Step |
Hyp |
Ref |
Expression |
1 |
|
stoweidlem52.1 |
⊢ Ⅎ 𝑡 𝑈 |
2 |
|
stoweidlem52.2 |
⊢ Ⅎ 𝑡 𝜑 |
3 |
|
stoweidlem52.3 |
⊢ Ⅎ 𝑡 𝑃 |
4 |
|
stoweidlem52.4 |
⊢ 𝐾 = ( topGen ‘ ran (,) ) |
5 |
|
stoweidlem52.5 |
⊢ 𝑉 = { 𝑡 ∈ 𝑇 ∣ ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) } |
6 |
|
stoweidlem52.7 |
⊢ 𝑇 = ∪ 𝐽 |
7 |
|
stoweidlem52.8 |
⊢ 𝐶 = ( 𝐽 Cn 𝐾 ) |
8 |
|
stoweidlem52.9 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 ) |
9 |
|
stoweidlem52.10 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
10 |
|
stoweidlem52.11 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
11 |
|
stoweidlem52.12 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( 𝑡 ∈ 𝑇 ↦ 𝑎 ) ∈ 𝐴 ) |
12 |
|
stoweidlem52.13 |
⊢ ( 𝜑 → 𝐷 ∈ ℝ+ ) |
13 |
|
stoweidlem52.14 |
⊢ ( 𝜑 → 𝐷 < 1 ) |
14 |
|
stoweidlem52.15 |
⊢ ( 𝜑 → 𝑈 ∈ 𝐽 ) |
15 |
|
stoweidlem52.16 |
⊢ ( 𝜑 → 𝑍 ∈ 𝑈 ) |
16 |
|
stoweidlem52.17 |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
17 |
|
stoweidlem52.18 |
⊢ ( 𝜑 → ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑃 ‘ 𝑡 ) ∧ ( 𝑃 ‘ 𝑡 ) ≤ 1 ) ) |
18 |
|
stoweidlem52.19 |
⊢ ( 𝜑 → ( 𝑃 ‘ 𝑍 ) = 0 ) |
19 |
|
stoweidlem52.20 |
⊢ ( 𝜑 → ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) 𝐷 ≤ ( 𝑃 ‘ 𝑡 ) ) |
20 |
|
nfcv |
⊢ Ⅎ 𝑡 ( 𝐷 / 2 ) |
21 |
12
|
rpred |
⊢ ( 𝜑 → 𝐷 ∈ ℝ ) |
22 |
21
|
rehalfcld |
⊢ ( 𝜑 → ( 𝐷 / 2 ) ∈ ℝ ) |
23 |
22
|
rexrd |
⊢ ( 𝜑 → ( 𝐷 / 2 ) ∈ ℝ* ) |
24 |
8 7
|
sseqtrdi |
⊢ ( 𝜑 → 𝐴 ⊆ ( 𝐽 Cn 𝐾 ) ) |
25 |
24 16
|
sseldd |
⊢ ( 𝜑 → 𝑃 ∈ ( 𝐽 Cn 𝐾 ) ) |
26 |
20 3 2 4 6 5 23 25
|
rfcnpre2 |
⊢ ( 𝜑 → 𝑉 ∈ 𝐽 ) |
27 |
|
elssuni |
⊢ ( 𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽 ) |
28 |
14 27
|
syl |
⊢ ( 𝜑 → 𝑈 ⊆ ∪ 𝐽 ) |
29 |
28 6
|
sseqtrrdi |
⊢ ( 𝜑 → 𝑈 ⊆ 𝑇 ) |
30 |
29 15
|
sseldd |
⊢ ( 𝜑 → 𝑍 ∈ 𝑇 ) |
31 |
|
2re |
⊢ 2 ∈ ℝ |
32 |
31
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℝ ) |
33 |
12
|
rpgt0d |
⊢ ( 𝜑 → 0 < 𝐷 ) |
34 |
|
2pos |
⊢ 0 < 2 |
35 |
34
|
a1i |
⊢ ( 𝜑 → 0 < 2 ) |
36 |
21 32 33 35
|
divgt0d |
⊢ ( 𝜑 → 0 < ( 𝐷 / 2 ) ) |
37 |
18 36
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝑃 ‘ 𝑍 ) < ( 𝐷 / 2 ) ) |
38 |
|
nfcv |
⊢ Ⅎ 𝑡 𝑍 |
39 |
|
nfcv |
⊢ Ⅎ 𝑡 𝑇 |
40 |
3 38
|
nffv |
⊢ Ⅎ 𝑡 ( 𝑃 ‘ 𝑍 ) |
41 |
|
nfcv |
⊢ Ⅎ 𝑡 < |
42 |
40 41 20
|
nfbr |
⊢ Ⅎ 𝑡 ( 𝑃 ‘ 𝑍 ) < ( 𝐷 / 2 ) |
43 |
|
fveq2 |
⊢ ( 𝑡 = 𝑍 → ( 𝑃 ‘ 𝑡 ) = ( 𝑃 ‘ 𝑍 ) ) |
44 |
43
|
breq1d |
⊢ ( 𝑡 = 𝑍 → ( ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) ↔ ( 𝑃 ‘ 𝑍 ) < ( 𝐷 / 2 ) ) ) |
45 |
38 39 42 44
|
elrabf |
⊢ ( 𝑍 ∈ { 𝑡 ∈ 𝑇 ∣ ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) } ↔ ( 𝑍 ∈ 𝑇 ∧ ( 𝑃 ‘ 𝑍 ) < ( 𝐷 / 2 ) ) ) |
46 |
30 37 45
|
sylanbrc |
⊢ ( 𝜑 → 𝑍 ∈ { 𝑡 ∈ 𝑇 ∣ ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) } ) |
47 |
46 5
|
eleqtrrdi |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
48 |
|
nfrab1 |
⊢ Ⅎ 𝑡 { 𝑡 ∈ 𝑇 ∣ ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) } |
49 |
5 48
|
nfcxfr |
⊢ Ⅎ 𝑡 𝑉 |
50 |
8 16
|
sseldd |
⊢ ( 𝜑 → 𝑃 ∈ 𝐶 ) |
51 |
4 6 7 50
|
fcnre |
⊢ ( 𝜑 → 𝑃 : 𝑇 ⟶ ℝ ) |
52 |
51
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → 𝑃 : 𝑇 ⟶ ℝ ) |
53 |
5
|
rabeq2i |
⊢ ( 𝑡 ∈ 𝑉 ↔ ( 𝑡 ∈ 𝑇 ∧ ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) ) ) |
54 |
53
|
biimpi |
⊢ ( 𝑡 ∈ 𝑉 → ( 𝑡 ∈ 𝑇 ∧ ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) ) ) |
55 |
54
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 𝑡 ∈ 𝑇 ∧ ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) ) ) |
56 |
55
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → 𝑡 ∈ 𝑇 ) |
57 |
52 56
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 𝑃 ‘ 𝑡 ) ∈ ℝ ) |
58 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 𝐷 / 2 ) ∈ ℝ ) |
59 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → 𝐷 ∈ ℝ ) |
60 |
55
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) ) |
61 |
|
halfpos |
⊢ ( 𝐷 ∈ ℝ → ( 0 < 𝐷 ↔ ( 𝐷 / 2 ) < 𝐷 ) ) |
62 |
21 61
|
syl |
⊢ ( 𝜑 → ( 0 < 𝐷 ↔ ( 𝐷 / 2 ) < 𝐷 ) ) |
63 |
33 62
|
mpbid |
⊢ ( 𝜑 → ( 𝐷 / 2 ) < 𝐷 ) |
64 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 𝐷 / 2 ) < 𝐷 ) |
65 |
57 58 59 60 64
|
lttrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → ( 𝑃 ‘ 𝑡 ) < 𝐷 ) |
66 |
65
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) ∧ ¬ 𝑡 ∈ 𝑈 ) → ( 𝑃 ‘ 𝑡 ) < 𝐷 ) |
67 |
21
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) ∧ ¬ 𝑡 ∈ 𝑈 ) → 𝐷 ∈ ℝ ) |
68 |
57
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) ∧ ¬ 𝑡 ∈ 𝑈 ) → ( 𝑃 ‘ 𝑡 ) ∈ ℝ ) |
69 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) ∧ ¬ 𝑡 ∈ 𝑈 ) → ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) 𝐷 ≤ ( 𝑃 ‘ 𝑡 ) ) |
70 |
56
|
anim1i |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) ∧ ¬ 𝑡 ∈ 𝑈 ) → ( 𝑡 ∈ 𝑇 ∧ ¬ 𝑡 ∈ 𝑈 ) ) |
71 |
|
eldif |
⊢ ( 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ↔ ( 𝑡 ∈ 𝑇 ∧ ¬ 𝑡 ∈ 𝑈 ) ) |
72 |
70 71
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) ∧ ¬ 𝑡 ∈ 𝑈 ) → 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ) |
73 |
|
rsp |
⊢ ( ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) 𝐷 ≤ ( 𝑃 ‘ 𝑡 ) → ( 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) → 𝐷 ≤ ( 𝑃 ‘ 𝑡 ) ) ) |
74 |
69 72 73
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) ∧ ¬ 𝑡 ∈ 𝑈 ) → 𝐷 ≤ ( 𝑃 ‘ 𝑡 ) ) |
75 |
67 68 74
|
lensymd |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) ∧ ¬ 𝑡 ∈ 𝑈 ) → ¬ ( 𝑃 ‘ 𝑡 ) < 𝐷 ) |
76 |
66 75
|
condan |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑉 ) → 𝑡 ∈ 𝑈 ) |
77 |
76
|
ex |
⊢ ( 𝜑 → ( 𝑡 ∈ 𝑉 → 𝑡 ∈ 𝑈 ) ) |
78 |
2 49 1 77
|
ssrd |
⊢ ( 𝜑 → 𝑉 ⊆ 𝑈 ) |
79 |
|
nfv |
⊢ Ⅎ 𝑡 𝑒 ∈ ℝ+ |
80 |
2 79
|
nfan |
⊢ Ⅎ 𝑡 ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) |
81 |
|
nfv |
⊢ Ⅎ 𝑡 𝑦 ∈ 𝐴 |
82 |
80 81
|
nfan |
⊢ Ⅎ 𝑡 ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) |
83 |
|
nfra1 |
⊢ Ⅎ 𝑡 ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) |
84 |
|
nfra1 |
⊢ Ⅎ 𝑡 ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) |
85 |
|
nfra1 |
⊢ Ⅎ 𝑡 ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 |
86 |
83 84 85
|
nf3an |
⊢ Ⅎ 𝑡 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) |
87 |
82 86
|
nfan |
⊢ Ⅎ 𝑡 ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) |
88 |
|
eqid |
⊢ ( 𝑡 ∈ 𝑇 ↦ ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) = ( 𝑡 ∈ 𝑇 ↦ ( 1 − ( 𝑦 ‘ 𝑡 ) ) ) |
89 |
|
eqid |
⊢ ( 𝑡 ∈ 𝑇 ↦ 1 ) = ( 𝑡 ∈ 𝑇 ↦ 1 ) |
90 |
|
ssrab2 |
⊢ { 𝑡 ∈ 𝑇 ∣ ( 𝑃 ‘ 𝑡 ) < ( 𝐷 / 2 ) } ⊆ 𝑇 |
91 |
5 90
|
eqsstri |
⊢ 𝑉 ⊆ 𝑇 |
92 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) → 𝑦 ∈ 𝐴 ) |
93 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) → 𝜑 ) |
94 |
8
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ 𝐶 ) |
95 |
4 6 7 94
|
fcnre |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝑦 : 𝑇 ⟶ ℝ ) |
96 |
93 92 95
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) → 𝑦 : 𝑇 ⟶ ℝ ) |
97 |
8
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝑓 ∈ 𝐶 ) |
98 |
4 6 7 97
|
fcnre |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝑓 : 𝑇 ⟶ ℝ ) |
99 |
93 98
|
sylan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) ∧ 𝑓 ∈ 𝐴 ) → 𝑓 : 𝑇 ⟶ ℝ ) |
100 |
93 9
|
syl3an1 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
101 |
93 10
|
syl3an1 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
102 |
93 11
|
sylan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) ∧ 𝑎 ∈ ℝ ) → ( 𝑡 ∈ 𝑇 ↦ 𝑎 ) ∈ 𝐴 ) |
103 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) → 𝑒 ∈ ℝ+ ) |
104 |
|
simpr1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) → ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ) |
105 |
|
simpr2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) → ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ) |
106 |
|
simpr3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) → ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) |
107 |
87 88 89 91 92 96 99 100 101 102 103 104 105 106
|
stoweidlem41 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝐴 ) ∧ ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) |
108 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝐷 ∈ ℝ+ ) |
109 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝐷 < 1 ) |
110 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝑃 ∈ 𝐴 ) |
111 |
51
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝑃 : 𝑇 ⟶ ℝ ) |
112 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑃 ‘ 𝑡 ) ∧ ( 𝑃 ‘ 𝑡 ) ≤ 1 ) ) |
113 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) 𝐷 ≤ ( 𝑃 ‘ 𝑡 ) ) |
114 |
98
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑓 ∈ 𝐴 ) → 𝑓 : 𝑇 ⟶ ℝ ) |
115 |
9
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) + ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
116 |
10
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( 𝑡 ∈ 𝑇 ↦ ( ( 𝑓 ‘ 𝑡 ) · ( 𝑔 ‘ 𝑡 ) ) ) ∈ 𝐴 ) |
117 |
11
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑎 ∈ ℝ ) → ( 𝑡 ∈ 𝑇 ↦ 𝑎 ) ∈ 𝐴 ) |
118 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝑒 ∈ ℝ+ ) |
119 |
3 80 5 108 109 110 111 112 113 114 115 116 117 118
|
stoweidlem49 |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑦 ‘ 𝑡 ) ∧ ( 𝑦 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 1 − 𝑒 ) < ( 𝑦 ‘ 𝑡 ) ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 𝑦 ‘ 𝑡 ) < 𝑒 ) ) |
120 |
107 119
|
r19.29a |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) |
121 |
120
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑒 ∈ ℝ+ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) |
122 |
47 78 121
|
jca31 |
⊢ ( 𝜑 → ( ( 𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈 ) ∧ ∀ 𝑒 ∈ ℝ+ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) ) |
123 |
|
eleq2 |
⊢ ( 𝑣 = 𝑉 → ( 𝑍 ∈ 𝑣 ↔ 𝑍 ∈ 𝑉 ) ) |
124 |
|
sseq1 |
⊢ ( 𝑣 = 𝑉 → ( 𝑣 ⊆ 𝑈 ↔ 𝑉 ⊆ 𝑈 ) ) |
125 |
123 124
|
anbi12d |
⊢ ( 𝑣 = 𝑉 → ( ( 𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈 ) ↔ ( 𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈 ) ) ) |
126 |
|
nfcv |
⊢ Ⅎ 𝑡 𝑣 |
127 |
126 49
|
raleqf |
⊢ ( 𝑣 = 𝑉 → ( ∀ 𝑡 ∈ 𝑣 ( 𝑥 ‘ 𝑡 ) < 𝑒 ↔ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝑒 ) ) |
128 |
127
|
3anbi2d |
⊢ ( 𝑣 = 𝑉 → ( ( ∀ 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 ) ) ) ) |
129 |
128
|
rexbidv |
⊢ ( 𝑣 = 𝑉 → ( ∃ 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 ) ) ) ) |
130 |
129
|
ralbidv |
⊢ ( 𝑣 = 𝑉 → ( ∀ 𝑒 ∈ ℝ+ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑣 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ↔ ∀ 𝑒 ∈ ℝ+ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) ) |
131 |
125 130
|
anbi12d |
⊢ ( 𝑣 = 𝑉 → ( ( ( 𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈 ) ∧ ∀ 𝑒 ∈ ℝ+ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑣 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) ↔ ( ( 𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈 ) ∧ ∀ 𝑒 ∈ ℝ+ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) ) ) |
132 |
131
|
rspcev |
⊢ ( ( 𝑉 ∈ 𝐽 ∧ ( ( 𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈 ) ∧ ∀ 𝑒 ∈ ℝ+ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑉 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) ) → ∃ 𝑣 ∈ 𝐽 ( ( 𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈 ) ∧ ∀ 𝑒 ∈ ℝ+ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑣 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) ) |
133 |
26 122 132
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑣 ∈ 𝐽 ( ( 𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈 ) ∧ ∀ 𝑒 ∈ ℝ+ ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑡 ∈ 𝑇 ( 0 ≤ ( 𝑥 ‘ 𝑡 ) ∧ ( 𝑥 ‘ 𝑡 ) ≤ 1 ) ∧ ∀ 𝑡 ∈ 𝑣 ( 𝑥 ‘ 𝑡 ) < 𝑒 ∧ ∀ 𝑡 ∈ ( 𝑇 ∖ 𝑈 ) ( 1 − 𝑒 ) < ( 𝑥 ‘ 𝑡 ) ) ) ) |