Step | Hyp | Ref
| Expression |
1 | | polval2.o |
. . 3
β’ β₯ =
(ocβπΎ) |
2 | | polval2.a |
. . 3
β’ π΄ = (AtomsβπΎ) |
3 | | polval2.m |
. . 3
β’ π = (pmapβπΎ) |
4 | | polval2.p |
. . 3
β’ π =
(β₯πβπΎ) |
5 | 1, 2, 3, 4 | polvalN 38371 |
. 2
β’ ((πΎ β HL β§ π β π΄) β (πβπ) = (π΄ β© β©
π β π (πβ( β₯ βπ)))) |
6 | | hlop 37827 |
. . . . . 6
β’ (πΎ β HL β πΎ β OP) |
7 | 6 | ad2antrr 725 |
. . . . 5
β’ (((πΎ β HL β§ π β π΄) β§ π β π) β πΎ β OP) |
8 | | ssel2 3940 |
. . . . . . 7
β’ ((π β π΄ β§ π β π) β π β π΄) |
9 | 8 | adantll 713 |
. . . . . 6
β’ (((πΎ β HL β§ π β π΄) β§ π β π) β π β π΄) |
10 | | eqid 2737 |
. . . . . . 7
β’
(BaseβπΎ) =
(BaseβπΎ) |
11 | 10, 2 | atbase 37754 |
. . . . . 6
β’ (π β π΄ β π β (BaseβπΎ)) |
12 | 9, 11 | syl 17 |
. . . . 5
β’ (((πΎ β HL β§ π β π΄) β§ π β π) β π β (BaseβπΎ)) |
13 | 10, 1 | opoccl 37659 |
. . . . 5
β’ ((πΎ β OP β§ π β (BaseβπΎ)) β ( β₯ βπ) β (BaseβπΎ)) |
14 | 7, 12, 13 | syl2anc 585 |
. . . 4
β’ (((πΎ β HL β§ π β π΄) β§ π β π) β ( β₯ βπ) β (BaseβπΎ)) |
15 | 14 | ralrimiva 3144 |
. . 3
β’ ((πΎ β HL β§ π β π΄) β βπ β π ( β₯ βπ) β (BaseβπΎ)) |
16 | | eqid 2737 |
. . . 4
β’
(glbβπΎ) =
(glbβπΎ) |
17 | 10, 16, 2, 3 | pmapglb2xN 38238 |
. . 3
β’ ((πΎ β HL β§ βπ β π ( β₯ βπ) β (BaseβπΎ)) β (πβ((glbβπΎ)β{π₯ β£ βπ β π π₯ = ( β₯ βπ)})) = (π΄ β© β©
π β π (πβ( β₯ βπ)))) |
18 | 15, 17 | syldan 592 |
. 2
β’ ((πΎ β HL β§ π β π΄) β (πβ((glbβπΎ)β{π₯ β£ βπ β π π₯ = ( β₯ βπ)})) = (π΄ β© β©
π β π (πβ( β₯ βπ)))) |
19 | | polval2.u |
. . . . . 6
β’ π = (lubβπΎ) |
20 | 10, 19, 16, 1 | glbconxN 37844 |
. . . . 5
β’ ((πΎ β HL β§ βπ β π ( β₯ βπ) β (BaseβπΎ)) β ((glbβπΎ)β{π₯ β£ βπ β π π₯ = ( β₯ βπ)}) = ( β₯ β(πβ{π₯ β£ βπ β π π₯ = ( β₯ β( β₯
βπ))}))) |
21 | 15, 20 | syldan 592 |
. . . 4
β’ ((πΎ β HL β§ π β π΄) β ((glbβπΎ)β{π₯ β£ βπ β π π₯ = ( β₯ βπ)}) = ( β₯ β(πβ{π₯ β£ βπ β π π₯ = ( β₯ β( β₯
βπ))}))) |
22 | 10, 1 | opococ 37660 |
. . . . . . . . . . 11
β’ ((πΎ β OP β§ π β (BaseβπΎ)) β ( β₯ β( β₯
βπ)) = π) |
23 | 7, 12, 22 | syl2anc 585 |
. . . . . . . . . 10
β’ (((πΎ β HL β§ π β π΄) β§ π β π) β ( β₯ β( β₯
βπ)) = π) |
24 | 23 | eqeq2d 2748 |
. . . . . . . . 9
β’ (((πΎ β HL β§ π β π΄) β§ π β π) β (π₯ = ( β₯ β( β₯
βπ)) β π₯ = π)) |
25 | 24 | rexbidva 3174 |
. . . . . . . 8
β’ ((πΎ β HL β§ π β π΄) β (βπ β π π₯ = ( β₯ β( β₯
βπ)) β
βπ β π π₯ = π)) |
26 | 25 | abbidv 2806 |
. . . . . . 7
β’ ((πΎ β HL β§ π β π΄) β {π₯ β£ βπ β π π₯ = ( β₯ β( β₯
βπ))} = {π₯ β£ βπ β π π₯ = π}) |
27 | | df-rex 3075 |
. . . . . . . . . 10
β’
(βπ β
π π₯ = π β βπ(π β π β§ π₯ = π)) |
28 | | equcom 2022 |
. . . . . . . . . . . 12
β’ (π₯ = π β π = π₯) |
29 | 28 | anbi1ci 627 |
. . . . . . . . . . 11
β’ ((π β π β§ π₯ = π) β (π = π₯ β§ π β π)) |
30 | 29 | exbii 1851 |
. . . . . . . . . 10
β’
(βπ(π β π β§ π₯ = π) β βπ(π = π₯ β§ π β π)) |
31 | | eleq1w 2821 |
. . . . . . . . . . 11
β’ (π = π₯ β (π β π β π₯ β π)) |
32 | 31 | equsexvw 2009 |
. . . . . . . . . 10
β’
(βπ(π = π₯ β§ π β π) β π₯ β π) |
33 | 27, 30, 32 | 3bitri 297 |
. . . . . . . . 9
β’
(βπ β
π π₯ = π β π₯ β π) |
34 | 33 | abbii 2807 |
. . . . . . . 8
β’ {π₯ β£ βπ β π π₯ = π} = {π₯ β£ π₯ β π} |
35 | | abid2 2876 |
. . . . . . . 8
β’ {π₯ β£ π₯ β π} = π |
36 | 34, 35 | eqtri 2765 |
. . . . . . 7
β’ {π₯ β£ βπ β π π₯ = π} = π |
37 | 26, 36 | eqtrdi 2793 |
. . . . . 6
β’ ((πΎ β HL β§ π β π΄) β {π₯ β£ βπ β π π₯ = ( β₯ β( β₯
βπ))} = π) |
38 | 37 | fveq2d 6847 |
. . . . 5
β’ ((πΎ β HL β§ π β π΄) β (πβ{π₯ β£ βπ β π π₯ = ( β₯ β( β₯
βπ))}) = (πβπ)) |
39 | 38 | fveq2d 6847 |
. . . 4
β’ ((πΎ β HL β§ π β π΄) β ( β₯ β(πβ{π₯ β£ βπ β π π₯ = ( β₯ β( β₯
βπ))})) = ( β₯
β(πβπ))) |
40 | 21, 39 | eqtrd 2777 |
. . 3
β’ ((πΎ β HL β§ π β π΄) β ((glbβπΎ)β{π₯ β£ βπ β π π₯ = ( β₯ βπ)}) = ( β₯ β(πβπ))) |
41 | 40 | fveq2d 6847 |
. 2
β’ ((πΎ β HL β§ π β π΄) β (πβ((glbβπΎ)β{π₯ β£ βπ β π π₯ = ( β₯ βπ)})) = (πβ( β₯ β(πβπ)))) |
42 | 5, 18, 41 | 3eqtr2d 2783 |
1
β’ ((πΎ β HL β§ π β π΄) β (πβπ) = (πβ( β₯ β(πβπ)))) |