π΅ β (Diophβπ)) β (βπ β
(mzPolyββ)π΄ =
{π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} β§ βπ β (mzPolyββ)π΅ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}))) |
15 | | reeanv 3216 |
. . . . 5
β’
(βπ β
(mzPolyββ)βπ β (mzPolyββ)(π΄ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} β§ π΅ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) β (βπ β (mzPolyββ)π΄ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} β§ βπ β (mzPolyββ)π΅ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)})) |
16 | | unab 4262 |
. . . . . . . . 9
β’ ({π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} βͺ {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) = {π β£ (βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0) β¨ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0))} |
17 | | r19.43 3122 |
. . . . . . . . . . 11
β’
(βπ β
(β0 βm β)((π = (π βΎ (1...π)) β§ (πβπ) = 0) β¨ (π = (π βΎ (1...π)) β§ (πβπ) = 0)) β (βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0) β¨ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0))) |
18 | | andi 1007 |
. . . . . . . . . . . . 13
β’ ((π = (π βΎ (1...π)) β§ ((πβπ) = 0 β¨ (πβπ) = 0)) β ((π = (π βΎ (1...π)) β§ (πβπ) = 0) β¨ (π = (π βΎ (1...π)) β§ (πβπ) = 0))) |
19 | | zex 12516 |
. . . . . . . . . . . . . . . . . . . 20
β’ β€
β V |
20 | | nn0ssz 12530 |
. . . . . . . . . . . . . . . . . . . 20
β’
β0 β β€ |
21 | | mapss 8833 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((β€
β V β§ β0 β β€) β (β0
βm β) β (β€ βm
β)) |
22 | 19, 20, 21 | mp2an 691 |
. . . . . . . . . . . . . . . . . . 19
β’
(β0 βm β) β (β€
βm β) |
23 | 22 | sseli 3944 |
. . . . . . . . . . . . . . . . . 18
β’ (π β (β0
βm β) β π β (β€ βm
β)) |
24 | 23 | adantl 483 |
. . . . . . . . . . . . . . . . 17
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β π β (β€ βm
β)) |
25 | | fveq2 6846 |
. . . . . . . . . . . . . . . . . . 19
β’ (π = π β (πβπ) = (πβπ)) |
26 | | fveq2 6846 |
. . . . . . . . . . . . . . . . . . 19
β’ (π = π β (πβπ) = (πβπ)) |
27 | 25, 26 | oveq12d 7379 |
. . . . . . . . . . . . . . . . . 18
β’ (π = π β ((πβπ) Β· (πβπ)) = ((πβπ) Β· (πβπ))) |
28 | | eqid 2733 |
. . . . . . . . . . . . . . . . . 18
β’ (π β (β€
βm β) β¦ ((πβπ) Β· (πβπ))) = (π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ))) |
29 | | ovex 7394 |
. . . . . . . . . . . . . . . . . 18
β’ ((πβπ) Β· (πβπ)) β V |
30 | 27, 28, 29 | fvmpt 6952 |
. . . . . . . . . . . . . . . . 17
β’ (π β (β€
βm β) β ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = ((πβπ) Β· (πβπ))) |
31 | 24, 30 | syl 17 |
. . . . . . . . . . . . . . . 16
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = ((πβπ) Β· (πβπ))) |
32 | 31 | eqeq1d 2735 |
. . . . . . . . . . . . . . 15
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β (((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0 β ((πβπ) Β· (πβπ)) = 0)) |
33 | | simplrl 776 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β π β
(mzPolyββ)) |
34 | | mzpf 41106 |
. . . . . . . . . . . . . . . . . . 19
β’ (π β (mzPolyββ)
β π:(β€
βm β)βΆβ€) |
35 | 33, 34 | syl 17 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β π:(β€ βm
β)βΆβ€) |
36 | 35, 24 | ffvelcdmd 7040 |
. . . . . . . . . . . . . . . . 17
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β (πβπ) β β€) |
37 | 36 | zcnd 12616 |
. . . . . . . . . . . . . . . 16
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β (πβπ) β β) |
38 | | simplrr 777 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β π β
(mzPolyββ)) |
39 | | mzpf 41106 |
. . . . . . . . . . . . . . . . . . 19
β’ (π β (mzPolyββ)
β π:(β€
βm β)βΆβ€) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β π:(β€ βm
β)βΆβ€) |
41 | 40, 24 | ffvelcdmd 7040 |
. . . . . . . . . . . . . . . . 17
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β (πβπ) β β€) |
42 | 41 | zcnd 12616 |
. . . . . . . . . . . . . . . 16
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β (πβπ) β β) |
43 | 37, 42 | mul0ord 11813 |
. . . . . . . . . . . . . . 15
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β (((πβπ) Β· (πβπ)) = 0 β ((πβπ) = 0 β¨ (πβπ) = 0))) |
44 | 32, 43 | bitr2d 280 |
. . . . . . . . . . . . . 14
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β (((πβπ) = 0 β¨ (πβπ) = 0) β ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0)) |
45 | 44 | anbi2d 630 |
. . . . . . . . . . . . 13
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β ((π = (π βΎ (1...π)) β§ ((πβπ) = 0 β¨ (πβπ) = 0)) β (π = (π βΎ (1...π)) β§ ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0))) |
46 | 18, 45 | bitr3id 285 |
. . . . . . . . . . . 12
β’ (((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β§ π β (β0
βm β)) β (((π = (π βΎ (1...π)) β§ (πβπ) = 0) β¨ (π = (π βΎ (1...π)) β§ (πβπ) = 0)) β (π = (π βΎ (1...π)) β§ ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0))) |
47 | 46 | rexbidva 3170 |
. . . . . . . . . . 11
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
(βπ β
(β0 βm β)((π = (π βΎ (1...π)) β§ (πβπ) = 0) β¨ (π = (π βΎ (1...π)) β§ (πβπ) = 0)) β βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0))) |
48 | 17, 47 | bitr3id 285 |
. . . . . . . . . 10
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
((βπ β
(β0 βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0) β¨ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)) β βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0))) |
49 | 48 | abbidv 2802 |
. . . . . . . . 9
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
{π β£ (βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0) β¨ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0))} = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0)}) |
50 | 16, 49 | eqtrid 2785 |
. . . . . . . 8
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
({π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} βͺ {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0)}) |
51 | | simpl 484 |
. . . . . . . . 9
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β π β
β0) |
52 | 2, 9 | pm3.2i 472 |
. . . . . . . . . 10
β’ (β
β V β§ (1...π)
β β) |
53 | 52 | a1i 11 |
. . . . . . . . 9
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
(β β V β§ (1...π) β β)) |
54 | | simprl 770 |
. . . . . . . . . . . . 13
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β π β
(mzPolyββ)) |
55 | 54, 34 | syl 17 |
. . . . . . . . . . . 12
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β π:(β€ βm
β)βΆβ€) |
56 | 55 | feqmptd 6914 |
. . . . . . . . . . 11
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β π = (π β (β€ βm β)
β¦ (πβπ))) |
57 | 56, 54 | eqeltrrd 2835 |
. . . . . . . . . 10
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
(π β (β€
βm β) β¦ (πβπ)) β
(mzPolyββ)) |
58 | | simprr 772 |
. . . . . . . . . . . . 13
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β π β
(mzPolyββ)) |
59 | 58, 39 | syl 17 |
. . . . . . . . . . . 12
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β π:(β€ βm
β)βΆβ€) |
60 | 59 | feqmptd 6914 |
. . . . . . . . . . 11
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β π = (π β (β€ βm β)
β¦ (πβπ))) |
61 | 60, 58 | eqeltrrd 2835 |
. . . . . . . . . 10
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
(π β (β€
βm β) β¦ (πβπ)) β
(mzPolyββ)) |
62 | | mzpmulmpt 41112 |
. . . . . . . . . 10
β’ (((π β (β€
βm β) β¦ (πβπ)) β (mzPolyββ) β§ (π β (β€
βm β) β¦ (πβπ)) β (mzPolyββ)) β
(π β (β€
βm β) β¦ ((πβπ) Β· (πβπ))) β
(mzPolyββ)) |
63 | 57, 61, 62 | syl2anc 585 |
. . . . . . . . 9
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
(π β (β€
βm β) β¦ ((πβπ) Β· (πβπ))) β
(mzPolyββ)) |
64 | | eldioph2 41132 |
. . . . . . . . 9
β’ ((π β β0
β§ (β β V β§ (1...π) β β) β§ (π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ))) β (mzPolyββ)) β
{π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0)} β (Diophβπ)) |
65 | 51, 53, 63, 64 | syl3anc 1372 |
. . . . . . . 8
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
{π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ ((π β (β€ βm β)
β¦ ((πβπ) Β· (πβπ)))βπ) = 0)} β (Diophβπ)) |
66 | 50, 65 | eqeltrd 2834 |
. . . . . . 7
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
({π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} βͺ {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) β (Diophβπ)) |
67 | | uneq12 4122 |
. . . . . . . 8
β’ ((π΄ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} β§ π΅ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) β (π΄ βͺ π΅) = ({π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} βͺ {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)})) |
68 | 67 | eleq1d 2819 |
. . . . . . 7
β’ ((π΄ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} β§ π΅ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) β ((π΄ βͺ π΅) β (Diophβπ) β ({π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} βͺ {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) β (Diophβπ))) |
69 | 66, 68 | syl5ibrcom 247 |
. . . . . 6
β’ ((π β β0
β§ (π β
(mzPolyββ) β§ π β (mzPolyββ))) β
((π΄ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} β§ π΅ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) β (π΄ βͺ π΅) β (Diophβπ))) |
70 | 69 | rexlimdvva 3202 |
. . . . 5
β’ (π β β0
β (βπ β
(mzPolyββ)βπ β (mzPolyββ)(π΄ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} β§ π΅ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) β (π΄ βͺ π΅) β (Diophβπ))) |
71 | 15, 70 | biimtrrid 242 |
. . . 4
β’ (π β β0
β ((βπ β
(mzPolyββ)π΄ =
{π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)} β§ βπ β (mzPolyββ)π΅ = {π β£ βπ β (β0
βm β)(π = (π βΎ (1...π)) β§ (πβπ) = 0)}) β (π΄ βͺ π΅) β (Diophβπ))) |
72 | 14, 71 | sylbid 239 |
. . 3
β’ (π β β0
β ((π΄ β
(Diophβπ) β§ π΅ β (Diophβπ)) β (π΄ βͺ π΅) β (Diophβπ))) |
73 | 1, 72 | syl 17 |
. 2
β’ (π΄ β (Diophβπ) β ((π΄ β (Diophβπ) β§ π΅ β (Diophβπ)) β (π΄ βͺ π΅) β (Diophβπ))) |
74 | 73 | anabsi5 668 |
1
β’ ((π΄ β (Diophβπ) β§ π΅ β (Diophβπ)) β (π΄ βͺ π΅) β (Diophβπ)) |