Step | Hyp | Ref
| Expression |
1 | | df-perpg 27947 |
. . . . . 6
β’ βG
= (π β V β¦
{β¨π, πβ© β£ ((π β ran (LineGβπ) β§ π β ran (LineGβπ)) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπ))}) |
2 | | simpr 486 |
. . . . . . . . . . . . 13
β’ ((π β§ π = πΊ) β π = πΊ) |
3 | 2 | fveq2d 6896 |
. . . . . . . . . . . 12
β’ ((π β§ π = πΊ) β (LineGβπ) = (LineGβπΊ)) |
4 | | perpln.l |
. . . . . . . . . . . 12
β’ πΏ = (LineGβπΊ) |
5 | 3, 4 | eqtr4di 2791 |
. . . . . . . . . . 11
β’ ((π β§ π = πΊ) β (LineGβπ) = πΏ) |
6 | 5 | rneqd 5938 |
. . . . . . . . . 10
β’ ((π β§ π = πΊ) β ran (LineGβπ) = ran πΏ) |
7 | 6 | eleq2d 2820 |
. . . . . . . . 9
β’ ((π β§ π = πΊ) β (π β ran (LineGβπ) β π β ran πΏ)) |
8 | 6 | eleq2d 2820 |
. . . . . . . . 9
β’ ((π β§ π = πΊ) β (π β ran (LineGβπ) β π β ran πΏ)) |
9 | 7, 8 | anbi12d 632 |
. . . . . . . 8
β’ ((π β§ π = πΊ) β ((π β ran (LineGβπ) β§ π β ran (LineGβπ)) β (π β ran πΏ β§ π β ran πΏ))) |
10 | 2 | fveq2d 6896 |
. . . . . . . . . . 11
β’ ((π β§ π = πΊ) β (βGβπ) = (βGβπΊ)) |
11 | 10 | eleq2d 2820 |
. . . . . . . . . 10
β’ ((π β§ π = πΊ) β (β¨βπ’π₯π£ββ© β (βGβπ) β β¨βπ’π₯π£ββ© β (βGβπΊ))) |
12 | 11 | ralbidv 3178 |
. . . . . . . . 9
β’ ((π β§ π = πΊ) β (βπ£ β π β¨βπ’π₯π£ββ© β (βGβπ) β βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))) |
13 | 12 | rexralbidv 3221 |
. . . . . . . 8
β’ ((π β§ π = πΊ) β (βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπ) β βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))) |
14 | 9, 13 | anbi12d 632 |
. . . . . . 7
β’ ((π β§ π = πΊ) β (((π β ran (LineGβπ) β§ π β ran (LineGβπ)) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπ)) β ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ)))) |
15 | 14 | opabbidv 5215 |
. . . . . 6
β’ ((π β§ π = πΊ) β {β¨π, πβ© β£ ((π β ran (LineGβπ) β§ π β ran (LineGβπ)) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπ))} = {β¨π, πβ© β£ ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))}) |
16 | | perpln.1 |
. . . . . . 7
β’ (π β πΊ β TarskiG) |
17 | 16 | elexd 3495 |
. . . . . 6
β’ (π β πΊ β V) |
18 | 4 | fvexi 6906 |
. . . . . . . . 9
β’ πΏ β V |
19 | | rnexg 7895 |
. . . . . . . . 9
β’ (πΏ β V β ran πΏ β V) |
20 | 18, 19 | mp1i 13 |
. . . . . . . 8
β’ (π β ran πΏ β V) |
21 | 20, 20 | xpexd 7738 |
. . . . . . 7
β’ (π β (ran πΏ Γ ran πΏ) β V) |
22 | | opabssxp 5769 |
. . . . . . . 8
β’
{β¨π, πβ© β£ ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))} β (ran πΏ Γ ran πΏ) |
23 | 22 | a1i 11 |
. . . . . . 7
β’ (π β {β¨π, πβ© β£ ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))} β (ran πΏ Γ ran πΏ)) |
24 | 21, 23 | ssexd 5325 |
. . . . . 6
β’ (π β {β¨π, πβ© β£ ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))} β V) |
25 | 1, 15, 17, 24 | fvmptd2 7007 |
. . . . 5
β’ (π β (βGβπΊ) = {β¨π, πβ© β£ ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))}) |
26 | 25 | rneqd 5938 |
. . . 4
β’ (π β ran (βGβπΊ) = ran {β¨π, πβ© β£ ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))}) |
27 | 22 | rnssi 5940 |
. . . 4
β’ ran
{β¨π, πβ© β£ ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))} β ran (ran πΏ Γ ran πΏ) |
28 | 26, 27 | eqsstrdi 4037 |
. . 3
β’ (π β ran (βGβπΊ) β ran (ran πΏ Γ ran πΏ)) |
29 | | rnxpss 6172 |
. . 3
β’ ran (ran
πΏ Γ ran πΏ) β ran πΏ |
30 | 28, 29 | sstrdi 3995 |
. 2
β’ (π β ran (βGβπΊ) β ran πΏ) |
31 | | relopabv 5822 |
. . . . . 6
β’ Rel
{β¨π, πβ© β£ ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))} |
32 | 25 | releqd 5779 |
. . . . . 6
β’ (π β (Rel
(βGβπΊ) β
Rel {β¨π, πβ© β£ ((π β ran πΏ β§ π β ran πΏ) β§ βπ₯ β (π β© π)βπ’ β π βπ£ β π β¨βπ’π₯π£ββ© β (βGβπΊ))})) |
33 | 31, 32 | mpbiri 258 |
. . . . 5
β’ (π β Rel (βGβπΊ)) |
34 | | perpln.2 |
. . . . 5
β’ (π β π΄(βGβπΊ)π΅) |
35 | | brrelex12 5729 |
. . . . 5
β’ ((Rel
(βGβπΊ) β§
π΄(βGβπΊ)π΅) β (π΄ β V β§ π΅ β V)) |
36 | 33, 34, 35 | syl2anc 585 |
. . . 4
β’ (π β (π΄ β V β§ π΅ β V)) |
37 | 36 | simpld 496 |
. . 3
β’ (π β π΄ β V) |
38 | 36 | simprd 497 |
. . 3
β’ (π β π΅ β V) |
39 | | brelrng 5941 |
. . 3
β’ ((π΄ β V β§ π΅ β V β§ π΄(βGβπΊ)π΅) β π΅ β ran (βGβπΊ)) |
40 | 37, 38, 34, 39 | syl3anc 1372 |
. 2
β’ (π β π΅ β ran (βGβπΊ)) |
41 | 30, 40 | sseldd 3984 |
1
β’ (π β π΅ β ran πΏ) |