Step | Hyp | Ref
| Expression |
1 | | pjpm.v |
. . . . 5
β’ π = (Baseβπ) |
2 | | pjpm.l |
. . . . 5
β’ πΏ = (LSubSpβπ) |
3 | | eqid 2733 |
. . . . 5
β’
(ocvβπ) =
(ocvβπ) |
4 | | eqid 2733 |
. . . . 5
β’
(proj1βπ) = (proj1βπ) |
5 | | pjpm.k |
. . . . 5
β’ πΎ = (projβπ) |
6 | 1, 2, 3, 4, 5 | pjfval 21128 |
. . . 4
β’ πΎ = ((π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) β© (V Γ (π βm π))) |
7 | | inss1 4189 |
. . . 4
β’ ((π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) β© (V Γ (π βm π))) β (π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) |
8 | 6, 7 | eqsstri 3979 |
. . 3
β’ πΎ β (π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) |
9 | | funmpt 6540 |
. . 3
β’ Fun
(π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) |
10 | | funss 6521 |
. . 3
β’ (πΎ β (π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) β (Fun (π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) β Fun πΎ)) |
11 | 8, 9, 10 | mp2 9 |
. 2
β’ Fun πΎ |
12 | | eqid 2733 |
. . . . . 6
β’ (π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) = (π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) |
13 | | ovexd 7393 |
. . . . . 6
β’ (π₯ β πΏ β (π₯(proj1βπ)((ocvβπ)βπ₯)) β V) |
14 | 12, 13 | fmpti 7061 |
. . . . 5
β’ (π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))):πΏβΆV |
15 | | fssxp 6697 |
. . . . 5
β’ ((π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))):πΏβΆV β (π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) β (πΏ Γ V)) |
16 | | ssrin 4194 |
. . . . 5
β’ ((π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) β (πΏ Γ V) β ((π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) β© (V Γ (π βm π))) β ((πΏ Γ V) β© (V Γ (π βm π)))) |
17 | 14, 15, 16 | mp2b 10 |
. . . 4
β’ ((π₯ β πΏ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) β© (V Γ (π βm π))) β ((πΏ Γ V) β© (V Γ (π βm π))) |
18 | 6, 17 | eqsstri 3979 |
. . 3
β’ πΎ β ((πΏ Γ V) β© (V Γ (π βm π))) |
19 | | inxp 5789 |
. . . 4
β’ ((πΏ Γ V) β© (V Γ
(π βm π))) = ((πΏ β© V) Γ (V β© (π βm π))) |
20 | | inv1 4355 |
. . . . 5
β’ (πΏ β© V) = πΏ |
21 | | incom 4162 |
. . . . . 6
β’ (V β©
(π βm π)) = ((π βm π) β© V) |
22 | | inv1 4355 |
. . . . . 6
β’ ((π βm π) β© V) = (π βm π) |
23 | 21, 22 | eqtri 2761 |
. . . . 5
β’ (V β©
(π βm π)) = (π βm π) |
24 | 20, 23 | xpeq12i 5662 |
. . . 4
β’ ((πΏ β© V) Γ (V β©
(π βm π))) = (πΏ Γ (π βm π)) |
25 | 19, 24 | eqtri 2761 |
. . 3
β’ ((πΏ Γ V) β© (V Γ
(π βm π))) = (πΏ Γ (π βm π)) |
26 | 18, 25 | sseqtri 3981 |
. 2
β’ πΎ β (πΏ Γ (π βm π)) |
27 | | ovex 7391 |
. . 3
β’ (π βm π) β V |
28 | 2 | fvexi 6857 |
. . 3
β’ πΏ β V |
29 | 27, 28 | elpm 8814 |
. 2
β’ (πΎ β ((π βm π) βpm πΏ) β (Fun πΎ β§ πΎ β (πΏ Γ (π βm π)))) |
30 | 11, 26, 29 | mpbir2an 710 |
1
β’ πΎ β ((π βm π) βpm πΏ) |