Step | Hyp | Ref
| Expression |
1 | | opnvonmbllem2.x |
. . . . . . . . . . 11
β’ (π β π β Fin) |
2 | | eqid 2733 |
. . . . . . . . . . . 12
β’
(distβ(β^βπ)) = (distβ(β^βπ)) |
3 | 2 | rrxmetfi 24799 |
. . . . . . . . . . 11
β’ (π β Fin β
(distβ(β^βπ)) β (Metβ(β
βm π))) |
4 | 1, 3 | syl 17 |
. . . . . . . . . 10
β’ (π β
(distβ(β^βπ)) β (Metβ(β
βm π))) |
5 | | metxmet 23710 |
. . . . . . . . . 10
β’
((distβ(β^βπ)) β (Metβ(β
βm π))
β (distβ(β^βπ)) β (βMetβ(β
βm π))) |
6 | 4, 5 | syl 17 |
. . . . . . . . 9
β’ (π β
(distβ(β^βπ)) β (βMetβ(β
βm π))) |
7 | 6 | adantr 482 |
. . . . . . . 8
β’ ((π β§ π₯ β πΊ) β (distβ(β^βπ)) β
(βMetβ(β βm π))) |
8 | | opnvonmbllem2.g |
. . . . . . . . . 10
β’ (π β πΊ β (TopOpenβ(β^βπ))) |
9 | | eqid 2733 |
. . . . . . . . . . . . . 14
β’
(β^βπ) =
(β^βπ) |
10 | 9 | rrxval 24774 |
. . . . . . . . . . . . 13
β’ (π β Fin β
(β^βπ) =
(toβPreHilβ(βfld freeLMod π))) |
11 | 1, 10 | syl 17 |
. . . . . . . . . . . 12
β’ (π β (β^βπ) =
(toβPreHilβ(βfld freeLMod π))) |
12 | 11 | fveq2d 6850 |
. . . . . . . . . . 11
β’ (π β
(TopOpenβ(β^βπ)) =
(TopOpenβ(toβPreHilβ(βfld freeLMod π)))) |
13 | | ovex 7394 |
. . . . . . . . . . . . 13
β’
(βfld freeLMod π) β V |
14 | | eqid 2733 |
. . . . . . . . . . . . . 14
β’
(toβPreHilβ(βfld freeLMod π)) =
(toβPreHilβ(βfld freeLMod π)) |
15 | | eqid 2733 |
. . . . . . . . . . . . . 14
β’
(distβ(toβPreHilβ(βfld freeLMod π))) =
(distβ(toβPreHilβ(βfld freeLMod π))) |
16 | | eqid 2733 |
. . . . . . . . . . . . . 14
β’
(TopOpenβ(toβPreHilβ(βfld freeLMod
π))) =
(TopOpenβ(toβPreHilβ(βfld freeLMod π))) |
17 | 14, 15, 16 | tcphtopn 24613 |
. . . . . . . . . . . . 13
β’
((βfld freeLMod π) β V β
(TopOpenβ(toβPreHilβ(βfld freeLMod π))) =
(MetOpenβ(distβ(toβPreHilβ(βfld
freeLMod π))))) |
18 | 13, 17 | ax-mp 5 |
. . . . . . . . . . . 12
β’
(TopOpenβ(toβPreHilβ(βfld freeLMod
π))) =
(MetOpenβ(distβ(toβPreHilβ(βfld
freeLMod π)))) |
19 | 18 | a1i 11 |
. . . . . . . . . . 11
β’ (π β
(TopOpenβ(toβPreHilβ(βfld freeLMod π))) =
(MetOpenβ(distβ(toβPreHilβ(βfld
freeLMod π))))) |
20 | 11 | eqcomd 2739 |
. . . . . . . . . . . . 13
β’ (π β
(toβPreHilβ(βfld freeLMod π)) = (β^βπ)) |
21 | 20 | fveq2d 6850 |
. . . . . . . . . . . 12
β’ (π β
(distβ(toβPreHilβ(βfld freeLMod π))) =
(distβ(β^βπ))) |
22 | 21 | fveq2d 6850 |
. . . . . . . . . . 11
β’ (π β
(MetOpenβ(distβ(toβPreHilβ(βfld
freeLMod π)))) =
(MetOpenβ(distβ(β^βπ)))) |
23 | 12, 19, 22 | 3eqtrd 2777 |
. . . . . . . . . 10
β’ (π β
(TopOpenβ(β^βπ)) =
(MetOpenβ(distβ(β^βπ)))) |
24 | 8, 23 | eleqtrd 2836 |
. . . . . . . . 9
β’ (π β πΊ β
(MetOpenβ(distβ(β^βπ)))) |
25 | 24 | adantr 482 |
. . . . . . . 8
β’ ((π β§ π₯ β πΊ) β πΊ β
(MetOpenβ(distβ(β^βπ)))) |
26 | | simpr 486 |
. . . . . . . 8
β’ ((π β§ π₯ β πΊ) β π₯ β πΊ) |
27 | | eqid 2733 |
. . . . . . . . 9
β’
(MetOpenβ(distβ(β^βπ))) =
(MetOpenβ(distβ(β^βπ))) |
28 | 27 | mopni2 23872 |
. . . . . . . 8
β’
(((distβ(β^βπ)) β (βMetβ(β
βm π))
β§ πΊ β
(MetOpenβ(distβ(β^βπ))) β§ π₯ β πΊ) β βπ β β+ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) |
29 | 7, 25, 26, 28 | syl3anc 1372 |
. . . . . . 7
β’ ((π β§ π₯ β πΊ) β βπ β β+ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) |
30 | 1 | ad2antrr 725 |
. . . . . . . . . . . 12
β’ (((π β§ π₯ β πΊ) β§ π β β+) β π β Fin) |
31 | | eqid 2733 |
. . . . . . . . . . . . . . . . . 18
β’
(TopOpenβ(β^βπ)) = (TopOpenβ(β^βπ)) |
32 | 31 | rrxtoponfi 44622 |
. . . . . . . . . . . . . . . . 17
β’ (π β Fin β
(TopOpenβ(β^βπ)) β (TopOnβ(β
βm π))) |
33 | 1, 32 | syl 17 |
. . . . . . . . . . . . . . . 16
β’ (π β
(TopOpenβ(β^βπ)) β (TopOnβ(β
βm π))) |
34 | | toponss 22299 |
. . . . . . . . . . . . . . . 16
β’
(((TopOpenβ(β^βπ)) β (TopOnβ(β
βm π))
β§ πΊ β
(TopOpenβ(β^βπ))) β πΊ β (β βm π)) |
35 | 33, 8, 34 | syl2anc 585 |
. . . . . . . . . . . . . . 15
β’ (π β πΊ β (β βm π)) |
36 | 35 | adantr 482 |
. . . . . . . . . . . . . 14
β’ ((π β§ π₯ β πΊ) β πΊ β (β βm π)) |
37 | 36, 26 | sseldd 3949 |
. . . . . . . . . . . . 13
β’ ((π β§ π₯ β πΊ) β π₯ β (β βm π)) |
38 | 37 | adantr 482 |
. . . . . . . . . . . 12
β’ (((π β§ π₯ β πΊ) β§ π β β+) β π₯ β (β
βm π)) |
39 | | simpr 486 |
. . . . . . . . . . . 12
β’ (((π β§ π₯ β πΊ) β§ π β β+) β π β
β+) |
40 | 30, 38, 39 | hoiqssbl 44956 |
. . . . . . . . . . 11
β’ (((π β§ π₯ β πΊ) β§ π β β+) β
βπ β (β
βm π)βπ β (β βm π)(π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) |
41 | 40 | 3adant3 1133 |
. . . . . . . . . 10
β’ (((π β§ π₯ β πΊ) β§ π β β+ β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β βπ β (β βm π)βπ β (β βm π)(π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) |
42 | | nfv 1918 |
. . . . . . . . . . . . . . . 16
β’
β²π(π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) |
43 | | nfv 1918 |
. . . . . . . . . . . . . . . 16
β’
β²π(π β (β
βm π) β§
π β (β
βm π)) |
44 | | nfcv 2904 |
. . . . . . . . . . . . . . . . . 18
β’
β²ππ₯ |
45 | | nfixp1 8862 |
. . . . . . . . . . . . . . . . . 18
β’
β²πXπ β
π ((πβπ)[,)(πβπ)) |
46 | 44, 45 | nfel 2918 |
. . . . . . . . . . . . . . . . 17
β’
β²π π₯ β Xπ β
π ((πβπ)[,)(πβπ)) |
47 | | nfcv 2904 |
. . . . . . . . . . . . . . . . . 18
β’
β²π(π₯(ballβ(distβ(β^βπ)))π) |
48 | 45, 47 | nfss 3940 |
. . . . . . . . . . . . . . . . 17
β’
β²πXπ β
π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π) |
49 | 46, 48 | nfan 1903 |
. . . . . . . . . . . . . . . 16
β’
β²π(π₯ β Xπ β
π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π)) |
50 | 42, 43, 49 | nf3an 1905 |
. . . . . . . . . . . . . . 15
β’
β²π((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β§ (π β (β βm π) β§ π β (β βm π)) β§ (π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) |
51 | 1 | adantr 482 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β π β Fin) |
52 | 51 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . 15
β’ (((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β§ (π β (β βm π) β§ π β (β βm π)) β§ (π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) β π β Fin) |
53 | | elmapi 8793 |
. . . . . . . . . . . . . . . . 17
β’ (π β (β
βm π)
β π:πβΆβ) |
54 | 53 | adantr 482 |
. . . . . . . . . . . . . . . 16
β’ ((π β (β
βm π) β§
π β (β
βm π))
β π:πβΆβ) |
55 | 54 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . 15
β’ (((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β§ (π β (β βm π) β§ π β (β βm π)) β§ (π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) β π:πβΆβ) |
56 | | elmapi 8793 |
. . . . . . . . . . . . . . . . 17
β’ (π β (β
βm π)
β π:πβΆβ) |
57 | 56 | adantl 483 |
. . . . . . . . . . . . . . . 16
β’ ((π β (β
βm π) β§
π β (β
βm π))
β π:πβΆβ) |
58 | 57 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . 15
β’ (((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β§ (π β (β βm π) β§ π β (β βm π)) β§ (π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) β π:πβΆβ) |
59 | | simp3r 1203 |
. . . . . . . . . . . . . . 15
β’ (((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β§ (π β (β βm π) β§ π β (β βm π)) β§ (π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) β Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π)) |
60 | | simp1r 1199 |
. . . . . . . . . . . . . . 15
β’ (((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β§ (π β (β βm π) β§ π β (β βm π)) β§ (π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) β (π₯(ballβ(distβ(β^βπ)))π) β πΊ) |
61 | | simp3l 1202 |
. . . . . . . . . . . . . . 15
β’ (((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β§ (π β (β βm π) β§ π β (β βm π)) β§ (π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) β π₯ β Xπ β π ((πβπ)[,)(πβπ))) |
62 | | opnvonmbl.k |
. . . . . . . . . . . . . . 15
β’ πΎ = {β β ((β Γ β)
βm π)
β£ Xπ β π (([,) β β)βπ) β πΊ} |
63 | | eqid 2733 |
. . . . . . . . . . . . . . 15
β’ (π β π β¦ β¨(πβπ), (πβπ)β©) = (π β π β¦ β¨(πβπ), (πβπ)β©) |
64 | 50, 52, 55, 58, 59, 60, 61, 62, 63 | opnvonmbllem1 44963 |
. . . . . . . . . . . . . 14
β’ (((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β§ (π β (β βm π) β§ π β (β βm π)) β§ (π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π))) β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ)) |
65 | 64 | 3exp 1120 |
. . . . . . . . . . . . 13
β’ ((π β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β ((π β (β βm π) β§ π β (β βm π)) β ((π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π)) β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ)))) |
66 | 65 | adantlr 714 |
. . . . . . . . . . . 12
β’ (((π β§ π₯ β πΊ) β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β ((π β (β βm π) β§ π β (β βm π)) β ((π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π)) β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ)))) |
67 | 66 | 3adant2 1132 |
. . . . . . . . . . 11
β’ (((π β§ π₯ β πΊ) β§ π β β+ β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β ((π β (β βm π) β§ π β (β βm π)) β ((π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π)) β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ)))) |
68 | 67 | rexlimdvv 3201 |
. . . . . . . . . 10
β’ (((π β§ π₯ β πΊ) β§ π β β+ β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β (βπ β (β βm π)βπ β (β βm π)(π₯ β Xπ β π ((πβπ)[,)(πβπ)) β§ Xπ β π ((πβπ)[,)(πβπ)) β (π₯(ballβ(distβ(β^βπ)))π)) β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ))) |
69 | 41, 68 | mpd 15 |
. . . . . . . . 9
β’ (((π β§ π₯ β πΊ) β§ π β β+ β§ (π₯(ballβ(distβ(β^βπ)))π) β πΊ) β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ)) |
70 | 69 | 3exp 1120 |
. . . . . . . 8
β’ ((π β§ π₯ β πΊ) β (π β β+ β ((π₯(ballβ(distβ(β^βπ)))π) β πΊ β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ)))) |
71 | 70 | rexlimdv 3147 |
. . . . . . 7
β’ ((π β§ π₯ β πΊ) β (βπ β β+ (π₯(ballβ(distβ(β^βπ)))π) β πΊ β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ))) |
72 | 29, 71 | mpd 15 |
. . . . . 6
β’ ((π β§ π₯ β πΊ) β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ)) |
73 | | eliun 4962 |
. . . . . 6
β’ (π₯ β βͺ β
β πΎ Xπ β
π (([,) β β)βπ) β ββ β πΎ π₯ β Xπ β π (([,) β β)βπ)) |
74 | 72, 73 | sylibr 233 |
. . . . 5
β’ ((π β§ π₯ β πΊ) β π₯ β βͺ
β β πΎ Xπ β π (([,) β β)βπ)) |
75 | 74 | ralrimiva 3140 |
. . . 4
β’ (π β βπ₯ β πΊ π₯ β βͺ
β β πΎ Xπ β π (([,) β β)βπ)) |
76 | | dfss3 3936 |
. . . 4
β’ (πΊ β βͺ β
β πΎ Xπ β
π (([,) β β)βπ) β βπ₯ β πΊ π₯ β βͺ
β β πΎ Xπ β π (([,) β β)βπ)) |
77 | 75, 76 | sylibr 233 |
. . 3
β’ (π β πΊ β βͺ
β β πΎ Xπ β π (([,) β β)βπ)) |
78 | 62 | eleq2i 2826 |
. . . . . . . . 9
β’ (β β πΎ β β β {β β ((β Γ β)
βm π)
β£ Xπ β π (([,) β β)βπ) β πΊ}) |
79 | 78 | biimpi 215 |
. . . . . . . 8
β’ (β β πΎ β β β {β β ((β Γ β)
βm π)
β£ Xπ β π (([,) β β)βπ) β πΊ}) |
80 | 79 | adantl 483 |
. . . . . . 7
β’ ((π β§ β β πΎ) β β β {β β ((β Γ β)
βm π)
β£ Xπ β π (([,) β β)βπ) β πΊ}) |
81 | | rabid 3426 |
. . . . . . 7
β’ (β β {β β ((β Γ β)
βm π)
β£ Xπ β π (([,) β β)βπ) β πΊ} β (β β ((β Γ β)
βm π) β§
Xπ
β π (([,) β
β)βπ) β πΊ)) |
82 | 80, 81 | sylib 217 |
. . . . . 6
β’ ((π β§ β β πΎ) β (β β ((β Γ β)
βm π) β§
Xπ
β π (([,) β
β)βπ) β πΊ)) |
83 | 82 | simprd 497 |
. . . . 5
β’ ((π β§ β β πΎ) β Xπ β π (([,) β β)βπ) β πΊ) |
84 | 83 | ralrimiva 3140 |
. . . 4
β’ (π β ββ β πΎ Xπ β π (([,) β β)βπ) β πΊ) |
85 | | iunss 5009 |
. . . 4
β’ (βͺ β
β πΎ Xπ β
π (([,) β β)βπ) β πΊ β ββ β πΎ Xπ β π (([,) β β)βπ) β πΊ) |
86 | 84, 85 | sylibr 233 |
. . 3
β’ (π β βͺ β
β πΎ Xπ β
π (([,) β β)βπ) β πΊ) |
87 | 77, 86 | eqssd 3965 |
. 2
β’ (π β πΊ = βͺ β β πΎ Xπ β π (([,) β β)βπ)) |
88 | | opnvonmbllem2.n |
. . . 4
β’ π = dom (volnβπ) |
89 | 1, 88 | dmovnsal 44943 |
. . 3
β’ (π β π β SAlg) |
90 | | ssrab2 4041 |
. . . . . 6
β’ {β β ((β Γ
β) βm π) β£ Xπ β π (([,) β β)βπ) β πΊ} β ((β Γ β)
βm π) |
91 | 62, 90 | eqsstri 3982 |
. . . . 5
β’ πΎ β ((β Γ
β) βm π) |
92 | 91 | a1i 11 |
. . . 4
β’ (π β πΎ β ((β Γ β)
βm π)) |
93 | | qct 43687 |
. . . . . . 7
β’ β
βΌ Ο |
94 | 93 | a1i 11 |
. . . . . 6
β’ (π β β βΌ
Ο) |
95 | | xpct 9960 |
. . . . . 6
β’ ((β
βΌ Ο β§ β βΌ Ο) β (β Γ β)
βΌ Ο) |
96 | 94, 94, 95 | syl2anc 585 |
. . . . 5
β’ (π β (β Γ β)
βΌ Ο) |
97 | 96, 1 | mpct 43513 |
. . . 4
β’ (π β ((β Γ β)
βm π)
βΌ Ο) |
98 | | ssct 9001 |
. . . 4
β’ ((πΎ β ((β Γ
β) βm π) β§ ((β Γ β)
βm π)
βΌ Ο) β πΎ
βΌ Ο) |
99 | 92, 97, 98 | syl2anc 585 |
. . 3
β’ (π β πΎ βΌ Ο) |
100 | | reex 11150 |
. . . . . . . . . 10
β’ β
β V |
101 | 100, 100 | xpex 7691 |
. . . . . . . . 9
β’ (β
Γ β) β V |
102 | | qssre 12892 |
. . . . . . . . . 10
β’ β
β β |
103 | | xpss12 5652 |
. . . . . . . . . 10
β’ ((β
β β β§ β β β) β (β Γ β)
β (β Γ β)) |
104 | 102, 102,
103 | mp2an 691 |
. . . . . . . . 9
β’ (β
Γ β) β (β Γ β) |
105 | | mapss 8833 |
. . . . . . . . 9
β’
(((β Γ β) β V β§ (β Γ β)
β (β Γ β)) β ((β Γ β)
βm π)
β ((β Γ β) βm π)) |
106 | 101, 104,
105 | mp2an 691 |
. . . . . . . 8
β’ ((β
Γ β) βm π) β ((β Γ β)
βm π) |
107 | 91 | sseli 3944 |
. . . . . . . 8
β’ (β β πΎ β β β ((β Γ β)
βm π)) |
108 | 106, 107 | sselid 3946 |
. . . . . . 7
β’ (β β πΎ β β β ((β Γ β)
βm π)) |
109 | | elmapi 8793 |
. . . . . . 7
β’ (β β ((β Γ
β) βm π) β β:πβΆ(β Γ
β)) |
110 | 108, 109 | syl 17 |
. . . . . 6
β’ (β β πΎ β β:πβΆ(β Γ
β)) |
111 | 110 | adantl 483 |
. . . . 5
β’ ((π β§ β β πΎ) β β:πβΆ(β Γ
β)) |
112 | | 2fveq3 6851 |
. . . . . 6
β’ (π = π β (1st β(ββπ)) = (1st β(ββπ))) |
113 | 112 | cbvmptv 5222 |
. . . . 5
β’ (π β π β¦ (1st β(ββπ))) = (π β π β¦ (1st β(ββπ))) |
114 | | 2fveq3 6851 |
. . . . . 6
β’ (π = π β (2nd β(ββπ)) = (2nd β(ββπ))) |
115 | 114 | cbvmptv 5222 |
. . . . 5
β’ (π β π β¦ (2nd β(ββπ))) = (π β π β¦ (2nd β(ββπ))) |
116 | 111, 113,
115 | hoicoto2 44936 |
. . . 4
β’ ((π β§ β β πΎ) β Xπ β π (([,) β β)βπ) = Xπ β π (((π β π β¦ (1st β(ββπ)))βπ)[,)((π β π β¦ (2nd β(ββπ)))βπ))) |
117 | 1 | adantr 482 |
. . . . 5
β’ ((π β§ β β πΎ) β π β Fin) |
118 | 111 | ffvelcdmda 7039 |
. . . . . . 7
β’ (((π β§ β β πΎ) β§ π β π) β (ββπ) β (β Γ
β)) |
119 | | xp1st 7957 |
. . . . . . 7
β’ ((ββπ) β (β Γ β) β
(1st β(ββπ)) β β) |
120 | 118, 119 | syl 17 |
. . . . . 6
β’ (((π β§ β β πΎ) β§ π β π) β (1st β(ββπ)) β β) |
121 | 120 | fmpttd 7067 |
. . . . 5
β’ ((π β§ β β πΎ) β (π β π β¦ (1st β(ββπ))):πβΆβ) |
122 | | xp2nd 7958 |
. . . . . . 7
β’ ((ββπ) β (β Γ β) β
(2nd β(ββπ)) β β) |
123 | 118, 122 | syl 17 |
. . . . . 6
β’ (((π β§ β β πΎ) β§ π β π) β (2nd β(ββπ)) β β) |
124 | 123 | fmpttd 7067 |
. . . . 5
β’ ((π β§ β β πΎ) β (π β π β¦ (2nd β(ββπ))):πβΆβ) |
125 | 117, 88, 121, 124 | hoimbl 44962 |
. . . 4
β’ ((π β§ β β πΎ) β Xπ β π (((π β π β¦ (1st β(ββπ)))βπ)[,)((π β π β¦ (2nd β(ββπ)))βπ)) β π) |
126 | 116, 125 | eqeltrd 2834 |
. . 3
β’ ((π β§ β β πΎ) β Xπ β π (([,) β β)βπ) β π) |
127 | 89, 99, 126 | saliuncl 44654 |
. 2
β’ (π β βͺ β
β πΎ Xπ β
π (([,) β β)βπ) β π) |
128 | 87, 127 | eqeltrd 2834 |
1
β’ (π β πΊ β π) |