Step |
Hyp |
Ref |
Expression |
1 |
|
stoweidlem59.1 |
⊢ Ⅎ 𝑡 𝐹 |
2 |
|
stoweidlem59.2 |
⊢ Ⅎ 𝑡 𝜑 |
3 |
|
stoweidlem59.3 |
⊢ 𝐾 = ( topGen ‘ ran (,) ) |
4 |
|
stoweidlem59.4 |
⊢ 𝑇 = ∪ 𝐽 |
5 |
|
stoweidlem59.5 |
⊢ 𝐶 = ( 𝐽 Cn 𝐾 ) |
6 |
|
stoweidlem59.6 |
⊢ 𝐷 = ( 𝑗 ∈ ( 0 ... 𝑁 ) ↦ { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ) |
7 |
|
stoweidlem59.7 |
⊢ 𝐵 = ( 𝑗 ∈ ( 0 ... 𝑁 ) ↦ { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ) |
8 |
|
stoweidlem59.8 |
⊢ 𝑌 = { 𝑦 ∈ 𝐴 ∣ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) } |
9 |
|
stoweidlem59.9 |
⊢ 𝐻 = ( 𝑗 ∈ ( 0 ... 𝑁 ) ↦ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) |
10 |
|
stoweidlem59.10 |
⊢ ( 𝜑 → 𝐽 ∈ Comp ) |
11 |
|
stoweidlem59.11 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 ) |
12 |
|
stoweidlem59.12 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
13 |
|
stoweidlem59.13 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
14 |
|
stoweidlem59.14 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑡 ∈ 𝑇 ↦ 𝑦 ) ∈ 𝐴 ) |
15 |
|
stoweidlem59.15 |
⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡 ) ) → ∃ 𝑞 ∈ 𝐴 ( 𝑞 ‘ 𝑟 ) ≠ ( 𝑞 ‘ 𝑡 ) ) |
16 |
|
stoweidlem59.16 |
⊢ ( 𝜑 → 𝐹 ∈ 𝐶 ) |
17 |
|
stoweidlem59.17 |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
18 |
|
stoweidlem59.18 |
⊢ ( 𝜑 → 𝐸 < ( 1 / 3 ) ) |
19 |
|
stoweidlem59.19 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
20 |
|
nfrab1 |
⊢ Ⅎ 𝑦 { 𝑦 ∈ 𝐴 ∣ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) } |
21 |
8 20
|
nfcxfr |
⊢ Ⅎ 𝑦 𝑌 |
22 |
|
nfcv |
⊢ Ⅎ 𝑧 𝑌 |
23 |
|
nfv |
⊢ Ⅎ 𝑧 ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) |
24 |
|
nfv |
⊢ Ⅎ 𝑦 ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑧 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑧 ‘ 𝑡 ) ) |
25 |
|
fveq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ‘ 𝑡 ) = ( 𝑧 ‘ 𝑡 ) ) |
26 |
25
|
breq1d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ↔ ( 𝑧 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) |
27 |
26
|
ralbidv |
⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ↔ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑧 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) |
28 |
25
|
breq2d |
⊢ ( 𝑦 = 𝑧 → ( ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ↔ ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑧 ‘ 𝑡 ) ) ) |
29 |
28
|
ralbidv |
⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ↔ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑧 ‘ 𝑡 ) ) ) |
30 |
27 29
|
anbi12d |
⊢ ( 𝑦 = 𝑧 → ( ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) ↔ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑧 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑧 ‘ 𝑡 ) ) ) ) |
31 |
21 22 23 24 30
|
cbvrabw |
⊢ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } = { 𝑧 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑧 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑧 ‘ 𝑡 ) ) } |
32 |
|
ovexd |
⊢ ( 𝜑 → ( 𝐽 Cn 𝐾 ) ∈ V ) |
33 |
11 5
|
sseqtrdi |
⊢ ( 𝜑 → 𝐴 ⊆ ( 𝐽 Cn 𝐾 ) ) |
34 |
32 33
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
35 |
8 34
|
rabexd |
⊢ ( 𝜑 → 𝑌 ∈ V ) |
36 |
31 35
|
rabexd |
⊢ ( 𝜑 → { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ∈ V ) |
37 |
36
|
ralrimivw |
⊢ ( 𝜑 → ∀ 𝑗 ∈ ( 0 ... 𝑁 ) { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ∈ V ) |
38 |
9
|
fnmpt |
⊢ ( ∀ 𝑗 ∈ ( 0 ... 𝑁 ) { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ∈ V → 𝐻 Fn ( 0 ... 𝑁 ) ) |
39 |
37 38
|
syl |
⊢ ( 𝜑 → 𝐻 Fn ( 0 ... 𝑁 ) ) |
40 |
|
fzfi |
⊢ ( 0 ... 𝑁 ) ∈ Fin |
41 |
|
fnfi |
⊢ ( ( 𝐻 Fn ( 0 ... 𝑁 ) ∧ ( 0 ... 𝑁 ) ∈ Fin ) → 𝐻 ∈ Fin ) |
42 |
39 40 41
|
sylancl |
⊢ ( 𝜑 → 𝐻 ∈ Fin ) |
43 |
|
rnfi |
⊢ ( 𝐻 ∈ Fin → ran 𝐻 ∈ Fin ) |
44 |
42 43
|
syl |
⊢ ( 𝜑 → ran 𝐻 ∈ Fin ) |
45 |
|
fnchoice |
⊢ ( ran 𝐻 ∈ Fin → ∃ ℎ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) |
46 |
44 45
|
syl |
⊢ ( 𝜑 → ∃ ℎ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) |
47 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ℎ Fn ran 𝐻 ) |
48 |
|
ovex |
⊢ ( 0 ... 𝑁 ) ∈ V |
49 |
48
|
mptex |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) ↦ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) ∈ V |
50 |
9 49
|
eqeltri |
⊢ 𝐻 ∈ V |
51 |
50
|
rnex |
⊢ ran 𝐻 ∈ V |
52 |
|
fnex |
⊢ ( ( ℎ Fn ran 𝐻 ∧ ran 𝐻 ∈ V ) → ℎ ∈ V ) |
53 |
47 51 52
|
sylancl |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ℎ ∈ V ) |
54 |
|
coexg |
⊢ ( ( ℎ ∈ V ∧ 𝐻 ∈ V ) → ( ℎ ∘ 𝐻 ) ∈ V ) |
55 |
53 50 54
|
sylancl |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ( ℎ ∘ 𝐻 ) ∈ V ) |
56 |
|
dffn3 |
⊢ ( ℎ Fn ran 𝐻 ↔ ℎ : ran 𝐻 ⟶ ran ℎ ) |
57 |
47 56
|
sylib |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ℎ : ran 𝐻 ⟶ ran ℎ ) |
58 |
|
nfv |
⊢ Ⅎ 𝑤 𝜑 |
59 |
|
nfv |
⊢ Ⅎ 𝑤 ℎ Fn ran 𝐻 |
60 |
|
nfra1 |
⊢ Ⅎ 𝑤 ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) |
61 |
59 60
|
nfan |
⊢ Ⅎ 𝑤 ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) |
62 |
58 61
|
nfan |
⊢ Ⅎ 𝑤 ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) |
63 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑤 ∈ ran 𝐻 ) → ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) |
64 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑤 ∈ ran 𝐻 ) → 𝑤 ∈ ran 𝐻 ) |
65 |
|
fvelrnb |
⊢ ( 𝐻 Fn ( 0 ... 𝑁 ) → ( 𝑤 ∈ ran 𝐻 ↔ ∃ 𝑎 ∈ ( 0 ... 𝑁 ) ( 𝐻 ‘ 𝑎 ) = 𝑤 ) ) |
66 |
|
nfv |
⊢ Ⅎ 𝑎 ( 𝐻 ‘ 𝑗 ) = 𝑤 |
67 |
|
nfmpt1 |
⊢ Ⅎ 𝑗 ( 𝑗 ∈ ( 0 ... 𝑁 ) ↦ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) |
68 |
9 67
|
nfcxfr |
⊢ Ⅎ 𝑗 𝐻 |
69 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑎 |
70 |
68 69
|
nffv |
⊢ Ⅎ 𝑗 ( 𝐻 ‘ 𝑎 ) |
71 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑤 |
72 |
70 71
|
nfeq |
⊢ Ⅎ 𝑗 ( 𝐻 ‘ 𝑎 ) = 𝑤 |
73 |
|
fveq2 |
⊢ ( 𝑗 = 𝑎 → ( 𝐻 ‘ 𝑗 ) = ( 𝐻 ‘ 𝑎 ) ) |
74 |
73
|
eqeq1d |
⊢ ( 𝑗 = 𝑎 → ( ( 𝐻 ‘ 𝑗 ) = 𝑤 ↔ ( 𝐻 ‘ 𝑎 ) = 𝑤 ) ) |
75 |
66 72 74
|
cbvrexw |
⊢ ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝐻 ‘ 𝑗 ) = 𝑤 ↔ ∃ 𝑎 ∈ ( 0 ... 𝑁 ) ( 𝐻 ‘ 𝑎 ) = 𝑤 ) |
76 |
65 75
|
bitr4di |
⊢ ( 𝐻 Fn ( 0 ... 𝑁 ) → ( 𝑤 ∈ ran 𝐻 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝐻 ‘ 𝑗 ) = 𝑤 ) ) |
77 |
39 76
|
syl |
⊢ ( 𝜑 → ( 𝑤 ∈ ran 𝐻 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝐻 ‘ 𝑗 ) = 𝑤 ) ) |
78 |
77
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran 𝐻 ) → ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝐻 ‘ 𝑗 ) = 𝑤 ) |
79 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ∧ ( 𝐻 ‘ 𝑗 ) = 𝑤 ) → ( 𝐻 ‘ 𝑗 ) = 𝑤 ) |
80 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝑗 ∈ ( 0 ... 𝑁 ) ) |
81 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ∈ V ) |
82 |
9
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑁 ) ∧ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ∈ V ) → ( 𝐻 ‘ 𝑗 ) = { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) |
83 |
80 81 82
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐻 ‘ 𝑗 ) = { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) |
84 |
|
nfcv |
⊢ Ⅎ 𝑡 ( 0 ... 𝑁 ) |
85 |
|
nfrab1 |
⊢ Ⅎ 𝑡 { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } |
86 |
84 85
|
nfmpt |
⊢ Ⅎ 𝑡 ( 𝑗 ∈ ( 0 ... 𝑁 ) ↦ { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ) |
87 |
6 86
|
nfcxfr |
⊢ Ⅎ 𝑡 𝐷 |
88 |
|
nfcv |
⊢ Ⅎ 𝑡 𝑗 |
89 |
87 88
|
nffv |
⊢ Ⅎ 𝑡 ( 𝐷 ‘ 𝑗 ) |
90 |
|
nfcv |
⊢ Ⅎ 𝑡 𝑇 |
91 |
|
nfrab1 |
⊢ Ⅎ 𝑡 { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } |
92 |
84 91
|
nfmpt |
⊢ Ⅎ 𝑡 ( 𝑗 ∈ ( 0 ... 𝑁 ) ↦ { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ) |
93 |
7 92
|
nfcxfr |
⊢ Ⅎ 𝑡 𝐵 |
94 |
93 88
|
nffv |
⊢ Ⅎ 𝑡 ( 𝐵 ‘ 𝑗 ) |
95 |
90 94
|
nfdif |
⊢ Ⅎ 𝑡 ( 𝑇 ∖ ( 𝐵 ‘ 𝑗 ) ) |
96 |
|
nfv |
⊢ Ⅎ 𝑡 𝑗 ∈ ( 0 ... 𝑁 ) |
97 |
2 96
|
nfan |
⊢ Ⅎ 𝑡 ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) |
98 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝐽 ∈ Comp ) |
99 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝐴 ⊆ 𝐶 ) |
100 |
12
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
101 |
13
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
102 |
14
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ℝ ) → ( 𝑡 ∈ 𝑇 ↦ 𝑦 ) ∈ 𝐴 ) |
103 |
15
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡 ) ) → ∃ 𝑞 ∈ 𝐴 ( 𝑞 ‘ 𝑟 ) ≠ ( 𝑞 ‘ 𝑡 ) ) |
104 |
10
|
uniexd |
⊢ ( 𝜑 → ∪ 𝐽 ∈ V ) |
105 |
4 104
|
eqeltrid |
⊢ ( 𝜑 → 𝑇 ∈ V ) |
106 |
105
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝑇 ∈ V ) |
107 |
|
rabexg |
⊢ ( 𝑇 ∈ V → { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ∈ V ) |
108 |
106 107
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ∈ V ) |
109 |
7
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑁 ) ∧ { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ∈ V ) → ( 𝐵 ‘ 𝑗 ) = { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ) |
110 |
80 108 109
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐵 ‘ 𝑗 ) = { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ) |
111 |
|
eqid |
⊢ { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } = { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } |
112 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → 𝑗 ∈ ℤ ) |
113 |
112
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → 𝑗 ∈ ℝ ) |
114 |
|
3re |
⊢ 3 ∈ ℝ |
115 |
|
3ne0 |
⊢ 3 ≠ 0 |
116 |
114 115
|
rereccli |
⊢ ( 1 / 3 ) ∈ ℝ |
117 |
|
readdcl |
⊢ ( ( 𝑗 ∈ ℝ ∧ ( 1 / 3 ) ∈ ℝ ) → ( 𝑗 + ( 1 / 3 ) ) ∈ ℝ ) |
118 |
113 116 117
|
sylancl |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 𝑗 + ( 1 / 3 ) ) ∈ ℝ ) |
119 |
118
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝑗 + ( 1 / 3 ) ) ∈ ℝ ) |
120 |
17
|
rpred |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
121 |
120
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝐸 ∈ ℝ ) |
122 |
119 121
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ∈ ℝ ) |
123 |
16 5
|
eleqtrdi |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) |
124 |
123
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) |
125 |
1 3 4 111 122 124
|
rfcnpre3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ∈ ( Clsd ‘ 𝐽 ) ) |
126 |
110 125
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐵 ‘ 𝑗 ) ∈ ( Clsd ‘ 𝐽 ) ) |
127 |
|
rabexg |
⊢ ( 𝑇 ∈ V → { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ∈ V ) |
128 |
106 127
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ∈ V ) |
129 |
6
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑁 ) ∧ { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ∈ V ) → ( 𝐷 ‘ 𝑗 ) = { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ) |
130 |
80 128 129
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐷 ‘ 𝑗 ) = { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ) |
131 |
|
eqid |
⊢ { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } = { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } |
132 |
|
resubcl |
⊢ ( ( 𝑗 ∈ ℝ ∧ ( 1 / 3 ) ∈ ℝ ) → ( 𝑗 − ( 1 / 3 ) ) ∈ ℝ ) |
133 |
113 116 132
|
sylancl |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 𝑗 − ( 1 / 3 ) ) ∈ ℝ ) |
134 |
133
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝑗 − ( 1 / 3 ) ) ∈ ℝ ) |
135 |
134 121
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) ∈ ℝ ) |
136 |
1 3 4 131 135 124
|
rfcnpre4 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ∈ ( Clsd ‘ 𝐽 ) ) |
137 |
130 136
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐷 ‘ 𝑗 ) ∈ ( Clsd ‘ 𝐽 ) ) |
138 |
135
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) ∈ ℝ ) |
139 |
122
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ∈ ℝ ) |
140 |
3 4 5 16
|
fcnre |
⊢ ( 𝜑 → 𝐹 : 𝑇 ⟶ ℝ ) |
141 |
140
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → 𝐹 : 𝑇 ⟶ ℝ ) |
142 |
|
ssrab2 |
⊢ { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ⊆ 𝑇 |
143 |
110 142
|
eqsstrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐵 ‘ 𝑗 ) ⊆ 𝑇 ) |
144 |
143
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → 𝑡 ∈ 𝑇 ) |
145 |
141 144
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑡 ) ∈ ℝ ) |
146 |
116 132
|
mpan2 |
⊢ ( 𝑗 ∈ ℝ → ( 𝑗 − ( 1 / 3 ) ) ∈ ℝ ) |
147 |
|
id |
⊢ ( 𝑗 ∈ ℝ → 𝑗 ∈ ℝ ) |
148 |
116 117
|
mpan2 |
⊢ ( 𝑗 ∈ ℝ → ( 𝑗 + ( 1 / 3 ) ) ∈ ℝ ) |
149 |
|
3pos |
⊢ 0 < 3 |
150 |
114 149
|
recgt0ii |
⊢ 0 < ( 1 / 3 ) |
151 |
116 150
|
elrpii |
⊢ ( 1 / 3 ) ∈ ℝ+ |
152 |
|
ltsubrp |
⊢ ( ( 𝑗 ∈ ℝ ∧ ( 1 / 3 ) ∈ ℝ+ ) → ( 𝑗 − ( 1 / 3 ) ) < 𝑗 ) |
153 |
151 152
|
mpan2 |
⊢ ( 𝑗 ∈ ℝ → ( 𝑗 − ( 1 / 3 ) ) < 𝑗 ) |
154 |
|
ltaddrp |
⊢ ( ( 𝑗 ∈ ℝ ∧ ( 1 / 3 ) ∈ ℝ+ ) → 𝑗 < ( 𝑗 + ( 1 / 3 ) ) ) |
155 |
151 154
|
mpan2 |
⊢ ( 𝑗 ∈ ℝ → 𝑗 < ( 𝑗 + ( 1 / 3 ) ) ) |
156 |
146 147 148 153 155
|
lttrd |
⊢ ( 𝑗 ∈ ℝ → ( 𝑗 − ( 1 / 3 ) ) < ( 𝑗 + ( 1 / 3 ) ) ) |
157 |
113 156
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 𝑗 − ( 1 / 3 ) ) < ( 𝑗 + ( 1 / 3 ) ) ) |
158 |
157
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝑗 − ( 1 / 3 ) ) < ( 𝑗 + ( 1 / 3 ) ) ) |
159 |
17
|
rpregt0d |
⊢ ( 𝜑 → ( 𝐸 ∈ ℝ ∧ 0 < 𝐸 ) ) |
160 |
159
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐸 ∈ ℝ ∧ 0 < 𝐸 ) ) |
161 |
|
ltmul1 |
⊢ ( ( ( 𝑗 − ( 1 / 3 ) ) ∈ ℝ ∧ ( 𝑗 + ( 1 / 3 ) ) ∈ ℝ ∧ ( 𝐸 ∈ ℝ ∧ 0 < 𝐸 ) ) → ( ( 𝑗 − ( 1 / 3 ) ) < ( 𝑗 + ( 1 / 3 ) ) ↔ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) < ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ) ) |
162 |
134 119 160 161
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑗 − ( 1 / 3 ) ) < ( 𝑗 + ( 1 / 3 ) ) ↔ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) < ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ) ) |
163 |
158 162
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) < ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ) |
164 |
163
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) < ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ) |
165 |
110
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ↔ 𝑡 ∈ { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ) ) |
166 |
165
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → 𝑡 ∈ { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ) |
167 |
|
rabid |
⊢ ( 𝑡 ∈ { 𝑡 ∈ 𝑇 ∣ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) } ↔ ( 𝑡 ∈ 𝑇 ∧ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) ) ) |
168 |
166 167
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ( 𝑡 ∈ 𝑇 ∧ ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) ) ) |
169 |
168
|
simprd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ( ( 𝑗 + ( 1 / 3 ) ) · 𝐸 ) ≤ ( 𝐹 ‘ 𝑡 ) ) |
170 |
138 139 145 164 169
|
ltletrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) < ( 𝐹 ‘ 𝑡 ) ) |
171 |
138 145
|
ltnled |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ( ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) < ( 𝐹 ‘ 𝑡 ) ↔ ¬ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) ) ) |
172 |
170 171
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ¬ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) ) |
173 |
172
|
intnand |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ¬ ( 𝑡 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) ) ) |
174 |
|
rabid |
⊢ ( 𝑡 ∈ { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ↔ ( 𝑡 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) ) ) |
175 |
173 174
|
sylnibr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ¬ 𝑡 ∈ { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ) |
176 |
130
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ( 𝐷 ‘ 𝑗 ) = { 𝑡 ∈ 𝑇 ∣ ( 𝐹 ‘ 𝑡 ) ≤ ( ( 𝑗 − ( 1 / 3 ) ) · 𝐸 ) } ) |
177 |
175 176
|
neleqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ) → ¬ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ) |
178 |
177
|
ex |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) → ¬ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ) ) |
179 |
97 178
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ¬ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ) |
180 |
|
disj |
⊢ ( ( ( 𝐵 ‘ 𝑗 ) ∩ ( 𝐷 ‘ 𝑗 ) ) = ∅ ↔ ∀ 𝑎 ∈ ( 𝐵 ‘ 𝑗 ) ¬ 𝑎 ∈ ( 𝐷 ‘ 𝑗 ) ) |
181 |
|
nfcv |
⊢ Ⅎ 𝑎 ( 𝐵 ‘ 𝑗 ) |
182 |
89
|
nfcri |
⊢ Ⅎ 𝑡 𝑎 ∈ ( 𝐷 ‘ 𝑗 ) |
183 |
182
|
nfn |
⊢ Ⅎ 𝑡 ¬ 𝑎 ∈ ( 𝐷 ‘ 𝑗 ) |
184 |
|
nfv |
⊢ Ⅎ 𝑎 ¬ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) |
185 |
|
eleq1 |
⊢ ( 𝑎 = 𝑡 → ( 𝑎 ∈ ( 𝐷 ‘ 𝑗 ) ↔ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ) ) |
186 |
185
|
notbid |
⊢ ( 𝑎 = 𝑡 → ( ¬ 𝑎 ∈ ( 𝐷 ‘ 𝑗 ) ↔ ¬ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ) ) |
187 |
181 94 183 184 186
|
cbvralfw |
⊢ ( ∀ 𝑎 ∈ ( 𝐵 ‘ 𝑗 ) ¬ 𝑎 ∈ ( 𝐷 ‘ 𝑗 ) ↔ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ¬ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ) |
188 |
180 187
|
bitri |
⊢ ( ( ( 𝐵 ‘ 𝑗 ) ∩ ( 𝐷 ‘ 𝑗 ) ) = ∅ ↔ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ¬ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ) |
189 |
179 188
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝐵 ‘ 𝑗 ) ∩ ( 𝐷 ‘ 𝑗 ) ) = ∅ ) |
190 |
|
eqid |
⊢ ( 𝑇 ∖ ( 𝐵 ‘ 𝑗 ) ) = ( 𝑇 ∖ ( 𝐵 ‘ 𝑗 ) ) |
191 |
19
|
nnrpd |
⊢ ( 𝜑 → 𝑁 ∈ ℝ+ ) |
192 |
17 191
|
rpdivcld |
⊢ ( 𝜑 → ( 𝐸 / 𝑁 ) ∈ ℝ+ ) |
193 |
192
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐸 / 𝑁 ) ∈ ℝ+ ) |
194 |
120 19
|
nndivred |
⊢ ( 𝜑 → ( 𝐸 / 𝑁 ) ∈ ℝ ) |
195 |
116
|
a1i |
⊢ ( 𝜑 → ( 1 / 3 ) ∈ ℝ ) |
196 |
19
|
nnge1d |
⊢ ( 𝜑 → 1 ≤ 𝑁 ) |
197 |
|
1re |
⊢ 1 ∈ ℝ |
198 |
|
0lt1 |
⊢ 0 < 1 |
199 |
197 198
|
pm3.2i |
⊢ ( 1 ∈ ℝ ∧ 0 < 1 ) |
200 |
199
|
a1i |
⊢ ( 𝜑 → ( 1 ∈ ℝ ∧ 0 < 1 ) ) |
201 |
19
|
nnred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
202 |
19
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑁 ) |
203 |
|
lediv2 |
⊢ ( ( ( 1 ∈ ℝ ∧ 0 < 1 ) ∧ ( 𝑁 ∈ ℝ ∧ 0 < 𝑁 ) ∧ ( 𝐸 ∈ ℝ ∧ 0 < 𝐸 ) ) → ( 1 ≤ 𝑁 ↔ ( 𝐸 / 𝑁 ) ≤ ( 𝐸 / 1 ) ) ) |
204 |
200 201 202 159 203
|
syl121anc |
⊢ ( 𝜑 → ( 1 ≤ 𝑁 ↔ ( 𝐸 / 𝑁 ) ≤ ( 𝐸 / 1 ) ) ) |
205 |
196 204
|
mpbid |
⊢ ( 𝜑 → ( 𝐸 / 𝑁 ) ≤ ( 𝐸 / 1 ) ) |
206 |
17
|
rpcnd |
⊢ ( 𝜑 → 𝐸 ∈ ℂ ) |
207 |
206
|
div1d |
⊢ ( 𝜑 → ( 𝐸 / 1 ) = 𝐸 ) |
208 |
205 207
|
breqtrd |
⊢ ( 𝜑 → ( 𝐸 / 𝑁 ) ≤ 𝐸 ) |
209 |
194 120 195 208 18
|
lelttrd |
⊢ ( 𝜑 → ( 𝐸 / 𝑁 ) < ( 1 / 3 ) ) |
210 |
209
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐸 / 𝑁 ) < ( 1 / 3 ) ) |
211 |
89 95 97 3 4 5 98 99 100 101 102 103 126 137 189 190 193 210
|
stoweidlem58 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) |
212 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) |
213 |
211 212
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) |
214 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) → 𝑥 ∈ 𝐴 ) |
215 |
|
simprr1 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) → ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ) |
216 |
|
fveq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ‘ 𝑡 ) = ( 𝑥 ‘ 𝑡 ) ) |
217 |
216
|
breq2d |
⊢ ( 𝑦 = 𝑥 → ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ↔ 0 ≤ ( 𝑥 ‘ 𝑡 ) ) ) |
218 |
216
|
breq1d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 ‘ 𝑡 ) ≤ 1 ↔ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ) |
219 |
217 218
|
anbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ↔ ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ) ) |
220 |
219
|
ralbidv |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ↔ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ) ) |
221 |
220 8
|
elrab2 |
⊢ ( 𝑥 ∈ 𝑌 ↔ ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ) ) |
222 |
214 215 221
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) → 𝑥 ∈ 𝑌 ) |
223 |
|
simprr2 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) → ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) |
224 |
|
simprr3 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) → ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) |
225 |
223 224
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) → ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) |
226 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑥 |
227 |
|
nfv |
⊢ Ⅎ 𝑦 ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) |
228 |
216
|
breq1d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ↔ ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) |
229 |
228
|
ralbidv |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ↔ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) |
230 |
216
|
breq2d |
⊢ ( 𝑦 = 𝑥 → ( ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ↔ ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) |
231 |
230
|
ralbidv |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ↔ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) |
232 |
229 231
|
anbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) ↔ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) |
233 |
226 21 227 232
|
elrabf |
⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ↔ ( 𝑥 ∈ 𝑌 ∧ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) |
234 |
222 225 233
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) ) → 𝑥 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) |
235 |
234
|
ex |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) → 𝑥 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) ) |
236 |
235
|
eximdv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑥 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑥 ‘ 𝑡 ) ) ) → ∃ 𝑥 𝑥 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) ) |
237 |
213 236
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ∃ 𝑥 𝑥 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) |
238 |
|
ne0i |
⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } → { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ≠ ∅ ) |
239 |
238
|
exlimiv |
⊢ ( ∃ 𝑥 𝑥 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } → { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ≠ ∅ ) |
240 |
237 239
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ≠ ∅ ) |
241 |
83 240
|
eqnetrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐻 ‘ 𝑗 ) ≠ ∅ ) |
242 |
241
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ∧ ( 𝐻 ‘ 𝑗 ) = 𝑤 ) → ( 𝐻 ‘ 𝑗 ) ≠ ∅ ) |
243 |
79 242
|
eqnetrrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ∧ ( 𝐻 ‘ 𝑗 ) = 𝑤 ) → 𝑤 ≠ ∅ ) |
244 |
243
|
3exp |
⊢ ( 𝜑 → ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( ( 𝐻 ‘ 𝑗 ) = 𝑤 → 𝑤 ≠ ∅ ) ) ) |
245 |
244
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝐻 ‘ 𝑗 ) = 𝑤 → 𝑤 ≠ ∅ ) ) |
246 |
245
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran 𝐻 ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝐻 ‘ 𝑗 ) = 𝑤 → 𝑤 ≠ ∅ ) ) |
247 |
78 246
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ran 𝐻 ) → 𝑤 ≠ ∅ ) |
248 |
247
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑤 ∈ ran 𝐻 ) → 𝑤 ≠ ∅ ) |
249 |
|
rsp |
⊢ ( ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) → ( 𝑤 ∈ ran 𝐻 → ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) |
250 |
63 64 248 249
|
syl3c |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑤 ∈ ran 𝐻 ) → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) |
251 |
250
|
ex |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ( 𝑤 ∈ ran 𝐻 → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) |
252 |
62 251
|
ralrimi |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ∀ 𝑤 ∈ ran 𝐻 ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) |
253 |
|
chfnrn |
⊢ ( ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) → ran ℎ ⊆ ∪ ran 𝐻 ) |
254 |
47 252 253
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ran ℎ ⊆ ∪ ran 𝐻 ) |
255 |
|
nfv |
⊢ Ⅎ 𝑦 𝜑 |
256 |
|
nfcv |
⊢ Ⅎ 𝑦 ℎ |
257 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 0 ... 𝑁 ) |
258 |
|
nfrab1 |
⊢ Ⅎ 𝑦 { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } |
259 |
257 258
|
nfmpt |
⊢ Ⅎ 𝑦 ( 𝑗 ∈ ( 0 ... 𝑁 ) ↦ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) |
260 |
9 259
|
nfcxfr |
⊢ Ⅎ 𝑦 𝐻 |
261 |
260
|
nfrn |
⊢ Ⅎ 𝑦 ran 𝐻 |
262 |
256 261
|
nffn |
⊢ Ⅎ 𝑦 ℎ Fn ran 𝐻 |
263 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) |
264 |
261 263
|
nfralw |
⊢ Ⅎ 𝑦 ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) |
265 |
262 264
|
nfan |
⊢ Ⅎ 𝑦 ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) |
266 |
255 265
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) |
267 |
261
|
nfuni |
⊢ Ⅎ 𝑦 ∪ ran 𝐻 |
268 |
|
fnunirn |
⊢ ( 𝐻 Fn ( 0 ... 𝑁 ) → ( 𝑦 ∈ ∪ ran 𝐻 ↔ ∃ 𝑧 ∈ ( 0 ... 𝑁 ) 𝑦 ∈ ( 𝐻 ‘ 𝑧 ) ) ) |
269 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑧 |
270 |
68 269
|
nffv |
⊢ Ⅎ 𝑗 ( 𝐻 ‘ 𝑧 ) |
271 |
270
|
nfcri |
⊢ Ⅎ 𝑗 𝑦 ∈ ( 𝐻 ‘ 𝑧 ) |
272 |
|
nfv |
⊢ Ⅎ 𝑧 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) |
273 |
|
fveq2 |
⊢ ( 𝑧 = 𝑗 → ( 𝐻 ‘ 𝑧 ) = ( 𝐻 ‘ 𝑗 ) ) |
274 |
273
|
eleq2d |
⊢ ( 𝑧 = 𝑗 → ( 𝑦 ∈ ( 𝐻 ‘ 𝑧 ) ↔ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) ) |
275 |
271 272 274
|
cbvrexw |
⊢ ( ∃ 𝑧 ∈ ( 0 ... 𝑁 ) 𝑦 ∈ ( 𝐻 ‘ 𝑧 ) ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) |
276 |
268 275
|
bitrdi |
⊢ ( 𝐻 Fn ( 0 ... 𝑁 ) → ( 𝑦 ∈ ∪ ran 𝐻 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) ) |
277 |
39 276
|
syl |
⊢ ( 𝜑 → ( 𝑦 ∈ ∪ ran 𝐻 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) ) |
278 |
277
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ∪ ran 𝐻 ) → ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) |
279 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
280 |
68
|
nfrn |
⊢ Ⅎ 𝑗 ran 𝐻 |
281 |
280
|
nfuni |
⊢ Ⅎ 𝑗 ∪ ran 𝐻 |
282 |
281
|
nfcri |
⊢ Ⅎ 𝑗 𝑦 ∈ ∪ ran 𝐻 |
283 |
279 282
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑦 ∈ ∪ ran 𝐻 ) |
284 |
|
nfv |
⊢ Ⅎ 𝑗 𝑦 ∈ 𝑌 |
285 |
|
simp1l |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ∪ ran 𝐻 ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → 𝜑 ) |
286 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ∪ ran 𝐻 ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → 𝑗 ∈ ( 0 ... 𝑁 ) ) |
287 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ∪ ran 𝐻 ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) |
288 |
83
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ↔ 𝑦 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) ) |
289 |
288
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → 𝑦 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) |
290 |
|
rabid |
⊢ ( 𝑦 ∈ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ↔ ( 𝑦 ∈ 𝑌 ∧ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) ) ) |
291 |
289 290
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → ( 𝑦 ∈ 𝑌 ∧ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) ) ) |
292 |
291
|
simpld |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → 𝑦 ∈ 𝑌 ) |
293 |
285 286 287 292
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ∪ ran 𝐻 ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → 𝑦 ∈ 𝑌 ) |
294 |
293
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ∪ ran 𝐻 ) → ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) → 𝑦 ∈ 𝑌 ) ) ) |
295 |
283 284 294
|
rexlimd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ∪ ran 𝐻 ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) → 𝑦 ∈ 𝑌 ) ) |
296 |
278 295
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ∪ ran 𝐻 ) → 𝑦 ∈ 𝑌 ) |
297 |
296
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑦 ∈ ∪ ran 𝐻 ) → 𝑦 ∈ 𝑌 ) |
298 |
297
|
ex |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ( 𝑦 ∈ ∪ ran 𝐻 → 𝑦 ∈ 𝑌 ) ) |
299 |
266 267 21 298
|
ssrd |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ∪ ran 𝐻 ⊆ 𝑌 ) |
300 |
|
ssrab2 |
⊢ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) } ⊆ 𝐴 |
301 |
8 300
|
eqsstri |
⊢ 𝑌 ⊆ 𝐴 |
302 |
299 301
|
sstrdi |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ∪ ran 𝐻 ⊆ 𝐴 ) |
303 |
254 302
|
sstrd |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ran ℎ ⊆ 𝐴 ) |
304 |
57 303
|
fssd |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ℎ : ran 𝐻 ⟶ 𝐴 ) |
305 |
|
dffn3 |
⊢ ( 𝐻 Fn ( 0 ... 𝑁 ) ↔ 𝐻 : ( 0 ... 𝑁 ) ⟶ ran 𝐻 ) |
306 |
39 305
|
sylib |
⊢ ( 𝜑 → 𝐻 : ( 0 ... 𝑁 ) ⟶ ran 𝐻 ) |
307 |
306
|
adantr |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → 𝐻 : ( 0 ... 𝑁 ) ⟶ ran 𝐻 ) |
308 |
|
fco |
⊢ ( ( ℎ : ran 𝐻 ⟶ 𝐴 ∧ 𝐻 : ( 0 ... 𝑁 ) ⟶ ran 𝐻 ) → ( ℎ ∘ 𝐻 ) : ( 0 ... 𝑁 ) ⟶ 𝐴 ) |
309 |
304 307 308
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ( ℎ ∘ 𝐻 ) : ( 0 ... 𝑁 ) ⟶ 𝐴 ) |
310 |
|
nfcv |
⊢ Ⅎ 𝑗 ℎ |
311 |
310 280
|
nffn |
⊢ Ⅎ 𝑗 ℎ Fn ran 𝐻 |
312 |
|
nfv |
⊢ Ⅎ 𝑗 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) |
313 |
280 312
|
nfralw |
⊢ Ⅎ 𝑗 ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) |
314 |
311 313
|
nfan |
⊢ Ⅎ 𝑗 ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) |
315 |
279 314
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) |
316 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝜑 ) |
317 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝑗 ∈ ( 0 ... 𝑁 ) ) |
318 |
39
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝐻 Fn ( 0 ... 𝑁 ) ) |
319 |
|
fvco2 |
⊢ ( ( 𝐻 Fn ( 0 ... 𝑁 ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) = ( ℎ ‘ ( 𝐻 ‘ 𝑗 ) ) ) |
320 |
318 319
|
sylancom |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) = ( ℎ ‘ ( 𝐻 ‘ 𝑗 ) ) ) |
321 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) |
322 |
|
fnfun |
⊢ ( 𝐻 Fn ( 0 ... 𝑁 ) → Fun 𝐻 ) |
323 |
39 322
|
syl |
⊢ ( 𝜑 → Fun 𝐻 ) |
324 |
323
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → Fun 𝐻 ) |
325 |
39
|
fndmd |
⊢ ( 𝜑 → dom 𝐻 = ( 0 ... 𝑁 ) ) |
326 |
325
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → dom 𝐻 = ( 0 ... 𝑁 ) ) |
327 |
80 326
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝑗 ∈ dom 𝐻 ) |
328 |
327
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝑗 ∈ dom 𝐻 ) |
329 |
|
fvelrn |
⊢ ( ( Fun 𝐻 ∧ 𝑗 ∈ dom 𝐻 ) → ( 𝐻 ‘ 𝑗 ) ∈ ran 𝐻 ) |
330 |
324 328 329
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐻 ‘ 𝑗 ) ∈ ran 𝐻 ) |
331 |
321 330
|
jca |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ∧ ( 𝐻 ‘ 𝑗 ) ∈ ran 𝐻 ) ) |
332 |
241
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐻 ‘ 𝑗 ) ≠ ∅ ) |
333 |
|
neeq1 |
⊢ ( 𝑤 = ( 𝐻 ‘ 𝑗 ) → ( 𝑤 ≠ ∅ ↔ ( 𝐻 ‘ 𝑗 ) ≠ ∅ ) ) |
334 |
|
fveq2 |
⊢ ( 𝑤 = ( 𝐻 ‘ 𝑗 ) → ( ℎ ‘ 𝑤 ) = ( ℎ ‘ ( 𝐻 ‘ 𝑗 ) ) ) |
335 |
|
id |
⊢ ( 𝑤 = ( 𝐻 ‘ 𝑗 ) → 𝑤 = ( 𝐻 ‘ 𝑗 ) ) |
336 |
334 335
|
eleq12d |
⊢ ( 𝑤 = ( 𝐻 ‘ 𝑗 ) → ( ( ℎ ‘ 𝑤 ) ∈ 𝑤 ↔ ( ℎ ‘ ( 𝐻 ‘ 𝑗 ) ) ∈ ( 𝐻 ‘ 𝑗 ) ) ) |
337 |
333 336
|
imbi12d |
⊢ ( 𝑤 = ( 𝐻 ‘ 𝑗 ) → ( ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ↔ ( ( 𝐻 ‘ 𝑗 ) ≠ ∅ → ( ℎ ‘ ( 𝐻 ‘ 𝑗 ) ) ∈ ( 𝐻 ‘ 𝑗 ) ) ) ) |
338 |
337
|
rspccva |
⊢ ( ( ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ∧ ( 𝐻 ‘ 𝑗 ) ∈ ran 𝐻 ) → ( ( 𝐻 ‘ 𝑗 ) ≠ ∅ → ( ℎ ‘ ( 𝐻 ‘ 𝑗 ) ) ∈ ( 𝐻 ‘ 𝑗 ) ) ) |
339 |
331 332 338
|
sylc |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ℎ ‘ ( 𝐻 ‘ 𝑗 ) ) ∈ ( 𝐻 ‘ 𝑗 ) ) |
340 |
320 339
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) |
341 |
256 260
|
nfco |
⊢ Ⅎ 𝑦 ( ℎ ∘ 𝐻 ) |
342 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑗 |
343 |
341 342
|
nffv |
⊢ Ⅎ 𝑦 ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) |
344 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) |
345 |
260 342
|
nffv |
⊢ Ⅎ 𝑦 ( 𝐻 ‘ 𝑗 ) |
346 |
343 345
|
nfel |
⊢ Ⅎ 𝑦 ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) |
347 |
344 346
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) |
348 |
343 21
|
nfel |
⊢ Ⅎ 𝑦 ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝑌 |
349 |
347 348
|
nfim |
⊢ Ⅎ 𝑦 ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝑌 ) |
350 |
|
eleq1 |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ↔ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) ) |
351 |
350
|
anbi2d |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) ) ) |
352 |
|
eleq1 |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( 𝑦 ∈ 𝑌 ↔ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝑌 ) ) |
353 |
351 352
|
imbi12d |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → 𝑦 ∈ 𝑌 ) ↔ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝑌 ) ) ) |
354 |
343 349 353 292
|
vtoclgf |
⊢ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) → ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝑌 ) ) |
355 |
354
|
anabsi7 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝑌 ) |
356 |
316 317 340 355
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝑌 ) |
357 |
8
|
eleq2i |
⊢ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝑌 ↔ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) } ) |
358 |
|
nfcv |
⊢ Ⅎ 𝑦 𝐴 |
359 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑇 |
360 |
|
nfcv |
⊢ Ⅎ 𝑦 0 |
361 |
|
nfcv |
⊢ Ⅎ 𝑦 ≤ |
362 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑡 |
363 |
343 362
|
nffv |
⊢ Ⅎ 𝑦 ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) |
364 |
360 361 363
|
nfbr |
⊢ Ⅎ 𝑦 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) |
365 |
|
nfcv |
⊢ Ⅎ 𝑦 1 |
366 |
363 361 365
|
nfbr |
⊢ Ⅎ 𝑦 ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 |
367 |
364 366
|
nfan |
⊢ Ⅎ 𝑦 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) |
368 |
359 367
|
nfralw |
⊢ Ⅎ 𝑦 ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) |
369 |
|
nfcv |
⊢ Ⅎ 𝑡 𝑦 |
370 |
|
nfcv |
⊢ Ⅎ 𝑡 ℎ |
371 |
|
nfra1 |
⊢ Ⅎ 𝑡 ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) |
372 |
|
nfra1 |
⊢ Ⅎ 𝑡 ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) |
373 |
371 372
|
nfan |
⊢ Ⅎ 𝑡 ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) |
374 |
|
nfra1 |
⊢ Ⅎ 𝑡 ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) |
375 |
|
nfcv |
⊢ Ⅎ 𝑡 𝐴 |
376 |
374 375
|
nfrabw |
⊢ Ⅎ 𝑡 { 𝑦 ∈ 𝐴 ∣ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) } |
377 |
8 376
|
nfcxfr |
⊢ Ⅎ 𝑡 𝑌 |
378 |
373 377
|
nfrabw |
⊢ Ⅎ 𝑡 { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } |
379 |
84 378
|
nfmpt |
⊢ Ⅎ 𝑡 ( 𝑗 ∈ ( 0 ... 𝑁 ) ↦ { 𝑦 ∈ 𝑌 ∣ ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) } ) |
380 |
9 379
|
nfcxfr |
⊢ Ⅎ 𝑡 𝐻 |
381 |
370 380
|
nfco |
⊢ Ⅎ 𝑡 ( ℎ ∘ 𝐻 ) |
382 |
381 88
|
nffv |
⊢ Ⅎ 𝑡 ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) |
383 |
369 382
|
nfeq |
⊢ Ⅎ 𝑡 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) |
384 |
|
fveq1 |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( 𝑦 ‘ 𝑡 ) = ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) |
385 |
384
|
breq2d |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ↔ 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) |
386 |
384
|
breq1d |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ( 𝑦 ‘ 𝑡 ) ≤ 1 ↔ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) |
387 |
385 386
|
anbi12d |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ↔ ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) ) |
388 |
383 387
|
ralbid |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ↔ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) ) |
389 |
343 358 368 388
|
elrabf |
⊢ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) } ↔ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝐴 ∧ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) ) |
390 |
357 389
|
bitri |
⊢ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝑌 ↔ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝐴 ∧ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) ) |
391 |
356 390
|
sylib |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ 𝐴 ∧ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) ) |
392 |
391
|
simprd |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) |
393 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝐷 ‘ 𝑗 ) |
394 |
|
nfcv |
⊢ Ⅎ 𝑦 < |
395 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝐸 / 𝑁 ) |
396 |
363 394 395
|
nfbr |
⊢ Ⅎ 𝑦 ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) |
397 |
393 396
|
nfralw |
⊢ Ⅎ 𝑦 ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) |
398 |
347 397
|
nfim |
⊢ Ⅎ 𝑦 ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) |
399 |
384
|
breq1d |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ↔ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) |
400 |
383 399
|
ralbid |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ↔ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) |
401 |
351 400
|
imbi12d |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ↔ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) ) |
402 |
291
|
simprd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) ) |
403 |
402
|
simpld |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( 𝑦 ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) |
404 |
343 398 401 403
|
vtoclgf |
⊢ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) → ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) |
405 |
404
|
anabsi7 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) |
406 |
316 317 340 405
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) |
407 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝐵 ‘ 𝑗 ) |
408 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 1 − ( 𝐸 / 𝑁 ) ) |
409 |
408 394 363
|
nfbr |
⊢ Ⅎ 𝑦 ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) |
410 |
407 409
|
nfralw |
⊢ Ⅎ 𝑦 ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) |
411 |
347 410
|
nfim |
⊢ Ⅎ 𝑦 ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) |
412 |
384
|
breq2d |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ↔ ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) |
413 |
383 412
|
ralbid |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ↔ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) |
414 |
351 413
|
imbi12d |
⊢ ( 𝑦 = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) → ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) ↔ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) ) |
415 |
402
|
simprd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑦 ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( 𝑦 ‘ 𝑡 ) ) |
416 |
343 411 414 415
|
vtoclgf |
⊢ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) → ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) |
417 |
416
|
anabsi7 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ∈ ( 𝐻 ‘ 𝑗 ) ) → ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) |
418 |
316 317 340 417
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) |
419 |
392 406 418
|
3jca |
⊢ ( ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) |
420 |
419
|
ex |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) ) |
421 |
315 420
|
ralrimi |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) |
422 |
309 421
|
jca |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ( ( ℎ ∘ 𝐻 ) : ( 0 ... 𝑁 ) ⟶ 𝐴 ∧ ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) ) |
423 |
|
feq1 |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( 𝑥 : ( 0 ... 𝑁 ) ⟶ 𝐴 ↔ ( ℎ ∘ 𝐻 ) : ( 0 ... 𝑁 ) ⟶ 𝐴 ) ) |
424 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑥 |
425 |
310 68
|
nfco |
⊢ Ⅎ 𝑗 ( ℎ ∘ 𝐻 ) |
426 |
424 425
|
nfeq |
⊢ Ⅎ 𝑗 𝑥 = ( ℎ ∘ 𝐻 ) |
427 |
|
nfcv |
⊢ Ⅎ 𝑡 𝑥 |
428 |
427 381
|
nfeq |
⊢ Ⅎ 𝑡 𝑥 = ( ℎ ∘ 𝐻 ) |
429 |
|
fveq1 |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( 𝑥 ‘ 𝑗 ) = ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ) |
430 |
429
|
fveq1d |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) = ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) |
431 |
430
|
breq2d |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( 0 ≤ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ↔ 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) |
432 |
430
|
breq1d |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ↔ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) |
433 |
431 432
|
anbi12d |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ( 0 ≤ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ↔ ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) ) |
434 |
428 433
|
ralbid |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ↔ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ) ) |
435 |
430
|
breq1d |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ↔ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) |
436 |
428 435
|
ralbid |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ↔ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ) ) |
437 |
430
|
breq2d |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ↔ ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) |
438 |
428 437
|
ralbid |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ↔ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) |
439 |
434 436 438
|
3anbi123d |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ) ↔ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) ) |
440 |
426 439
|
ralbid |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ) ↔ ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) ) |
441 |
423 440
|
anbi12d |
⊢ ( 𝑥 = ( ℎ ∘ 𝐻 ) → ( ( 𝑥 : ( 0 ... 𝑁 ) ⟶ 𝐴 ∧ ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ) ) ↔ ( ( ℎ ∘ 𝐻 ) : ( 0 ... 𝑁 ) ⟶ 𝐴 ∧ ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) ) ) |
442 |
441
|
spcegv |
⊢ ( ( ℎ ∘ 𝐻 ) ∈ V → ( ( ( ℎ ∘ 𝐻 ) : ( 0 ... 𝑁 ) ⟶ 𝐴 ∧ ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( ( ℎ ∘ 𝐻 ) ‘ 𝑗 ) ‘ 𝑡 ) ) ) → ∃ 𝑥 ( 𝑥 : ( 0 ... 𝑁 ) ⟶ 𝐴 ∧ ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ) ) ) ) |
443 |
55 422 442
|
sylc |
⊢ ( ( 𝜑 ∧ ( ℎ Fn ran 𝐻 ∧ ∀ 𝑤 ∈ ran 𝐻 ( 𝑤 ≠ ∅ → ( ℎ ‘ 𝑤 ) ∈ 𝑤 ) ) ) → ∃ 𝑥 ( 𝑥 : ( 0 ... 𝑁 ) ⟶ 𝐴 ∧ ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ) ) ) |
444 |
46 443
|
exlimddv |
⊢ ( 𝜑 → ∃ 𝑥 ( 𝑥 : ( 0 ... 𝑁 ) ⟶ 𝐴 ∧ ∀ 𝑗 ∈ ( 0 ... 𝑁 ) ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ∧ ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ ( 𝐷 ‘ 𝑗 ) ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) < ( 𝐸 / 𝑁 ) ∧ ∀ 𝑡 ∈ ( 𝐵 ‘ 𝑗 ) ( 1 − ( 𝐸 / 𝑁 ) ) < ( ( 𝑥 ‘ 𝑗 ) ‘ 𝑡 ) ) ) ) |