Step |
Hyp |
Ref |
Expression |
1 |
|
ioorrnopnxrlem.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
ioorrnopnxrlem.a |
⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ ℝ* ) |
3 |
|
ioorrnopnxrlem.b |
⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ* ) |
4 |
|
ioorrnopnxrlem.f |
⊢ ( 𝜑 → 𝐹 ∈ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
5 |
|
ioorrnopnxrlem.l |
⊢ 𝐿 = ( 𝑖 ∈ 𝑋 ↦ if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ) |
6 |
|
ioorrnopnxrlem.r |
⊢ 𝑅 = ( 𝑖 ∈ 𝑋 ↦ if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) ) |
7 |
|
ioorrnopnxrlem.v |
⊢ 𝑉 = X 𝑖 ∈ 𝑋 ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) |
8 |
7
|
a1i |
⊢ ( 𝜑 → 𝑉 = X 𝑖 ∈ 𝑋 ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ) |
9 |
|
iftrue |
⊢ ( ( 𝐴 ‘ 𝑖 ) = -∞ → if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) = ( ( 𝐹 ‘ 𝑖 ) − 1 ) ) |
10 |
9
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) = ( ( 𝐹 ‘ 𝑖 ) − 1 ) ) |
11 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → 𝐹 ∈ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → 𝑖 ∈ 𝑋 ) |
13 |
|
fvixp2 |
⊢ ( ( 𝐹 ∈ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑖 ) ∈ ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
14 |
11 12 13
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑖 ) ∈ ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
15 |
14
|
elioored |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℝ ) |
16 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → 1 ∈ ℝ ) |
17 |
15 16
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝑖 ) − 1 ) ∈ ℝ ) |
18 |
17
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( ( 𝐹 ‘ 𝑖 ) − 1 ) ∈ ℝ ) |
19 |
10 18
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ∈ ℝ ) |
20 |
|
iffalse |
⊢ ( ¬ ( 𝐴 ‘ 𝑖 ) = -∞ → if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) = ( 𝐴 ‘ 𝑖 ) ) |
21 |
20
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) = ( 𝐴 ‘ 𝑖 ) ) |
22 |
|
neqne |
⊢ ( ¬ ( 𝐴 ‘ 𝑖 ) = -∞ → ( 𝐴 ‘ 𝑖 ) ≠ -∞ ) |
23 |
22
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐴 ‘ 𝑖 ) ≠ -∞ ) |
24 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑖 ) ∈ ℝ* ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) ≠ -∞ ) → ( 𝐴 ‘ 𝑖 ) ∈ ℝ* ) |
26 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) ≠ -∞ ) → ( 𝐴 ‘ 𝑖 ) ≠ -∞ ) |
27 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
28 |
27
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → +∞ ∈ ℝ* ) |
29 |
15
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℝ* ) |
30 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑖 ) ∈ ℝ* ) |
31 |
|
ioogtlb |
⊢ ( ( ( 𝐴 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐵 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐹 ‘ 𝑖 ) ∈ ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) → ( 𝐴 ‘ 𝑖 ) < ( 𝐹 ‘ 𝑖 ) ) |
32 |
24 30 14 31
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑖 ) < ( 𝐹 ‘ 𝑖 ) ) |
33 |
15
|
ltpnfd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑖 ) < +∞ ) |
34 |
24 29 28 32 33
|
xrlttrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑖 ) < +∞ ) |
35 |
24 28 34
|
xrltned |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑖 ) ≠ +∞ ) |
36 |
35
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) ≠ -∞ ) → ( 𝐴 ‘ 𝑖 ) ≠ +∞ ) |
37 |
25 26 36
|
xrred |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) ≠ -∞ ) → ( 𝐴 ‘ 𝑖 ) ∈ ℝ ) |
38 |
23 37
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐴 ‘ 𝑖 ) ∈ ℝ ) |
39 |
21 38
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ∈ ℝ ) |
40 |
19 39
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ∈ ℝ ) |
41 |
40 5
|
fmptd |
⊢ ( 𝜑 → 𝐿 : 𝑋 ⟶ ℝ ) |
42 |
|
iftrue |
⊢ ( ( 𝐵 ‘ 𝑖 ) = +∞ → if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) = ( ( 𝐹 ‘ 𝑖 ) + 1 ) ) |
43 |
42
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) = ( ( 𝐹 ‘ 𝑖 ) + 1 ) ) |
44 |
15 16
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝑖 ) + 1 ) ∈ ℝ ) |
45 |
44
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( ( 𝐹 ‘ 𝑖 ) + 1 ) ∈ ℝ ) |
46 |
43 45
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) ∈ ℝ ) |
47 |
|
iffalse |
⊢ ( ¬ ( 𝐵 ‘ 𝑖 ) = +∞ → if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) = ( 𝐵 ‘ 𝑖 ) ) |
48 |
47
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) = ( 𝐵 ‘ 𝑖 ) ) |
49 |
|
neqne |
⊢ ( ¬ ( 𝐵 ‘ 𝑖 ) = +∞ → ( 𝐵 ‘ 𝑖 ) ≠ +∞ ) |
50 |
49
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝐵 ‘ 𝑖 ) ≠ +∞ ) |
51 |
30
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) ≠ +∞ ) → ( 𝐵 ‘ 𝑖 ) ∈ ℝ* ) |
52 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
53 |
52
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → -∞ ∈ ℝ* ) |
54 |
15
|
mnfltd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → -∞ < ( 𝐹 ‘ 𝑖 ) ) |
55 |
|
iooltub |
⊢ ( ( ( 𝐴 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐵 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐹 ‘ 𝑖 ) ∈ ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) → ( 𝐹 ‘ 𝑖 ) < ( 𝐵 ‘ 𝑖 ) ) |
56 |
24 30 14 55
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑖 ) < ( 𝐵 ‘ 𝑖 ) ) |
57 |
53 29 30 54 56
|
xrlttrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → -∞ < ( 𝐵 ‘ 𝑖 ) ) |
58 |
53 30 57
|
xrgtned |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑖 ) ≠ -∞ ) |
59 |
58
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) ≠ +∞ ) → ( 𝐵 ‘ 𝑖 ) ≠ -∞ ) |
60 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) ≠ +∞ ) → ( 𝐵 ‘ 𝑖 ) ≠ +∞ ) |
61 |
51 59 60
|
xrred |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) ≠ +∞ ) → ( 𝐵 ‘ 𝑖 ) ∈ ℝ ) |
62 |
50 61
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝐵 ‘ 𝑖 ) ∈ ℝ ) |
63 |
48 62
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) ∈ ℝ ) |
64 |
46 63
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) ∈ ℝ ) |
65 |
64 6
|
fmptd |
⊢ ( 𝜑 → 𝑅 : 𝑋 ⟶ ℝ ) |
66 |
1 41 65
|
ioorrnopn |
⊢ ( 𝜑 → X 𝑖 ∈ 𝑋 ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ∈ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) ) |
67 |
8 66
|
eqeltrd |
⊢ ( 𝜑 → 𝑉 ∈ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) ) |
68 |
4
|
elexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
69 |
|
ixpfn |
⊢ ( 𝐹 ∈ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) → 𝐹 Fn 𝑋 ) |
70 |
4 69
|
syl |
⊢ ( 𝜑 → 𝐹 Fn 𝑋 ) |
71 |
41
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐿 ‘ 𝑖 ) ∈ ℝ ) |
72 |
71
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐿 ‘ 𝑖 ) ∈ ℝ* ) |
73 |
65
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝑅 ‘ 𝑖 ) ∈ ℝ ) |
74 |
73
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝑅 ‘ 𝑖 ) ∈ ℝ* ) |
75 |
5
|
a1i |
⊢ ( 𝜑 → 𝐿 = ( 𝑖 ∈ 𝑋 ↦ if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ) ) |
76 |
40
|
elexd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ∈ V ) |
77 |
75 76
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐿 ‘ 𝑖 ) = if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ) |
78 |
77
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐿 ‘ 𝑖 ) = if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ) |
79 |
78 10
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐿 ‘ 𝑖 ) = ( ( 𝐹 ‘ 𝑖 ) − 1 ) ) |
80 |
15
|
ltm1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝑖 ) − 1 ) < ( 𝐹 ‘ 𝑖 ) ) |
81 |
80
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( ( 𝐹 ‘ 𝑖 ) − 1 ) < ( 𝐹 ‘ 𝑖 ) ) |
82 |
79 81
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐿 ‘ 𝑖 ) < ( 𝐹 ‘ 𝑖 ) ) |
83 |
77
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐿 ‘ 𝑖 ) = if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ) |
84 |
83 21
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐿 ‘ 𝑖 ) = ( 𝐴 ‘ 𝑖 ) ) |
85 |
32
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐴 ‘ 𝑖 ) < ( 𝐹 ‘ 𝑖 ) ) |
86 |
84 85
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐿 ‘ 𝑖 ) < ( 𝐹 ‘ 𝑖 ) ) |
87 |
82 86
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐿 ‘ 𝑖 ) < ( 𝐹 ‘ 𝑖 ) ) |
88 |
15
|
ltp1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑖 ) < ( ( 𝐹 ‘ 𝑖 ) + 1 ) ) |
89 |
88
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝐹 ‘ 𝑖 ) < ( ( 𝐹 ‘ 𝑖 ) + 1 ) ) |
90 |
6
|
a1i |
⊢ ( 𝜑 → 𝑅 = ( 𝑖 ∈ 𝑋 ↦ if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) ) ) |
91 |
64
|
elexd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) ∈ V ) |
92 |
90 91
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝑅 ‘ 𝑖 ) = if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) ) |
93 |
92
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝑅 ‘ 𝑖 ) = if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) ) |
94 |
93 43
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝑅 ‘ 𝑖 ) = ( ( 𝐹 ‘ 𝑖 ) + 1 ) ) |
95 |
94
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( ( 𝐹 ‘ 𝑖 ) + 1 ) = ( 𝑅 ‘ 𝑖 ) ) |
96 |
89 95
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝐹 ‘ 𝑖 ) < ( 𝑅 ‘ 𝑖 ) ) |
97 |
56
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝐹 ‘ 𝑖 ) < ( 𝐵 ‘ 𝑖 ) ) |
98 |
92
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝑅 ‘ 𝑖 ) = if ( ( 𝐵 ‘ 𝑖 ) = +∞ , ( ( 𝐹 ‘ 𝑖 ) + 1 ) , ( 𝐵 ‘ 𝑖 ) ) ) |
99 |
98 48
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝑅 ‘ 𝑖 ) = ( 𝐵 ‘ 𝑖 ) ) |
100 |
99
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝐵 ‘ 𝑖 ) = ( 𝑅 ‘ 𝑖 ) ) |
101 |
97 100
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝐹 ‘ 𝑖 ) < ( 𝑅 ‘ 𝑖 ) ) |
102 |
96 101
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑖 ) < ( 𝑅 ‘ 𝑖 ) ) |
103 |
72 74 15 87 102
|
eliood |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑖 ) ∈ ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ) |
104 |
103
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝑋 ( 𝐹 ‘ 𝑖 ) ∈ ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ) |
105 |
68 70 104
|
3jca |
⊢ ( 𝜑 → ( 𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀ 𝑖 ∈ 𝑋 ( 𝐹 ‘ 𝑖 ) ∈ ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ) ) |
106 |
|
elixp2 |
⊢ ( 𝐹 ∈ X 𝑖 ∈ 𝑋 ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ↔ ( 𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀ 𝑖 ∈ 𝑋 ( 𝐹 ‘ 𝑖 ) ∈ ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ) ) |
107 |
105 106
|
sylibr |
⊢ ( 𝜑 → 𝐹 ∈ X 𝑖 ∈ 𝑋 ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ) |
108 |
107 7
|
eleqtrrdi |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
109 |
24
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐴 ‘ 𝑖 ) ∈ ℝ* ) |
110 |
72
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐿 ‘ 𝑖 ) ∈ ℝ* ) |
111 |
19
|
mnfltd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → -∞ < if ( ( 𝐴 ‘ 𝑖 ) = -∞ , ( ( 𝐹 ‘ 𝑖 ) − 1 ) , ( 𝐴 ‘ 𝑖 ) ) ) |
112 |
111 10
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → -∞ < ( ( 𝐹 ‘ 𝑖 ) − 1 ) ) |
113 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐴 ‘ 𝑖 ) = -∞ ) |
114 |
113 79
|
breq12d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( ( 𝐴 ‘ 𝑖 ) < ( 𝐿 ‘ 𝑖 ) ↔ -∞ < ( ( 𝐹 ‘ 𝑖 ) − 1 ) ) ) |
115 |
112 114
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐴 ‘ 𝑖 ) < ( 𝐿 ‘ 𝑖 ) ) |
116 |
109 110 115
|
xrltled |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐴 ‘ 𝑖 ) ≤ ( 𝐿 ‘ 𝑖 ) ) |
117 |
84
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐴 ‘ 𝑖 ) = ( 𝐿 ‘ 𝑖 ) ) |
118 |
38 117
|
eqled |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐴 ‘ 𝑖 ) = -∞ ) → ( 𝐴 ‘ 𝑖 ) ≤ ( 𝐿 ‘ 𝑖 ) ) |
119 |
116 118
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑖 ) ≤ ( 𝐿 ‘ 𝑖 ) ) |
120 |
74
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝑅 ‘ 𝑖 ) ∈ ℝ* ) |
121 |
30
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝐵 ‘ 𝑖 ) ∈ ℝ* ) |
122 |
45
|
ltpnfd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( ( 𝐹 ‘ 𝑖 ) + 1 ) < +∞ ) |
123 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝐵 ‘ 𝑖 ) = +∞ ) |
124 |
94 123
|
breq12d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( ( 𝑅 ‘ 𝑖 ) < ( 𝐵 ‘ 𝑖 ) ↔ ( ( 𝐹 ‘ 𝑖 ) + 1 ) < +∞ ) ) |
125 |
122 124
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝑅 ‘ 𝑖 ) < ( 𝐵 ‘ 𝑖 ) ) |
126 |
120 121 125
|
xrltled |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝑅 ‘ 𝑖 ) ≤ ( 𝐵 ‘ 𝑖 ) ) |
127 |
73
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝑅 ‘ 𝑖 ) ∈ ℝ ) |
128 |
127 99
|
eqled |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) ∧ ¬ ( 𝐵 ‘ 𝑖 ) = +∞ ) → ( 𝑅 ‘ 𝑖 ) ≤ ( 𝐵 ‘ 𝑖 ) ) |
129 |
126 128
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( 𝑅 ‘ 𝑖 ) ≤ ( 𝐵 ‘ 𝑖 ) ) |
130 |
|
ioossioo |
⊢ ( ( ( ( 𝐴 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐵 ‘ 𝑖 ) ∈ ℝ* ) ∧ ( ( 𝐴 ‘ 𝑖 ) ≤ ( 𝐿 ‘ 𝑖 ) ∧ ( 𝑅 ‘ 𝑖 ) ≤ ( 𝐵 ‘ 𝑖 ) ) ) → ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ⊆ ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
131 |
24 30 119 129 130
|
syl22anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ⊆ ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
132 |
131
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝑋 ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ⊆ ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
133 |
|
ss2ixp |
⊢ ( ∀ 𝑖 ∈ 𝑋 ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ⊆ ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) → X 𝑖 ∈ 𝑋 ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
134 |
132 133
|
syl |
⊢ ( 𝜑 → X 𝑖 ∈ 𝑋 ( ( 𝐿 ‘ 𝑖 ) (,) ( 𝑅 ‘ 𝑖 ) ) ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
135 |
8 134
|
eqsstrd |
⊢ ( 𝜑 → 𝑉 ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) |
136 |
108 135
|
jca |
⊢ ( 𝜑 → ( 𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) ) |
137 |
|
eleq2 |
⊢ ( 𝑣 = 𝑉 → ( 𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉 ) ) |
138 |
|
sseq1 |
⊢ ( 𝑣 = 𝑉 → ( 𝑣 ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ↔ 𝑉 ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) ) |
139 |
137 138
|
anbi12d |
⊢ ( 𝑣 = 𝑉 → ( ( 𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) ↔ ( 𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) ) ) |
140 |
139
|
rspcev |
⊢ ( ( 𝑉 ∈ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) ) → ∃ 𝑣 ∈ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) ( 𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) ) |
141 |
67 136 140
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑣 ∈ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) ( 𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) (,) ( 𝐵 ‘ 𝑖 ) ) ) ) |