| Step | Hyp | Ref
| Expression |
| 1 | | 0red 11264 |
. . . 4
⊢
(((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) ∧ 𝑦 < 0) → 0 ∈
ℝ) |
| 2 | | simpllr 776 |
. . . 4
⊢
(((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) ∧ ¬ 𝑦 < 0) → 𝑦 ∈
ℝ) |
| 3 | 1, 2 | ifclda 4561 |
. . 3
⊢ ((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) → if(𝑦 < 0, 0, 𝑦) ∈ ℝ) |
| 4 | | ovif 7531 |
. . . . . 6
⊢ (if(𝑦 < 0, 0, 𝑦)(,]+∞) = if(𝑦 < 0, (0(,]+∞), (𝑦(,]+∞)) |
| 5 | | rexr 11307 |
. . . . . . 7
⊢ (𝑦 ∈ ℝ → 𝑦 ∈
ℝ*) |
| 6 | | 0xr 11308 |
. . . . . . . 8
⊢ 0 ∈
ℝ* |
| 7 | 6 | a1i 11 |
. . . . . . 7
⊢ (𝑦 ∈ ℝ → 0 ∈
ℝ*) |
| 8 | | pnfxr 11315 |
. . . . . . . 8
⊢ +∞
∈ ℝ* |
| 9 | 8 | a1i 11 |
. . . . . . 7
⊢ (𝑦 ∈ ℝ → +∞
∈ ℝ*) |
| 10 | | iocinif 32783 |
. . . . . . 7
⊢ ((𝑦 ∈ ℝ*
∧ 0 ∈ ℝ* ∧ +∞ ∈ ℝ*)
→ ((𝑦(,]+∞)
∩ (0(,]+∞)) = if(𝑦 < 0, (0(,]+∞), (𝑦(,]+∞))) |
| 11 | 5, 7, 9, 10 | syl3anc 1373 |
. . . . . 6
⊢ (𝑦 ∈ ℝ → ((𝑦(,]+∞) ∩
(0(,]+∞)) = if(𝑦 <
0, (0(,]+∞), (𝑦(,]+∞))) |
| 12 | 4, 11 | eqtr4id 2796 |
. . . . 5
⊢ (𝑦 ∈ ℝ → (if(𝑦 < 0, 0, 𝑦)(,]+∞) = ((𝑦(,]+∞) ∩
(0(,]+∞))) |
| 13 | 12 | ad2antlr 727 |
. . . 4
⊢ ((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) → (if(𝑦 < 0, 0, 𝑦)(,]+∞) = ((𝑦(,]+∞) ∩
(0(,]+∞))) |
| 14 | | iocssicc 13477 |
. . . . . 6
⊢
(0(,]+∞) ⊆ (0[,]+∞) |
| 15 | | sslin 4243 |
. . . . . 6
⊢
((0(,]+∞) ⊆ (0[,]+∞) → ((𝑦(,]+∞) ∩ (0(,]+∞)) ⊆
((𝑦(,]+∞) ∩
(0[,]+∞))) |
| 16 | 14, 15 | mp1i 13 |
. . . . 5
⊢ ((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) → ((𝑦(,]+∞) ∩
(0(,]+∞)) ⊆ ((𝑦(,]+∞) ∩
(0[,]+∞))) |
| 17 | | simpr 484 |
. . . . . 6
⊢ ((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) → (𝑦(,]+∞) ⊆ (𝐴 ∩
(0(,]+∞))) |
| 18 | | ssin 4239 |
. . . . . . . 8
⊢ (((𝑦(,]+∞) ⊆ 𝐴 ∧ (𝑦(,]+∞) ⊆ (0(,]+∞)) ↔
(𝑦(,]+∞) ⊆
(𝐴 ∩
(0(,]+∞))) |
| 19 | 18 | biimpri 228 |
. . . . . . 7
⊢ ((𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞)) →
((𝑦(,]+∞) ⊆
𝐴 ∧ (𝑦(,]+∞) ⊆
(0(,]+∞))) |
| 20 | 19 | simpld 494 |
. . . . . 6
⊢ ((𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞)) →
(𝑦(,]+∞) ⊆
𝐴) |
| 21 | | ssinss1 4246 |
. . . . . 6
⊢ ((𝑦(,]+∞) ⊆ 𝐴 → ((𝑦(,]+∞) ∩ (0[,]+∞)) ⊆
𝐴) |
| 22 | 17, 20, 21 | 3syl 18 |
. . . . 5
⊢ ((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) → ((𝑦(,]+∞) ∩
(0[,]+∞)) ⊆ 𝐴) |
| 23 | 16, 22 | sstrd 3994 |
. . . 4
⊢ ((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) → ((𝑦(,]+∞) ∩
(0(,]+∞)) ⊆ 𝐴) |
| 24 | 13, 23 | eqsstrd 4018 |
. . 3
⊢ ((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) → (if(𝑦 < 0, 0, 𝑦)(,]+∞) ⊆ 𝐴) |
| 25 | | oveq1 7438 |
. . . . 5
⊢ (𝑥 = if(𝑦 < 0, 0, 𝑦) → (𝑥(,]+∞) = (if(𝑦 < 0, 0, 𝑦)(,]+∞)) |
| 26 | 25 | sseq1d 4015 |
. . . 4
⊢ (𝑥 = if(𝑦 < 0, 0, 𝑦) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(𝑦 < 0, 0, 𝑦)(,]+∞) ⊆ 𝐴)) |
| 27 | 26 | rspcev 3622 |
. . 3
⊢
((if(𝑦 < 0, 0,
𝑦) ∈ ℝ ∧
(if(𝑦 < 0, 0, 𝑦)(,]+∞) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴) |
| 28 | 3, 24, 27 | syl2anc 584 |
. 2
⊢ ((((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴) |
| 29 | | letopon 23213 |
. . . . . . . . . 10
⊢
(ordTop‘ ≤ ) ∈
(TopOn‘ℝ*) |
| 30 | | iccssxr 13470 |
. . . . . . . . . 10
⊢
(0[,]+∞) ⊆ ℝ* |
| 31 | | resttopon 23169 |
. . . . . . . . . 10
⊢
(((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧
(0[,]+∞) ⊆ ℝ*) → ((ordTop‘ ≤ )
↾t (0[,]+∞)) ∈
(TopOn‘(0[,]+∞))) |
| 32 | 29, 30, 31 | mp2an 692 |
. . . . . . . . 9
⊢
((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈
(TopOn‘(0[,]+∞)) |
| 33 | 32 | topontopi 22921 |
. . . . . . . 8
⊢
((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈
Top |
| 34 | 33 | a1i 11 |
. . . . . . 7
⊢ (𝐴 ∈ 𝐽 → ((ordTop‘ ≤ )
↾t (0[,]+∞)) ∈ Top) |
| 35 | | ovex 7464 |
. . . . . . . 8
⊢
(0(,]+∞) ∈ V |
| 36 | 35 | a1i 11 |
. . . . . . 7
⊢ (𝐴 ∈ 𝐽 → (0(,]+∞) ∈
V) |
| 37 | | pnfneige0.j |
. . . . . . . . . 10
⊢ 𝐽 =
(TopOpen‘(ℝ*𝑠 ↾s
(0[,]+∞))) |
| 38 | | xrge0topn 33942 |
. . . . . . . . . 10
⊢
(TopOpen‘(ℝ*𝑠
↾s (0[,]+∞))) = ((ordTop‘ ≤ )
↾t (0[,]+∞)) |
| 39 | 37, 38 | eqtri 2765 |
. . . . . . . . 9
⊢ 𝐽 = ((ordTop‘ ≤ )
↾t (0[,]+∞)) |
| 40 | 39 | eleq2i 2833 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝐽 ↔ 𝐴 ∈ ((ordTop‘ ≤ )
↾t (0[,]+∞))) |
| 41 | 40 | biimpi 216 |
. . . . . . 7
⊢ (𝐴 ∈ 𝐽 → 𝐴 ∈ ((ordTop‘ ≤ )
↾t (0[,]+∞))) |
| 42 | | elrestr 17473 |
. . . . . . 7
⊢
((((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Top
∧ (0(,]+∞) ∈ V ∧ 𝐴 ∈ ((ordTop‘ ≤ )
↾t (0[,]+∞))) → (𝐴 ∩ (0(,]+∞)) ∈
(((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t
(0(,]+∞))) |
| 43 | 34, 36, 41, 42 | syl3anc 1373 |
. . . . . 6
⊢ (𝐴 ∈ 𝐽 → (𝐴 ∩ (0(,]+∞)) ∈
(((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t
(0(,]+∞))) |
| 44 | | letop 23214 |
. . . . . . 7
⊢
(ordTop‘ ≤ ) ∈ Top |
| 45 | | ovex 7464 |
. . . . . . 7
⊢
(0[,]+∞) ∈ V |
| 46 | | restabs 23173 |
. . . . . . 7
⊢
(((ordTop‘ ≤ ) ∈ Top ∧ (0(,]+∞) ⊆
(0[,]+∞) ∧ (0[,]+∞) ∈ V) → (((ordTop‘ ≤ )
↾t (0[,]+∞)) ↾t (0(,]+∞)) =
((ordTop‘ ≤ ) ↾t (0(,]+∞))) |
| 47 | 44, 14, 45, 46 | mp3an 1463 |
. . . . . 6
⊢
(((ordTop‘ ≤ ) ↾t (0[,]+∞))
↾t (0(,]+∞)) = ((ordTop‘ ≤ )
↾t (0(,]+∞)) |
| 48 | 43, 47 | eleqtrdi 2851 |
. . . . 5
⊢ (𝐴 ∈ 𝐽 → (𝐴 ∩ (0(,]+∞)) ∈
((ordTop‘ ≤ ) ↾t (0(,]+∞))) |
| 49 | 44 | a1i 11 |
. . . . . 6
⊢ (𝐴 ∈ 𝐽 → (ordTop‘ ≤ ) ∈
Top) |
| 50 | | iocpnfordt 23223 |
. . . . . . 7
⊢
(0(,]+∞) ∈ (ordTop‘ ≤ ) |
| 51 | 50 | a1i 11 |
. . . . . 6
⊢ (𝐴 ∈ 𝐽 → (0(,]+∞) ∈ (ordTop‘
≤ )) |
| 52 | | ssidd 4007 |
. . . . . 6
⊢ (𝐴 ∈ 𝐽 → (0(,]+∞) ⊆
(0(,]+∞)) |
| 53 | | inss2 4238 |
. . . . . . 7
⊢ (𝐴 ∩ (0(,]+∞)) ⊆
(0(,]+∞) |
| 54 | 53 | a1i 11 |
. . . . . 6
⊢ (𝐴 ∈ 𝐽 → (𝐴 ∩ (0(,]+∞)) ⊆
(0(,]+∞)) |
| 55 | | restopnb 23183 |
. . . . . 6
⊢
((((ordTop‘ ≤ ) ∈ Top ∧ (0(,]+∞) ∈ V) ∧
((0(,]+∞) ∈ (ordTop‘ ≤ ) ∧ (0(,]+∞) ⊆
(0(,]+∞) ∧ (𝐴
∩ (0(,]+∞)) ⊆ (0(,]+∞))) → ((𝐴 ∩ (0(,]+∞)) ∈ (ordTop‘
≤ ) ↔ (𝐴 ∩
(0(,]+∞)) ∈ ((ordTop‘ ≤ ) ↾t
(0(,]+∞)))) |
| 56 | 49, 36, 51, 52, 54, 55 | syl23anc 1379 |
. . . . 5
⊢ (𝐴 ∈ 𝐽 → ((𝐴 ∩ (0(,]+∞)) ∈ (ordTop‘
≤ ) ↔ (𝐴 ∩
(0(,]+∞)) ∈ ((ordTop‘ ≤ ) ↾t
(0(,]+∞)))) |
| 57 | 48, 56 | mpbird 257 |
. . . 4
⊢ (𝐴 ∈ 𝐽 → (𝐴 ∩ (0(,]+∞)) ∈ (ordTop‘
≤ )) |
| 58 | 57 | adantr 480 |
. . 3
⊢ ((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) → (𝐴 ∩ (0(,]+∞)) ∈ (ordTop‘
≤ )) |
| 59 | | simpr 484 |
. . . 4
⊢ ((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) → +∞ ∈ 𝐴) |
| 60 | | 0ltpnf 13164 |
. . . . . 6
⊢ 0 <
+∞ |
| 61 | | ubioc1 13440 |
. . . . . 6
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0
< +∞) → +∞ ∈ (0(,]+∞)) |
| 62 | 6, 8, 60, 61 | mp3an 1463 |
. . . . 5
⊢ +∞
∈ (0(,]+∞) |
| 63 | 62 | a1i 11 |
. . . 4
⊢ ((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) → +∞ ∈
(0(,]+∞)) |
| 64 | 59, 63 | elind 4200 |
. . 3
⊢ ((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) → +∞ ∈ (𝐴 ∩ (0(,]+∞))) |
| 65 | | pnfnei 23228 |
. . 3
⊢ (((𝐴 ∩ (0(,]+∞)) ∈
(ordTop‘ ≤ ) ∧ +∞ ∈ (𝐴 ∩ (0(,]+∞))) → ∃𝑦 ∈ ℝ (𝑦(,]+∞) ⊆ (𝐴 ∩
(0(,]+∞))) |
| 66 | 58, 64, 65 | syl2anc 584 |
. 2
⊢ ((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) → ∃𝑦 ∈ ℝ (𝑦(,]+∞) ⊆ (𝐴 ∩ (0(,]+∞))) |
| 67 | 28, 66 | r19.29a 3162 |
1
⊢ ((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴) |