Step | Hyp | Ref
| Expression |
1 | | islpln3.b |
. . 3
β’ π΅ = (BaseβπΎ) |
2 | | eqid 2737 |
. . 3
β’ ( β
βπΎ) = ( β
βπΎ) |
3 | | islpln3.n |
. . 3
β’ π = (LLinesβπΎ) |
4 | | islpln3.p |
. . 3
β’ π = (LPlanesβπΎ) |
5 | 1, 2, 3, 4 | islpln4 38023 |
. 2
β’ ((πΎ β HL β§ π β π΅) β (π β π β βπ¦ β π π¦( β βπΎ)π)) |
6 | | simpll 766 |
. . . . 5
β’ (((πΎ β HL β§ π β π΅) β§ π¦ β π) β πΎ β HL) |
7 | 1, 3 | llnbase 38001 |
. . . . . 6
β’ (π¦ β π β π¦ β π΅) |
8 | 7 | adantl 483 |
. . . . 5
β’ (((πΎ β HL β§ π β π΅) β§ π¦ β π) β π¦ β π΅) |
9 | | simplr 768 |
. . . . 5
β’ (((πΎ β HL β§ π β π΅) β§ π¦ β π) β π β π΅) |
10 | | islpln3.l |
. . . . . 6
β’ β€ =
(leβπΎ) |
11 | | islpln3.j |
. . . . . 6
β’ β¨ =
(joinβπΎ) |
12 | | islpln3.a |
. . . . . 6
β’ π΄ = (AtomsβπΎ) |
13 | 1, 10, 11, 2, 12 | cvrval3 37905 |
. . . . 5
β’ ((πΎ β HL β§ π¦ β π΅ β§ π β π΅) β (π¦( β βπΎ)π β βπ β π΄ (Β¬ π β€ π¦ β§ (π¦ β¨ π) = π))) |
14 | 6, 8, 9, 13 | syl3anc 1372 |
. . . 4
β’ (((πΎ β HL β§ π β π΅) β§ π¦ β π) β (π¦( β βπΎ)π β βπ β π΄ (Β¬ π β€ π¦ β§ (π¦ β¨ π) = π))) |
15 | | eqcom 2744 |
. . . . . . 7
β’ ((π¦ β¨ π) = π β π = (π¦ β¨ π)) |
16 | 15 | a1i 11 |
. . . . . 6
β’ ((((πΎ β HL β§ π β π΅) β§ π¦ β π) β§ π β π΄) β ((π¦ β¨ π) = π β π = (π¦ β¨ π))) |
17 | 16 | anbi2d 630 |
. . . . 5
β’ ((((πΎ β HL β§ π β π΅) β§ π¦ β π) β§ π β π΄) β ((Β¬ π β€ π¦ β§ (π¦ β¨ π) = π) β (Β¬ π β€ π¦ β§ π = (π¦ β¨ π)))) |
18 | 17 | rexbidva 3174 |
. . . 4
β’ (((πΎ β HL β§ π β π΅) β§ π¦ β π) β (βπ β π΄ (Β¬ π β€ π¦ β§ (π¦ β¨ π) = π) β βπ β π΄ (Β¬ π β€ π¦ β§ π = (π¦ β¨ π)))) |
19 | 14, 18 | bitrd 279 |
. . 3
β’ (((πΎ β HL β§ π β π΅) β§ π¦ β π) β (π¦( β βπΎ)π β βπ β π΄ (Β¬ π β€ π¦ β§ π = (π¦ β¨ π)))) |
20 | 19 | rexbidva 3174 |
. 2
β’ ((πΎ β HL β§ π β π΅) β (βπ¦ β π π¦( β βπΎ)π β βπ¦ β π βπ β π΄ (Β¬ π β€ π¦ β§ π = (π¦ β¨ π)))) |
21 | 5, 20 | bitrd 279 |
1
β’ ((πΎ β HL β§ π β π΅) β (π β π β βπ¦ β π βπ β π΄ (Β¬ π β€ π¦ β§ π = (π¦ β¨ π)))) |