Step | Hyp | Ref
| Expression |
1 | | bndth.4 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
2 | | bndth.1 |
. . . . . 6
⊢ 𝑋 = ∪
𝐽 |
3 | | bndth.2 |
. . . . . . . 8
⊢ 𝐾 = (topGen‘ran
(,)) |
4 | | retopon 22567 |
. . . . . . . 8
⊢
(topGen‘ran (,)) ∈ (TopOn‘ℝ) |
5 | 3, 4 | eqeltri 2697 |
. . . . . . 7
⊢ 𝐾 ∈
(TopOn‘ℝ) |
6 | 5 | toponunii 20721 |
. . . . . 6
⊢ ℝ =
∪ 𝐾 |
7 | 2, 6 | cnf 21050 |
. . . . 5
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶ℝ) |
8 | 1, 7 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
9 | | frn 6053 |
. . . 4
⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ) |
10 | 8, 9 | syl 17 |
. . 3
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
11 | | imassrn 5477 |
. . . . . 6
⊢ ((,)
“ ({-∞} × ℝ)) ⊆ ran (,) |
12 | | retopbas 22564 |
. . . . . . . 8
⊢ ran (,)
∈ TopBases |
13 | | bastg 20770 |
. . . . . . . 8
⊢ (ran (,)
∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) |
14 | 12, 13 | ax-mp 5 |
. . . . . . 7
⊢ ran (,)
⊆ (topGen‘ran (,)) |
15 | 14, 3 | sseqtr4i 3638 |
. . . . . 6
⊢ ran (,)
⊆ 𝐾 |
16 | 11, 15 | sstri 3612 |
. . . . 5
⊢ ((,)
“ ({-∞} × ℝ)) ⊆ 𝐾 |
17 | | retop 22565 |
. . . . . . . 8
⊢
(topGen‘ran (,)) ∈ Top |
18 | 3, 17 | eqeltri 2697 |
. . . . . . 7
⊢ 𝐾 ∈ Top |
19 | 18 | elexi 3213 |
. . . . . 6
⊢ 𝐾 ∈ V |
20 | 19 | elpw2 4828 |
. . . . 5
⊢ (((,)
“ ({-∞} × ℝ)) ∈ 𝒫 𝐾 ↔ ((,) “ ({-∞} ×
ℝ)) ⊆ 𝐾) |
21 | 16, 20 | mpbir 221 |
. . . 4
⊢ ((,)
“ ({-∞} × ℝ)) ∈ 𝒫 𝐾 |
22 | | bndth.3 |
. . . . . 6
⊢ (𝜑 → 𝐽 ∈ Comp) |
23 | | rncmp 21199 |
. . . . . 6
⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ↾t ran 𝐹) ∈ Comp) |
24 | 22, 1, 23 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → (𝐾 ↾t ran 𝐹) ∈ Comp) |
25 | 6 | cmpsub 21203 |
. . . . . 6
⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ℝ) → ((𝐾 ↾t ran 𝐹) ∈ Comp ↔
∀𝑢 ∈ 𝒫
𝐾(ran 𝐹 ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣))) |
26 | 18, 10, 25 | sylancr 695 |
. . . . 5
⊢ (𝜑 → ((𝐾 ↾t ran 𝐹) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐾(ran 𝐹 ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣))) |
27 | 24, 26 | mpbid 222 |
. . . 4
⊢ (𝜑 → ∀𝑢 ∈ 𝒫 𝐾(ran 𝐹 ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣)) |
28 | | unieq 4444 |
. . . . . . . 8
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → ∪ 𝑢 = ∪ ((,) “
({-∞} × ℝ))) |
29 | 11 | unissi 4461 |
. . . . . . . . . 10
⊢ ∪ ((,) “ ({-∞} × ℝ)) ⊆ ∪ ran (,) |
30 | | unirnioo 12273 |
. . . . . . . . . 10
⊢ ℝ =
∪ ran (,) |
31 | 29, 30 | sseqtr4i 3638 |
. . . . . . . . 9
⊢ ∪ ((,) “ ({-∞} × ℝ)) ⊆
ℝ |
32 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ) |
33 | | ltp1 10861 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1)) |
34 | | ressxr 10083 |
. . . . . . . . . . . . . 14
⊢ ℝ
⊆ ℝ* |
35 | | peano2re 10209 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
36 | 34, 35 | sseldi 3601 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ*) |
37 | | elioomnf 12268 |
. . . . . . . . . . . . 13
⊢ ((𝑥 + 1) ∈ ℝ*
→ (𝑥 ∈
(-∞(,)(𝑥 + 1)) ↔
(𝑥 ∈ ℝ ∧
𝑥 < (𝑥 + 1)))) |
38 | 36, 37 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → (𝑥 ∈ (-∞(,)(𝑥 + 1)) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < (𝑥 + 1)))) |
39 | 32, 33, 38 | mpbir2and 957 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → 𝑥 ∈ (-∞(,)(𝑥 + 1))) |
40 | | df-ov 6653 |
. . . . . . . . . . . 12
⊢
(-∞(,)(𝑥 + 1))
= ((,)‘〈-∞, (𝑥 + 1)〉) |
41 | | mnfxr 10096 |
. . . . . . . . . . . . . . . 16
⊢ -∞
∈ ℝ* |
42 | 41 | elexi 3213 |
. . . . . . . . . . . . . . 15
⊢ -∞
∈ V |
43 | 42 | snid 4208 |
. . . . . . . . . . . . . 14
⊢ -∞
∈ {-∞} |
44 | | opelxpi 5148 |
. . . . . . . . . . . . . 14
⊢
((-∞ ∈ {-∞} ∧ (𝑥 + 1) ∈ ℝ) → 〈-∞,
(𝑥 + 1)〉 ∈
({-∞} × ℝ)) |
45 | 43, 35, 44 | sylancr 695 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ →
〈-∞, (𝑥 +
1)〉 ∈ ({-∞} × ℝ)) |
46 | | ioof 12271 |
. . . . . . . . . . . . . . 15
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
47 | | ffun 6048 |
. . . . . . . . . . . . . . 15
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → Fun (,)) |
48 | 46, 47 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ Fun
(,) |
49 | | snssi 4339 |
. . . . . . . . . . . . . . . . 17
⊢ (-∞
∈ ℝ* → {-∞} ⊆
ℝ*) |
50 | 41, 49 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
{-∞} ⊆ ℝ* |
51 | | xpss12 5225 |
. . . . . . . . . . . . . . . 16
⊢
(({-∞} ⊆ ℝ* ∧ ℝ ⊆
ℝ*) → ({-∞} × ℝ) ⊆
(ℝ* × ℝ*)) |
52 | 50, 34, 51 | mp2an 708 |
. . . . . . . . . . . . . . 15
⊢
({-∞} × ℝ) ⊆ (ℝ* ×
ℝ*) |
53 | 46 | fdmi 6052 |
. . . . . . . . . . . . . . 15
⊢ dom (,) =
(ℝ* × ℝ*) |
54 | 52, 53 | sseqtr4i 3638 |
. . . . . . . . . . . . . 14
⊢
({-∞} × ℝ) ⊆ dom (,) |
55 | | funfvima2 6493 |
. . . . . . . . . . . . . 14
⊢ ((Fun (,)
∧ ({-∞} × ℝ) ⊆ dom (,)) → (〈-∞,
(𝑥 + 1)〉 ∈
({-∞} × ℝ) → ((,)‘〈-∞, (𝑥 + 1)〉) ∈ ((,) “
({-∞} × ℝ)))) |
56 | 48, 54, 55 | mp2an 708 |
. . . . . . . . . . . . 13
⊢
(〈-∞, (𝑥
+ 1)〉 ∈ ({-∞} × ℝ) →
((,)‘〈-∞, (𝑥 + 1)〉) ∈ ((,) “ ({-∞}
× ℝ))) |
57 | 45, 56 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ →
((,)‘〈-∞, (𝑥 + 1)〉) ∈ ((,) “ ({-∞}
× ℝ))) |
58 | 40, 57 | syl5eqel 2705 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ →
(-∞(,)(𝑥 + 1)) ∈
((,) “ ({-∞} × ℝ))) |
59 | | elunii 4441 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (-∞(,)(𝑥 + 1)) ∧ (-∞(,)(𝑥 + 1)) ∈ ((,) “
({-∞} × ℝ))) → 𝑥 ∈ ∪ ((,)
“ ({-∞} × ℝ))) |
60 | 39, 58, 59 | syl2anc 693 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ∪ ((,) “ ({-∞} ×
ℝ))) |
61 | 60 | ssriv 3607 |
. . . . . . . . 9
⊢ ℝ
⊆ ∪ ((,) “ ({-∞} ×
ℝ)) |
62 | 31, 61 | eqssi 3619 |
. . . . . . . 8
⊢ ∪ ((,) “ ({-∞} × ℝ)) =
ℝ |
63 | 28, 62 | syl6eq 2672 |
. . . . . . 7
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → ∪ 𝑢 = ℝ) |
64 | 63 | sseq2d 3633 |
. . . . . 6
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → (ran 𝐹 ⊆ ∪ 𝑢 ↔ ran 𝐹 ⊆ ℝ)) |
65 | | pweq 4161 |
. . . . . . . 8
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → 𝒫 𝑢 = 𝒫 ((,) “ ({-∞} ×
ℝ))) |
66 | 65 | ineq1d 3813 |
. . . . . . 7
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → (𝒫 𝑢 ∩ Fin) = (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin)) |
67 | 66 | rexeqdv 3145 |
. . . . . 6
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣)) |
68 | 64, 67 | imbi12d 334 |
. . . . 5
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → ((ran 𝐹 ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣) ↔ (ran 𝐹 ⊆ ℝ → ∃𝑣 ∈ (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣))) |
69 | 68 | rspcv 3305 |
. . . 4
⊢ (((,)
“ ({-∞} × ℝ)) ∈ 𝒫 𝐾 → (∀𝑢 ∈ 𝒫 𝐾(ran 𝐹 ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣) → (ran 𝐹 ⊆ ℝ → ∃𝑣 ∈ (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣))) |
70 | 21, 27, 69 | mpsyl 68 |
. . 3
⊢ (𝜑 → (ran 𝐹 ⊆ ℝ → ∃𝑣 ∈ (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣)) |
71 | 10, 70 | mpd 15 |
. 2
⊢ (𝜑 → ∃𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣) |
72 | | simpr 477 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) |
73 | | elin 3796 |
. . . . . . 7
⊢ (𝑣 ∈ (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin) ↔ (𝑣 ∈ 𝒫 ((,) “ ({-∞}
× ℝ)) ∧ 𝑣
∈ Fin)) |
74 | 72, 73 | sylib 208 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → (𝑣 ∈ 𝒫 ((,) “ ({-∞}
× ℝ)) ∧ 𝑣
∈ Fin)) |
75 | 74 | adantrr 753 |
. . . . 5
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → (𝑣 ∈ 𝒫 ((,) “ ({-∞}
× ℝ)) ∧ 𝑣
∈ Fin)) |
76 | 75 | simprd 479 |
. . . 4
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → 𝑣 ∈ Fin) |
77 | 74 | simpld 475 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → 𝑣 ∈ 𝒫 ((,) “ ({-∞}
× ℝ))) |
78 | 77 | elpwid 4170 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → 𝑣 ⊆ ((,) “ ({-∞} ×
ℝ))) |
79 | 50 | sseli 3599 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈ {-∞} → 𝑢 ∈
ℝ*) |
80 | 79 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → 𝑢 ∈
ℝ*) |
81 | 34 | sseli 3599 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ ℝ → 𝑤 ∈
ℝ*) |
82 | 81 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → 𝑤 ∈
ℝ*) |
83 | | mnflt 11957 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ ℝ → -∞
< 𝑤) |
84 | | xrltnle 10105 |
. . . . . . . . . . . . . . . 16
⊢
((-∞ ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (-∞
< 𝑤 ↔ ¬ 𝑤 ≤
-∞)) |
85 | 41, 81, 84 | sylancr 695 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ ℝ → (-∞
< 𝑤 ↔ ¬ 𝑤 ≤
-∞)) |
86 | 83, 85 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ ℝ → ¬
𝑤 ≤
-∞) |
87 | 86 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → ¬
𝑤 ≤
-∞) |
88 | | elsni 4194 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 ∈ {-∞} → 𝑢 = -∞) |
89 | 88 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → 𝑢 = -∞) |
90 | 89 | breq2d 4665 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → (𝑤 ≤ 𝑢 ↔ 𝑤 ≤ -∞)) |
91 | 87, 90 | mtbird 315 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → ¬
𝑤 ≤ 𝑢) |
92 | | ioo0 12200 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ ℝ*
∧ 𝑤 ∈
ℝ*) → ((𝑢(,)𝑤) = ∅ ↔ 𝑤 ≤ 𝑢)) |
93 | 79, 81, 92 | syl2an 494 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → ((𝑢(,)𝑤) = ∅ ↔ 𝑤 ≤ 𝑢)) |
94 | 93 | necon3abid 2830 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → ((𝑢(,)𝑤) ≠ ∅ ↔ ¬ 𝑤 ≤ 𝑢)) |
95 | 91, 94 | mpbird 247 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → (𝑢(,)𝑤) ≠ ∅) |
96 | | df-ioo 12179 |
. . . . . . . . . . . 12
⊢ (,) =
(𝑦 ∈
ℝ*, 𝑧
∈ ℝ* ↦ {𝑣 ∈ ℝ* ∣ (𝑦 < 𝑣 ∧ 𝑣 < 𝑧)}) |
97 | | idd 24 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ*
∧ 𝑤 ∈
ℝ*) → (𝑥 < 𝑤 → 𝑥 < 𝑤)) |
98 | | xrltle 11982 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ*
∧ 𝑤 ∈
ℝ*) → (𝑥 < 𝑤 → 𝑥 ≤ 𝑤)) |
99 | | idd 24 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ ℝ*
∧ 𝑥 ∈
ℝ*) → (𝑢 < 𝑥 → 𝑢 < 𝑥)) |
100 | | xrltle 11982 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ ℝ*
∧ 𝑥 ∈
ℝ*) → (𝑢 < 𝑥 → 𝑢 ≤ 𝑥)) |
101 | 96, 97, 98, 99, 100 | ixxub 12196 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ ℝ*
∧ 𝑤 ∈
ℝ* ∧ (𝑢(,)𝑤) ≠ ∅) → sup((𝑢(,)𝑤), ℝ*, < ) = 𝑤) |
102 | 80, 82, 95, 101 | syl3anc 1326 |
. . . . . . . . . 10
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) →
sup((𝑢(,)𝑤), ℝ*, < ) = 𝑤) |
103 | | simpr 477 |
. . . . . . . . . 10
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → 𝑤 ∈
ℝ) |
104 | 102, 103 | eqeltrd 2701 |
. . . . . . . . 9
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) →
sup((𝑢(,)𝑤), ℝ*, < ) ∈
ℝ) |
105 | 104 | rgen2 2975 |
. . . . . . . 8
⊢
∀𝑢 ∈
{-∞}∀𝑤 ∈
ℝ sup((𝑢(,)𝑤), ℝ*, < )
∈ ℝ |
106 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑧 = 〈𝑢, 𝑤〉 → ((,)‘𝑧) = ((,)‘〈𝑢, 𝑤〉)) |
107 | | df-ov 6653 |
. . . . . . . . . . . 12
⊢ (𝑢(,)𝑤) = ((,)‘〈𝑢, 𝑤〉) |
108 | 106, 107 | syl6eqr 2674 |
. . . . . . . . . . 11
⊢ (𝑧 = 〈𝑢, 𝑤〉 → ((,)‘𝑧) = (𝑢(,)𝑤)) |
109 | 108 | supeq1d 8352 |
. . . . . . . . . 10
⊢ (𝑧 = 〈𝑢, 𝑤〉 → sup(((,)‘𝑧), ℝ*, < ) =
sup((𝑢(,)𝑤), ℝ*, <
)) |
110 | 109 | eleq1d 2686 |
. . . . . . . . 9
⊢ (𝑧 = 〈𝑢, 𝑤〉 → (sup(((,)‘𝑧), ℝ*, < )
∈ ℝ ↔ sup((𝑢(,)𝑤), ℝ*, < ) ∈
ℝ)) |
111 | 110 | ralxp 5263 |
. . . . . . . 8
⊢
(∀𝑧 ∈
({-∞} × ℝ)sup(((,)‘𝑧), ℝ*, < ) ∈ ℝ
↔ ∀𝑢 ∈
{-∞}∀𝑤 ∈
ℝ sup((𝑢(,)𝑤), ℝ*, < )
∈ ℝ) |
112 | 105, 111 | mpbir 221 |
. . . . . . 7
⊢
∀𝑧 ∈
({-∞} × ℝ)sup(((,)‘𝑧), ℝ*, < ) ∈
ℝ |
113 | | ffn 6045 |
. . . . . . . . 9
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → (,) Fn (ℝ* ×
ℝ*)) |
114 | 46, 113 | ax-mp 5 |
. . . . . . . 8
⊢ (,) Fn
(ℝ* × ℝ*) |
115 | | supeq1 8351 |
. . . . . . . . . 10
⊢ (𝑤 = ((,)‘𝑧) → sup(𝑤, ℝ*, < ) =
sup(((,)‘𝑧),
ℝ*, < )) |
116 | 115 | eleq1d 2686 |
. . . . . . . . 9
⊢ (𝑤 = ((,)‘𝑧) → (sup(𝑤, ℝ*, < ) ∈ ℝ
↔ sup(((,)‘𝑧),
ℝ*, < ) ∈ ℝ)) |
117 | 116 | ralima 6498 |
. . . . . . . 8
⊢ (((,) Fn
(ℝ* × ℝ*) ∧ ({-∞} ×
ℝ) ⊆ (ℝ* × ℝ*)) →
(∀𝑤 ∈ ((,)
“ ({-∞} × ℝ))sup(𝑤, ℝ*, < ) ∈ ℝ
↔ ∀𝑧 ∈
({-∞} × ℝ)sup(((,)‘𝑧), ℝ*, < ) ∈
ℝ)) |
118 | 114, 52, 117 | mp2an 708 |
. . . . . . 7
⊢
(∀𝑤 ∈
((,) “ ({-∞} × ℝ))sup(𝑤, ℝ*, < ) ∈ ℝ
↔ ∀𝑧 ∈
({-∞} × ℝ)sup(((,)‘𝑧), ℝ*, < ) ∈
ℝ) |
119 | 112, 118 | mpbir 221 |
. . . . . 6
⊢
∀𝑤 ∈
((,) “ ({-∞} × ℝ))sup(𝑤, ℝ*, < ) ∈
ℝ |
120 | | ssralv 3666 |
. . . . . 6
⊢ (𝑣 ⊆ ((,) “
({-∞} × ℝ)) → (∀𝑤 ∈ ((,) “ ({-∞} ×
ℝ))sup(𝑤,
ℝ*, < ) ∈ ℝ → ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ∈
ℝ)) |
121 | 78, 119, 120 | mpisyl 21 |
. . . . 5
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ∈
ℝ) |
122 | 121 | adantrr 753 |
. . . 4
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ∈
ℝ) |
123 | | fimaxre3 10970 |
. . . 4
⊢ ((𝑣 ∈ Fin ∧ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ∈ ℝ)
→ ∃𝑥 ∈
ℝ ∀𝑤 ∈
𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥) |
124 | 76, 122, 123 | syl2anc 693 |
. . 3
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥) |
125 | | simplrr 801 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → ran 𝐹 ⊆ ∪ 𝑣) |
126 | 125 | sselda 3603 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ran 𝐹) → 𝑧 ∈ ∪ 𝑣) |
127 | | eluni2 4440 |
. . . . . . . 8
⊢ (𝑧 ∈ ∪ 𝑣
↔ ∃𝑤 ∈
𝑣 𝑧 ∈ 𝑤) |
128 | | r19.29r 3073 |
. . . . . . . . . 10
⊢
((∃𝑤 ∈
𝑣 𝑧 ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥) → ∃𝑤 ∈ 𝑣 (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) |
129 | | sspwuni 4611 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((,)
“ ({-∞} × ℝ)) ⊆ 𝒫 ℝ ↔ ∪ ((,) “ ({-∞} × ℝ)) ⊆
ℝ) |
130 | 31, 129 | mpbir 221 |
. . . . . . . . . . . . . . . . . 18
⊢ ((,)
“ ({-∞} × ℝ)) ⊆ 𝒫
ℝ |
131 | 78 | 3ad2ant1 1082 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑣 ⊆ ((,) “ ({-∞} ×
ℝ))) |
132 | | simp2r 1088 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ∈ 𝑣) |
133 | 131, 132 | sseldd 3604 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ∈ ((,) “ ({-∞} ×
ℝ))) |
134 | 130, 133 | sseldi 3601 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ∈ 𝒫 ℝ) |
135 | 134 | elpwid 4170 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ⊆ ℝ) |
136 | | simp3l 1089 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑧 ∈ 𝑤) |
137 | 135, 136 | sseldd 3604 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑧 ∈ ℝ) |
138 | 121 | r19.21bi 2932 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ 𝑤 ∈ 𝑣) → sup(𝑤, ℝ*, < ) ∈
ℝ) |
139 | 138 | adantrl 752 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣)) → sup(𝑤, ℝ*, < ) ∈
ℝ) |
140 | 139 | 3adant3 1081 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → sup(𝑤, ℝ*, < ) ∈
ℝ) |
141 | | simp2l 1087 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑥 ∈ ℝ) |
142 | 135, 34 | syl6ss 3615 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ⊆
ℝ*) |
143 | | supxrub 12154 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑤 ⊆ ℝ*
∧ 𝑧 ∈ 𝑤) → 𝑧 ≤ sup(𝑤, ℝ*, <
)) |
144 | 142, 136,
143 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑧 ≤ sup(𝑤, ℝ*, <
)) |
145 | | simp3r 1090 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → sup(𝑤, ℝ*, < ) ≤ 𝑥) |
146 | 137, 140,
141, 144, 145 | letrd 10194 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑧 ≤ 𝑥) |
147 | 146 | 3expia 1267 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣)) → ((𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
148 | 147 | anassrs 680 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ 𝑣) → ((𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
149 | 148 | rexlimdva 3031 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ 𝑥 ∈ ℝ) → (∃𝑤 ∈ 𝑣 (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
150 | 149 | adantlrr 757 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → (∃𝑤 ∈ 𝑣 (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
151 | 128, 150 | syl5 34 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → ((∃𝑤 ∈ 𝑣 𝑧 ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
152 | 151 | expdimp 453 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) ∧ ∃𝑤 ∈ 𝑣 𝑧 ∈ 𝑤) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → 𝑧 ≤ 𝑥)) |
153 | 127, 152 | sylan2b 492 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ∪ 𝑣) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → 𝑧 ≤ 𝑥)) |
154 | 126, 153 | syldan 487 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ran 𝐹) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → 𝑧 ≤ 𝑥)) |
155 | 154 | ralrimdva 2969 |
. . . . 5
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥)) |
156 | | ffn 6045 |
. . . . . . . 8
⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋) |
157 | 8, 156 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐹 Fn 𝑋) |
158 | 157 | ad2antrr 762 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn 𝑋) |
159 | | breq1 4656 |
. . . . . . 7
⊢ (𝑧 = (𝐹‘𝑦) → (𝑧 ≤ 𝑥 ↔ (𝐹‘𝑦) ≤ 𝑥)) |
160 | 159 | ralrn 6362 |
. . . . . 6
⊢ (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)) |
161 | 158, 160 | syl 17 |
. . . . 5
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)) |
162 | 155, 161 | sylibd 229 |
. . . 4
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)) |
163 | 162 | reximdva 3017 |
. . 3
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → (∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)) |
164 | 124, 163 | mpd 15 |
. 2
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥) |
165 | 71, 164 | rexlimddv 3035 |
1
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥) |