Step | Hyp | Ref
| Expression |
1 | | simpr 486 |
. . . . . . 7
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) |
2 | | simplrr 777 |
. . . . . . . . 9
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β π β (LIdealβπ
)) |
3 | | prmidlval.1 |
. . . . . . . . . 10
β’ π΅ = (Baseβπ
) |
4 | | eqid 2733 |
. . . . . . . . . 10
β’
(LIdealβπ
) =
(LIdealβπ
) |
5 | 3, 4 | lidlss 20696 |
. . . . . . . . 9
β’ (π β (LIdealβπ
) β π β π΅) |
6 | 2, 5 | syl 17 |
. . . . . . . 8
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β π β π΅) |
7 | | simplrl 776 |
. . . . . . . . . 10
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β π β (LIdealβπ
)) |
8 | 3, 4 | lidlss 20696 |
. . . . . . . . . 10
β’ (π β (LIdealβπ
) β π β π΅) |
9 | 7, 8 | syl 17 |
. . . . . . . . 9
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β π β π΅) |
10 | | simpllr 775 |
. . . . . . . . 9
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) |
11 | | ssralv 4011 |
. . . . . . . . 9
β’ (π β π΅ β (βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π)) β βπ₯ β π βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π)))) |
12 | 9, 10, 11 | sylc 65 |
. . . . . . . 8
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β βπ₯ β π βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) |
13 | | ssralv 4011 |
. . . . . . . . 9
β’ (π β π΅ β (βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π)) β βπ¦ β π ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π)))) |
14 | 13 | ralimdv 3163 |
. . . . . . . 8
β’ (π β π΅ β (βπ₯ β π βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π)) β βπ₯ β π βπ¦ β π ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π)))) |
15 | 6, 12, 14 | sylc 65 |
. . . . . . 7
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β βπ₯ β π βπ¦ β π ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) |
16 | | r19.26-2 3132 |
. . . . . . . 8
β’
(βπ₯ β
π βπ¦ β π ((π₯ Β· π¦) β π β§ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β (βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π β§ βπ₯ β π βπ¦ β π ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π)))) |
17 | | pm3.35 802 |
. . . . . . . . 9
β’ (((π₯ Β· π¦) β π β§ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β (π₯ β π β¨ π¦ β π)) |
18 | 17 | 2ralimi 3123 |
. . . . . . . 8
β’
(βπ₯ β
π βπ¦ β π ((π₯ Β· π¦) β π β§ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β βπ₯ β π βπ¦ β π (π₯ β π β¨ π¦ β π)) |
19 | 16, 18 | sylbir 234 |
. . . . . . 7
β’
((βπ₯ β
π βπ¦ β π (π₯ Β· π¦) β π β§ βπ₯ β π βπ¦ β π ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β βπ₯ β π βπ¦ β π (π₯ β π β¨ π¦ β π)) |
20 | 1, 15, 19 | syl2anc 585 |
. . . . . 6
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β βπ₯ β π βπ¦ β π (π₯ β π β¨ π¦ β π)) |
21 | | 2ralor 3218 |
. . . . . . 7
β’
(βπ₯ β
π βπ¦ β π (π₯ β π β¨ π¦ β π) β (βπ₯ β π π₯ β π β¨ βπ¦ β π π¦ β π)) |
22 | | dfss3 3933 |
. . . . . . . 8
β’ (π β π β βπ₯ β π π₯ β π) |
23 | | dfss3 3933 |
. . . . . . . 8
β’ (π β π β βπ¦ β π π¦ β π) |
24 | 22, 23 | orbi12i 914 |
. . . . . . 7
β’ ((π β π β¨ π β π) β (βπ₯ β π π₯ β π β¨ βπ¦ β π π¦ β π)) |
25 | 21, 24 | sylbb2 237 |
. . . . . 6
β’
(βπ₯ β
π βπ¦ β π (π₯ β π β¨ π¦ β π) β (π β π β¨ π β π)) |
26 | 20, 25 | syl 17 |
. . . . 5
β’
((((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β§ βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π) β (π β π β¨ π β π)) |
27 | 26 | ex 414 |
. . . 4
β’
(((((π
β Ring
β§ π β
(LIdealβπ
)) β§
π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β§ (π β (LIdealβπ
) β§ π β (LIdealβπ
))) β (βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π β (π β π β¨ π β π))) |
28 | 27 | ralrimivva 3194 |
. . 3
β’ ((((π
β Ring β§ π β (LIdealβπ
)) β§ π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β βπ β (LIdealβπ
)βπ β (LIdealβπ
)(βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π β (π β π β¨ π β π))) |
29 | | prmidlval.2 |
. . . . . 6
β’ Β· =
(.rβπ
) |
30 | 3, 29 | isprmidl 32258 |
. . . . 5
β’ (π
β Ring β (π β (PrmIdealβπ
) β (π β (LIdealβπ
) β§ π β π΅ β§ βπ β (LIdealβπ
)βπ β (LIdealβπ
)(βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π β (π β π β¨ π β π))))) |
31 | 30 | biimpar 479 |
. . . 4
β’ ((π
β Ring β§ (π β (LIdealβπ
) β§ π β π΅ β§ βπ β (LIdealβπ
)βπ β (LIdealβπ
)(βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π β (π β π β¨ π β π)))) β π β (PrmIdealβπ
)) |
32 | 31 | 3anassrs 1361 |
. . 3
β’ ((((π
β Ring β§ π β (LIdealβπ
)) β§ π β π΅) β§ βπ β (LIdealβπ
)βπ β (LIdealβπ
)(βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π β (π β π β¨ π β π))) β π β (PrmIdealβπ
)) |
33 | 28, 32 | syldan 592 |
. 2
β’ ((((π
β Ring β§ π β (LIdealβπ
)) β§ π β π΅) β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π))) β π β (PrmIdealβπ
)) |
34 | 33 | anasss 468 |
1
β’ (((π
β Ring β§ π β (LIdealβπ
)) β§ (π β π΅ β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯ Β· π¦) β π β (π₯ β π β¨ π¦ β π)))) β π β (PrmIdealβπ
)) |