Step |
Hyp |
Ref |
Expression |
1 |
|
stoweidlem41.1 |
⊢ Ⅎ 𝑡 𝜑 |
2 |
|
stoweidlem41.2 |
⊢ 𝑋 = ( 𝑡 ∈ 𝑇 ↦ ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
3 |
|
stoweidlem41.3 |
⊢ 𝐹 = ( 𝑡 ∈ 𝑇 ↦ 1 ) |
4 |
|
stoweidlem41.4 |
⊢ 𝑉 ⊆ 𝑇 |
5 |
|
stoweidlem41.5 |
⊢ ( 𝜑 → 𝑦 ∈ 𝐴 ) |
6 |
|
stoweidlem41.6 |
⊢ ( 𝜑 → 𝑦 : 𝑇 ⟶ ℝ ) |
7 |
|
stoweidlem41.7 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝑓 : 𝑇 ⟶ ℝ ) |
8 |
|
stoweidlem41.8 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
9 |
|
stoweidlem41.9 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
10 |
|
stoweidlem41.10 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) → ( 𝑡 ∈ 𝑇 ↦ 𝑤 ) ∈ 𝐴 ) |
11 |
|
stoweidlem41.11 |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
12 |
|
stoweidlem41.12 |
⊢ ( 𝜑 → ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ) |
13 |
|
stoweidlem41.13 |
⊢ ( 𝜑 → ∀ 𝑡 ∈ 𝑉 ( 1 − 𝐸 ) < ( 𝑦 ‘ 𝑡 ) ) |
14 |
|
stoweidlem41.14 |
⊢ ( 𝜑 → ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝐸 ) |
15 |
|
1re |
⊢ 1 ∈ ℝ |
16 |
3
|
fvmpt2 |
⊢ ( ( 𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ ) → ( 𝐹 ‘ 𝑡 ) = 1 ) |
17 |
15 16
|
mpan2 |
⊢ ( 𝑡 ∈ 𝑇 → ( 𝐹 ‘ 𝑡 ) = 1 ) |
18 |
17
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 𝐹 ‘ 𝑡 ) = 1 ) |
19 |
18
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑡 ) − ( 𝑦 ‘ 𝑡 ) ) = ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
20 |
1 19
|
mpteq2da |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) − ( 𝑦 ‘ 𝑡 ) ) ) = ( 𝑡 ∈ 𝑇 ↦ ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) ) |
21 |
20 2
|
eqtr4di |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) − ( 𝑦 ‘ 𝑡 ) ) ) = 𝑋 ) |
22 |
10
|
stoweidlem4 |
⊢ ( ( 𝜑 ∧ 1 ∈ ℝ ) → ( 𝑡 ∈ 𝑇 ↦ 1 ) ∈ 𝐴 ) |
23 |
15 22
|
mpan2 |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ 1 ) ∈ 𝐴 ) |
24 |
3 23
|
eqeltrid |
⊢ ( 𝜑 → 𝐹 ∈ 𝐴 ) |
25 |
|
nfmpt1 |
⊢ Ⅎ 𝑡 ( 𝑡 ∈ 𝑇 ↦ 1 ) |
26 |
3 25
|
nfcxfr |
⊢ Ⅎ 𝑡 𝐹 |
27 |
|
nfcv |
⊢ Ⅎ 𝑡 𝑦 |
28 |
26 27 1 7 8 9 10
|
stoweidlem33 |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) − ( 𝑦 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
29 |
24 5 28
|
mpd3an23 |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝐹 ‘ 𝑡 ) − ( 𝑦 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
30 |
21 29
|
eqeltrrd |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
31 |
6
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 𝑦 ‘ 𝑡 ) ∈ ℝ ) |
32 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → 1 ∈ ℝ ) |
33 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → 0 ∈ ℝ ) |
34 |
12
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ) |
35 |
34
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 𝑦 ‘ 𝑡 ) ≤ 1 ) |
36 |
|
1m0e1 |
⊢ ( 1 − 0 ) = 1 |
37 |
35 36
|
breqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 𝑦 ‘ 𝑡 ) ≤ ( 1 − 0 ) ) |
38 |
31 32 33 37
|
lesubd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → 0 ≤ ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
39 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → 𝑡 ∈ 𝑇 ) |
40 |
32 31
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 1 − ( 𝑦 ‘ 𝑡 ) ) ∈ ℝ ) |
41 |
2
|
fvmpt2 |
⊢ ( ( 𝑡 ∈ 𝑇 ∧ ( 1 − ( 𝑦 ‘ 𝑡 ) ) ∈ ℝ ) → ( 𝑋 ‘ 𝑡 ) = ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
42 |
39 40 41
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 𝑋 ‘ 𝑡 ) = ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
43 |
38 42
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → 0 ≤ ( 𝑋 ‘ 𝑡 ) ) |
44 |
34
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → 0 ≤ ( 𝑦 ‘ 𝑡 ) ) |
45 |
33 31 32 44
|
lesub2dd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 1 − ( 𝑦 ‘ 𝑡 ) ) ≤ ( 1 − 0 ) ) |
46 |
45 36
|
breqtrdi |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 1 − ( 𝑦 ‘ 𝑡 ) ) ≤ 1 ) |
47 |
42 46
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 𝑋 ‘ 𝑡 ) ≤ 1 ) |
48 |
43 47
|
jca |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 0 ≤ ( 𝑋 ‘ 𝑡 ) ∧ ( 𝑋 ‘ 𝑡 ) ≤ 1 ) ) |
49 |
48
|
ex |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑇 → ( 0 ≤ ( 𝑋 ‘ 𝑡 ) ∧ ( 𝑋 ‘ 𝑡 ) ≤ 1 ) ) ) |
50 |
1 49
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑋 ‘ 𝑡 ) ∧ ( 𝑋 ‘ 𝑡 ) ≤ 1 ) ) |
51 |
4
|
sseli |
⊢ ( 𝑡 ∈ 𝑉 → 𝑡 ∈ 𝑇 ) |
52 |
51 42
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 𝑋 ‘ 𝑡 ) = ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
53 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → 1 ∈ ℝ ) |
54 |
11
|
rpred |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
55 |
54
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → 𝐸 ∈ ℝ ) |
56 |
51 31
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 𝑦 ‘ 𝑡 ) ∈ ℝ ) |
57 |
13
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 1 − 𝐸 ) < ( 𝑦 ‘ 𝑡 ) ) |
58 |
53 55 56 57
|
ltsub23d |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 1 − ( 𝑦 ‘ 𝑡 ) ) < 𝐸 ) |
59 |
52 58
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 𝑋 ‘ 𝑡 ) < 𝐸 ) |
60 |
59
|
ex |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑉 → ( 𝑋 ‘ 𝑡 ) < 𝐸 ) ) |
61 |
1 60
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑡 ∈ 𝑉 ( 𝑋 ‘ 𝑡 ) < 𝐸 ) |
62 |
|
eldifi |
⊢ ( 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) → 𝑡 ∈ 𝑇 ) |
63 |
62 31
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ) → ( 𝑦 ‘ 𝑡 ) ∈ ℝ ) |
64 |
54
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ) → 𝐸 ∈ ℝ ) |
65 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ) → 1 ∈ ℝ ) |
66 |
14
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ) → ( 𝑦 ‘ 𝑡 ) < 𝐸 ) |
67 |
63 64 65 66
|
ltsub2dd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ) → ( 1 − 𝐸 ) < ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
68 |
62 42
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ) → ( 𝑋 ‘ 𝑡 ) = ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
69 |
67 68
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ) → ( 1 − 𝐸 ) < ( 𝑋 ‘ 𝑡 ) ) |
70 |
69
|
ex |
⊢ ( 𝜑 → ( 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) → ( 1 − 𝐸 ) < ( 𝑋 ‘ 𝑡 ) ) ) |
71 |
1 70
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝐸 ) < ( 𝑋 ‘ 𝑡 ) ) |
72 |
|
nfmpt1 |
⊢ Ⅎ 𝑡 ( 𝑡 ∈ 𝑇 ↦ ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
73 |
2 72
|
nfcxfr |
⊢ Ⅎ 𝑡 𝑋 |
74 |
73
|
nfeq2 |
⊢ Ⅎ 𝑡 𝑥 = 𝑋 |
75 |
|
fveq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ‘ 𝑡 ) = ( 𝑋 ‘ 𝑡 ) ) |
76 |
75
|
breq2d |
⊢ ( 𝑥 = 𝑋 → ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ↔ 0 ≤ ( 𝑋 ‘ 𝑡 ) ) ) |
77 |
75
|
breq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ‘ 𝑡 ) ≤ 1 ↔ ( 𝑋 ‘ 𝑡 ) ≤ 1 ) ) |
78 |
76 77
|
anbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ↔ ( 0 ≤ ( 𝑋 ‘ 𝑡 ) ∧ ( 𝑋 ‘ 𝑡 ) ≤ 1 ) ) ) |
79 |
74 78
|
ralbid |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ↔ ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑋 ‘ 𝑡 ) ∧ ( 𝑋 ‘ 𝑡 ) ≤ 1 ) ) ) |
80 |
75
|
breq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ‘ 𝑡 ) < 𝐸 ↔ ( 𝑋 ‘ 𝑡 ) < 𝐸 ) ) |
81 |
74 80
|
ralbid |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝐸 ↔ ∀ 𝑡 ∈ 𝑉 ( 𝑋 ‘ 𝑡 ) < 𝐸 ) ) |
82 |
75
|
breq2d |
⊢ ( 𝑥 = 𝑋 → ( ( 1 − 𝐸 ) < ( 𝑥 ‘ 𝑡 ) ↔ ( 1 − 𝐸 ) < ( 𝑋 ‘ 𝑡 ) ) ) |
83 |
74 82
|
ralbid |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝐸 ) < ( 𝑥 ‘ 𝑡 ) ↔ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝐸 ) < ( 𝑋 ‘ 𝑡 ) ) ) |
84 |
79 81 83
|
3anbi123d |
⊢ ( 𝑥 = 𝑋 → ( ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝐸 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝐸 ) < ( 𝑥 ‘ 𝑡 ) ) ↔ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑋 ‘ 𝑡 ) ∧ ( 𝑋 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑋 ‘ 𝑡 ) < 𝐸 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝐸 ) < ( 𝑋 ‘ 𝑡 ) ) ) ) |
85 |
84
|
rspcev |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑋 ‘ 𝑡 ) ∧ ( 𝑋 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑋 ‘ 𝑡 ) < 𝐸 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝐸 ) < ( 𝑋 ‘ 𝑡 ) ) ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝐸 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝐸 ) < ( 𝑥 ‘ 𝑡 ) ) ) |
86 |
30 50 61 71 85
|
syl13anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝐸 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝐸 ) < ( 𝑥 ‘ 𝑡 ) ) ) |