Step | Hyp | Ref
| Expression |
1 | | simpl2 1193 |
. . 3
β’ (((πΎ β π· β§ π β π΄ β§ π
β π΄) β§ (π β π
β§ π = {π β π΄ β£ π β€ (π β¨ π
)})) β π β π΄) |
2 | | simpl3 1194 |
. . 3
β’ (((πΎ β π· β§ π β π΄ β§ π
β π΄) β§ (π β π
β§ π = {π β π΄ β£ π β€ (π β¨ π
)})) β π
β π΄) |
3 | | simpr 486 |
. . 3
β’ (((πΎ β π· β§ π β π΄ β§ π
β π΄) β§ (π β π
β§ π = {π β π΄ β£ π β€ (π β¨ π
)})) β (π β π
β§ π = {π β π΄ β£ π β€ (π β¨ π
)})) |
4 | | neeq1 3007 |
. . . . 5
β’ (π = π β (π β π β π β π)) |
5 | | oveq1 7369 |
. . . . . . . 8
β’ (π = π β (π β¨ π) = (π β¨ π)) |
6 | 5 | breq2d 5122 |
. . . . . . 7
β’ (π = π β (π β€ (π β¨ π) β π β€ (π β¨ π))) |
7 | 6 | rabbidv 3418 |
. . . . . 6
β’ (π = π β {π β π΄ β£ π β€ (π β¨ π)} = {π β π΄ β£ π β€ (π β¨ π)}) |
8 | 7 | eqeq2d 2748 |
. . . . 5
β’ (π = π β (π = {π β π΄ β£ π β€ (π β¨ π)} β π = {π β π΄ β£ π β€ (π β¨ π)})) |
9 | 4, 8 | anbi12d 632 |
. . . 4
β’ (π = π β ((π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}) β (π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}))) |
10 | | neeq2 3008 |
. . . . 5
β’ (π = π
β (π β π β π β π
)) |
11 | | oveq2 7370 |
. . . . . . . 8
β’ (π = π
β (π β¨ π) = (π β¨ π
)) |
12 | 11 | breq2d 5122 |
. . . . . . 7
β’ (π = π
β (π β€ (π β¨ π) β π β€ (π β¨ π
))) |
13 | 12 | rabbidv 3418 |
. . . . . 6
β’ (π = π
β {π β π΄ β£ π β€ (π β¨ π)} = {π β π΄ β£ π β€ (π β¨ π
)}) |
14 | 13 | eqeq2d 2748 |
. . . . 5
β’ (π = π
β (π = {π β π΄ β£ π β€ (π β¨ π)} β π = {π β π΄ β£ π β€ (π β¨ π
)})) |
15 | 10, 14 | anbi12d 632 |
. . . 4
β’ (π = π
β ((π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}) β (π β π
β§ π = {π β π΄ β£ π β€ (π β¨ π
)}))) |
16 | 9, 15 | rspc2ev 3595 |
. . 3
β’ ((π β π΄ β§ π
β π΄ β§ (π β π
β§ π = {π β π΄ β£ π β€ (π β¨ π
)})) β βπ β π΄ βπ β π΄ (π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)})) |
17 | 1, 2, 3, 16 | syl3anc 1372 |
. 2
β’ (((πΎ β π· β§ π β π΄ β§ π
β π΄) β§ (π β π
β§ π = {π β π΄ β£ π β€ (π β¨ π
)})) β βπ β π΄ βπ β π΄ (π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)})) |
18 | | simpl1 1192 |
. . 3
β’ (((πΎ β π· β§ π β π΄ β§ π
β π΄) β§ (π β π
β§ π = {π β π΄ β£ π β€ (π β¨ π
)})) β πΎ β π·) |
19 | | isline.l |
. . . 4
β’ β€ =
(leβπΎ) |
20 | | isline.j |
. . . 4
β’ β¨ =
(joinβπΎ) |
21 | | isline.a |
. . . 4
β’ π΄ = (AtomsβπΎ) |
22 | | isline.n |
. . . 4
β’ π = (LinesβπΎ) |
23 | 19, 20, 21, 22 | isline 38231 |
. . 3
β’ (πΎ β π· β (π β π β βπ β π΄ βπ β π΄ (π β π β§ π = {π β π΄ β£ π β€ (π β¨ π)}))) |
24 | 18, 23 | syl 17 |
. 2
β’ (((πΎ β π· β§ π β π΄ β§ π
β π΄) β§ (π β π
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 = {π β π΄ β£ π β€ (π β¨ π)}))) |
25 | 17, 24 | mpbird 257 |
1
β’ (((πΎ β π· β§ π β π΄ β§ π
β π΄) β§ (π β π
β§ π = {π β π΄ β£ π β€ (π β¨ π
)})) β π β π) |