Step |
Hyp |
Ref |
Expression |
1 |
|
icccncfext.1 |
⊢ Ⅎ 𝑥 𝐹 |
2 |
|
icccncfext.2 |
⊢ 𝐽 = ( topGen ‘ ran (,) ) |
3 |
|
icccncfext.3 |
⊢ 𝑌 = ∪ 𝐾 |
4 |
|
icccncfext.4 |
⊢ 𝐺 = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
5 |
|
icccncfext.5 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
6 |
|
icccncfext.6 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
7 |
|
icccncfext.7 |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
8 |
|
icccncfext.8 |
⊢ ( 𝜑 → 𝐾 ∈ Top ) |
9 |
|
icccncfext.9 |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ) |
10 |
|
retopon |
⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) |
11 |
2 10
|
eqeltri |
⊢ 𝐽 ∈ ( TopOn ‘ ℝ ) |
12 |
5 6
|
iccssred |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
13 |
|
resttopon |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) → ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ∈ ( TopOn ‘ ( 𝐴 [,] 𝐵 ) ) ) |
14 |
11 12 13
|
sylancr |
⊢ ( 𝜑 → ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ∈ ( TopOn ‘ ( 𝐴 [,] 𝐵 ) ) ) |
15 |
8 3
|
jctir |
⊢ ( 𝜑 → ( 𝐾 ∈ Top ∧ 𝑌 = ∪ 𝐾 ) ) |
16 |
|
istopon |
⊢ ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ↔ ( 𝐾 ∈ Top ∧ 𝑌 = ∪ 𝐾 ) ) |
17 |
15 16
|
sylibr |
⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) |
18 |
|
cnf2 |
⊢ ( ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ∈ ( TopOn ‘ ( 𝐴 [,] 𝐵 ) ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ 𝑌 ) |
19 |
14 17 9 18
|
syl3anc |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ 𝑌 ) |
20 |
19
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
21 |
|
dffn3 |
⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) ↔ 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ran 𝐹 ) |
22 |
20 21
|
sylib |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ran 𝐹 ) |
23 |
22
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ran 𝐹 ) |
24 |
|
fnfun |
⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) → Fun 𝐹 ) |
25 |
20 24
|
syl |
⊢ ( 𝜑 → Fun 𝐹 ) |
26 |
5
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
27 |
6
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
28 |
|
lbicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
29 |
26 27 7 28
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
30 |
20
|
fndmd |
⊢ ( 𝜑 → dom 𝐹 = ( 𝐴 [,] 𝐵 ) ) |
31 |
30
|
eqcomd |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) = dom 𝐹 ) |
32 |
29 31
|
eleqtrd |
⊢ ( 𝜑 → 𝐴 ∈ dom 𝐹 ) |
33 |
|
fvelrn |
⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝐴 ) ∈ ran 𝐹 ) |
34 |
25 32 33
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ ran 𝐹 ) |
35 |
|
ubicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
36 |
26 27 7 35
|
syl3anc |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
37 |
36 31
|
eleqtrd |
⊢ ( 𝜑 → 𝐵 ∈ dom 𝐹 ) |
38 |
|
fvelrn |
⊢ ( ( Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 ) |
39 |
25 37 38
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 ) |
40 |
34 39
|
ifcld |
⊢ ( 𝜑 → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ ran 𝐹 ) |
41 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ ran 𝐹 ) |
42 |
23 41
|
ifclda |
⊢ ( 𝜑 → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ ran 𝐹 ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ ran 𝐹 ) |
44 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 ∈ ( 𝐴 [,] 𝐵 ) |
45 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑥 ) |
46 |
|
nfcv |
⊢ Ⅎ 𝑦 if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) |
47 |
44 45 46
|
nfif |
⊢ Ⅎ 𝑦 if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
48 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 ∈ ( 𝐴 [,] 𝐵 ) |
49 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑦 |
50 |
1 49
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 ) |
51 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 < 𝐴 |
52 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐴 |
53 |
1 52
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝐴 ) |
54 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐵 |
55 |
1 54
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝐵 ) |
56 |
51 53 55
|
nfif |
⊢ Ⅎ 𝑥 if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) |
57 |
48 50 56
|
nfif |
⊢ Ⅎ 𝑥 if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
58 |
|
eleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
59 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) |
60 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 < 𝐴 ↔ 𝑦 < 𝐴 ) ) |
61 |
60
|
ifbid |
⊢ ( 𝑥 = 𝑦 → if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
62 |
58 59 61
|
ifbieq12d |
⊢ ( 𝑥 = 𝑦 → if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
63 |
47 57 62
|
cbvmpt |
⊢ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
64 |
4 63
|
eqtri |
⊢ 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
65 |
43 64
|
fmptd |
⊢ ( 𝜑 → 𝐺 : ℝ ⟶ ran 𝐹 ) |
66 |
65
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐺 : ℝ ⟶ ran 𝐹 ) |
67 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → 𝜑 ) |
68 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) |
69 |
67 68
|
jca |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ) |
70 |
|
ssidd |
⊢ ( 𝜑 → ran 𝐹 ⊆ ran 𝐹 ) |
71 |
19
|
frnd |
⊢ ( 𝜑 → ran 𝐹 ⊆ 𝑌 ) |
72 |
|
cnrest2 |
⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝑌 ) → ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ↔ 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ) ) |
73 |
17 70 71 72
|
syl3anc |
⊢ ( 𝜑 → ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ↔ 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ) ) |
74 |
9 73
|
mpbid |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ) |
75 |
74
|
anim1i |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) → ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ) |
76 |
|
cnima |
⊢ ( ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) → ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ) |
77 |
69 75 76
|
3syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ) |
78 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
79 |
2 78
|
eqeltri |
⊢ 𝐽 ∈ Top |
80 |
79
|
a1i |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
81 |
|
reex |
⊢ ℝ ∈ V |
82 |
81
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
83 |
82 12
|
ssexd |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ∈ V ) |
84 |
80 83
|
jca |
⊢ ( 𝜑 → ( 𝐽 ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) ) |
85 |
67 84
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( 𝐽 ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) ) |
86 |
|
elrest |
⊢ ( ( 𝐽 ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) → ( ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ↔ ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) |
87 |
85 86
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ↔ ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) |
88 |
77 87
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
89 |
67
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝜑 ) |
90 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → 𝑦 ∈ ℝ ) |
91 |
90
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑦 ∈ ℝ ) |
92 |
|
simp1r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
93 |
89 91 92
|
jca31 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ) |
94 |
|
simpll2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 ) |
95 |
|
iooretop |
⊢ ( -∞ (,) 𝐴 ) ∈ ( topGen ‘ ran (,) ) |
96 |
95 2
|
eleqtrri |
⊢ ( -∞ (,) 𝐴 ) ∈ 𝐽 |
97 |
|
iooretop |
⊢ ( 𝐵 (,) +∞ ) ∈ ( topGen ‘ ran (,) ) |
98 |
97 2
|
eleqtrri |
⊢ ( 𝐵 (,) +∞ ) ∈ 𝐽 |
99 |
|
unopn |
⊢ ( ( 𝐽 ∈ Top ∧ ( -∞ (,) 𝐴 ) ∈ 𝐽 ∧ ( 𝐵 (,) +∞ ) ∈ 𝐽 ) → ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ) |
100 |
79 96 98 99
|
mp3an |
⊢ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 |
101 |
|
unopn |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ) → ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 ) |
102 |
79 100 101
|
mp3an13 |
⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 ) |
103 |
94 102
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 ) |
104 |
|
simpl1l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) ) |
105 |
104
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) ) |
106 |
|
simpl1r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
107 |
106
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
108 |
|
simpll3 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
109 |
|
difreicc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) = ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) |
110 |
5 6 109
|
syl2anc |
⊢ ( 𝜑 → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) = ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) |
111 |
110
|
eqcomd |
⊢ ( 𝜑 → ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) = ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ) |
112 |
111
|
eleq2d |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ) ) |
113 |
112
|
notbid |
⊢ ( 𝜑 → ( ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ¬ 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ) ) |
114 |
113
|
biimpa |
⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ¬ 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ) |
115 |
|
eldif |
⊢ ( 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
116 |
114 115
|
sylnib |
⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ¬ ( 𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
117 |
|
imnan |
⊢ ( ( 𝑦 ∈ ℝ → ¬ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ¬ ( 𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
118 |
116 117
|
sylibr |
⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ ℝ → ¬ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
119 |
118
|
imp |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑦 ∈ ℝ ) → ¬ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
120 |
119
|
notnotrd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
121 |
120
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
122 |
121
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
123 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝜑 ) |
124 |
12
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ℝ ) |
125 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ 𝑌 ) |
126 |
125
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑌 ) |
127 |
19 29
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 ) |
128 |
127
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 ) |
129 |
19 36
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 ) |
130 |
129
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 ) |
131 |
128 130
|
ifclda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑌 ) |
132 |
126 131
|
ifclda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 ) |
133 |
64
|
fvmpt2 |
⊢ ( ( 𝑦 ∈ ℝ ∧ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
134 |
124 132 133
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
135 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
136 |
135
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝑦 ) ) |
137 |
134 136
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
138 |
137
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
139 |
123 122 138
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
140 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
141 |
139 140
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) |
142 |
123 20
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
143 |
|
elpreima |
⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) ) |
144 |
142 143
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) ) |
145 |
122 141 144
|
mpbir2and |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
146 |
145
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
147 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
148 |
146 147
|
eleqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
149 |
|
elin |
⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
150 |
148 149
|
sylib |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
151 |
150
|
simpld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ 𝑤 ) |
152 |
151
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ 𝑤 ) ) |
153 |
152
|
orrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∨ 𝑦 ∈ 𝑤 ) ) |
154 |
153
|
orcomd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑦 ∈ 𝑤 ∨ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) |
155 |
|
elun |
⊢ ( 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ↔ ( 𝑦 ∈ 𝑤 ∨ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) |
156 |
154 155
|
sylibr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) |
157 |
105 107 108 156
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) |
158 |
|
imaundi |
⊢ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) = ( ( 𝐺 “ 𝑤 ) ∪ ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) |
159 |
105
|
simpld |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 ) |
160 |
|
toponss |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ 𝑤 ∈ 𝐽 ) → 𝑤 ⊆ ℝ ) |
161 |
11 94 160
|
sylancr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ⊆ ℝ ) |
162 |
159 161 108
|
jca31 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) |
163 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
164 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
165 |
4
|
funmpt2 |
⊢ Fun 𝐺 |
166 |
165
|
a1i |
⊢ ( 𝜑 → Fun 𝐺 ) |
167 |
166
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → Fun 𝐺 ) |
168 |
|
fvelima |
⊢ ( ( Fun 𝐺 ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → ∃ 𝑧 ∈ 𝑤 ( 𝐺 ‘ 𝑧 ) = 𝑦 ) |
169 |
167 168
|
sylancom |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → ∃ 𝑧 ∈ 𝑤 ( 𝐺 ‘ 𝑧 ) = 𝑦 ) |
170 |
|
eqcom |
⊢ ( ( 𝐺 ‘ 𝑧 ) = 𝑦 ↔ 𝑦 = ( 𝐺 ‘ 𝑧 ) ) |
171 |
170
|
biimpi |
⊢ ( ( 𝐺 ‘ 𝑧 ) = 𝑦 → 𝑦 = ( 𝐺 ‘ 𝑧 ) ) |
172 |
171
|
3ad2ant3 |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 = ( 𝐺 ‘ 𝑧 ) ) |
173 |
|
simp1ll |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ) |
174 |
|
simp1lr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
175 |
|
simp2 |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑧 ∈ 𝑤 ) |
176 |
|
simp-5l |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ) |
177 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
178 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ 𝑤 ) |
179 |
176 177 178
|
jca31 |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ) |
180 |
|
eleq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
181 |
180
|
anbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ) ) |
182 |
|
fveq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) ) |
183 |
|
fveq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) |
184 |
182 183
|
eqeq12d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) ) |
185 |
181 184
|
imbi12d |
⊢ ( 𝑦 = 𝑧 → ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) ) ) |
186 |
185 137
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
187 |
186
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
188 |
187
|
adantl3r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
189 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝜑 ) |
190 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑤 ⊆ ℝ ) |
191 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ 𝑤 ) |
192 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) |
193 |
191 192
|
elind |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
194 |
|
eqcom |
⊢ ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) ) |
195 |
194
|
biimpi |
⊢ ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) ) |
196 |
195
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) ) |
197 |
193 196
|
eleqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
198 |
197
|
adantl3r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
199 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
200 |
20
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
201 |
|
elpreima |
⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) → ( 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) ) ) |
202 |
200 201
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) ) ) |
203 |
199 202
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) ) |
204 |
203
|
simprd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) |
205 |
189 190 198 204
|
syl21anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) |
206 |
188 205
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
207 |
179 206
|
sylancom |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
208 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝜑 ) |
209 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
210 |
208 209
|
jca |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ) |
211 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
212 |
|
simp-5r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑤 ⊆ ℝ ) |
213 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ 𝑤 ) |
214 |
212 213
|
sseldd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ℝ ) |
215 |
210 211 214
|
jca31 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ) |
216 |
64
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) ) |
217 |
|
breq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 < 𝐴 ↔ 𝑧 < 𝐴 ) ) |
218 |
217
|
ifbid |
⊢ ( 𝑦 = 𝑧 → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
219 |
180 183 218
|
ifbieq12d |
⊢ ( 𝑦 = 𝑧 → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
220 |
219
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 = 𝑧 ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
221 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ℝ ) |
222 |
|
iffalse |
⊢ ( ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
223 |
222
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
224 |
|
simp-5r |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
225 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑧 < 𝐴 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
226 |
224 225
|
ifclda |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑢 ) |
227 |
223 226
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑢 ) |
228 |
216 220 221 227
|
fvmptd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
229 |
228 223
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
230 |
229 226
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
231 |
215 230
|
sylancom |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
232 |
231
|
adantl4r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
233 |
207 232
|
pm2.61dan |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
234 |
173 174 175 233
|
syl21anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
235 |
172 234
|
eqeltrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 ∈ 𝑢 ) |
236 |
235
|
rexlimdv3a |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → ( ∃ 𝑧 ∈ 𝑤 ( 𝐺 ‘ 𝑧 ) = 𝑦 → 𝑦 ∈ 𝑢 ) ) |
237 |
169 236
|
mpd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → 𝑦 ∈ 𝑢 ) |
238 |
237
|
ex |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) ) |
239 |
238
|
alrimiv |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∀ 𝑦 ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) ) |
240 |
162 163 164 239
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∀ 𝑦 ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) ) |
241 |
|
dfss2 |
⊢ ( ( 𝐺 “ 𝑤 ) ⊆ 𝑢 ↔ ∀ 𝑦 ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) ) |
242 |
240 241
|
sylibr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ 𝑤 ) ⊆ 𝑢 ) |
243 |
|
imaundi |
⊢ ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) = ( ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) |
244 |
165
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → Fun 𝐺 ) |
245 |
|
fvelima |
⊢ ( ( Fun 𝐺 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → ∃ 𝑧 ∈ ( -∞ (,) 𝐴 ) ( 𝐺 ‘ 𝑧 ) = 𝑡 ) |
246 |
244 245
|
sylancom |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → ∃ 𝑧 ∈ ( -∞ (,) 𝐴 ) ( 𝐺 ‘ 𝑧 ) = 𝑡 ) |
247 |
|
simp1l |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝜑 ) |
248 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑧 ∈ ( -∞ (,) 𝐴 ) ) |
249 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 ) |
250 |
64
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) ) |
251 |
219
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ 𝑦 = 𝑧 ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
252 |
|
elioore |
⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → 𝑧 ∈ ℝ ) |
253 |
252
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ℝ ) |
254 |
|
elioo3g |
⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ) |
255 |
254
|
biimpi |
⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ) |
256 |
255
|
simprrd |
⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → 𝑧 < 𝐴 ) |
257 |
256
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 < 𝐴 ) |
258 |
|
ltnle |
⊢ ( ( 𝑧 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝑧 < 𝐴 ↔ ¬ 𝐴 ≤ 𝑧 ) ) |
259 |
252 5 258
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝑧 < 𝐴 ↔ ¬ 𝐴 ≤ 𝑧 ) ) |
260 |
257 259
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝐴 ≤ 𝑧 ) |
261 |
260
|
intn3an2d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) |
262 |
5 6
|
jca |
⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
263 |
262
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
264 |
|
elicc2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) ) |
265 |
263 264
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) ) |
266 |
261 265
|
mtbird |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) |
267 |
266
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
268 |
256
|
iftrued |
⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐴 ) ) |
269 |
268
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐴 ) ) |
270 |
267 269
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐴 ) ) |
271 |
127
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 ) |
272 |
270 271
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 ) |
273 |
250 251 253 272
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
274 |
273
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
275 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 ) |
276 |
270
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐴 ) ) |
277 |
274 275 276
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐴 ) ) |
278 |
247 248 249 277
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐴 ) ) |
279 |
278
|
rexlimdv3a |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → ( ∃ 𝑧 ∈ ( -∞ (,) 𝐴 ) ( 𝐺 ‘ 𝑧 ) = 𝑡 → 𝑡 = ( 𝐹 ‘ 𝐴 ) ) ) |
280 |
246 279
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → 𝑡 = ( 𝐹 ‘ 𝐴 ) ) |
281 |
|
velsn |
⊢ ( 𝑡 ∈ { ( 𝐹 ‘ 𝐴 ) } ↔ 𝑡 = ( 𝐹 ‘ 𝐴 ) ) |
282 |
280 281
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐴 ) } ) |
283 |
282
|
ex |
⊢ ( 𝜑 → ( 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐴 ) } ) ) |
284 |
283
|
ssrdv |
⊢ ( 𝜑 → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ { ( 𝐹 ‘ 𝐴 ) } ) |
285 |
284
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ { ( 𝐹 ‘ 𝐴 ) } ) |
286 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
287 |
286
|
snssd |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → { ( 𝐹 ‘ 𝐴 ) } ⊆ 𝑢 ) |
288 |
285 287
|
sstrd |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ 𝑢 ) |
289 |
288
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ 𝑢 ) |
290 |
|
fvelima |
⊢ ( ( Fun 𝐺 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → ∃ 𝑧 ∈ ( 𝐵 (,) +∞ ) ( 𝐺 ‘ 𝑧 ) = 𝑡 ) |
291 |
166 290
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → ∃ 𝑧 ∈ ( 𝐵 (,) +∞ ) ( 𝐺 ‘ 𝑧 ) = 𝑡 ) |
292 |
|
simp1l |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝜑 ) |
293 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑧 ∈ ( 𝐵 (,) +∞ ) ) |
294 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 ) |
295 |
64
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) ) |
296 |
219
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ 𝑦 = 𝑧 ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
297 |
|
elioore |
⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) → 𝑧 ∈ ℝ ) |
298 |
297
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝑧 ∈ ℝ ) |
299 |
19
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑌 ) |
300 |
299
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑌 ) |
301 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ∈ ℝ ) |
302 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 ∈ ℝ ) |
303 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ≤ 𝐵 ) |
304 |
|
elioo3g |
⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) ↔ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( 𝐵 < 𝑧 ∧ 𝑧 < +∞ ) ) ) |
305 |
304
|
biimpi |
⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) → ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( 𝐵 < 𝑧 ∧ 𝑧 < +∞ ) ) ) |
306 |
305
|
simprld |
⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) → 𝐵 < 𝑧 ) |
307 |
306
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 < 𝑧 ) |
308 |
301 302 298 303 307
|
lelttrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 < 𝑧 ) |
309 |
301 298 308
|
ltnsymd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑧 < 𝐴 ) |
310 |
309
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐵 ) ) |
311 |
129
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 ) |
312 |
310 311
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑌 ) |
313 |
312
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑌 ) |
314 |
300 313
|
ifclda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 ) |
315 |
295 296 298 314
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
316 |
315
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
317 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 ) |
318 |
302 298
|
ltnled |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐵 < 𝑧 ↔ ¬ 𝑧 ≤ 𝐵 ) ) |
319 |
307 318
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑧 ≤ 𝐵 ) |
320 |
319
|
intn3an3d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) |
321 |
262
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
322 |
321 264
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) ) |
323 |
320 322
|
mtbird |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) |
324 |
323
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
325 |
324 310
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐵 ) ) |
326 |
325
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐵 ) ) |
327 |
316 317 326
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐵 ) ) |
328 |
292 293 294 327
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐵 ) ) |
329 |
328
|
rexlimdv3a |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → ( ∃ 𝑧 ∈ ( 𝐵 (,) +∞ ) ( 𝐺 ‘ 𝑧 ) = 𝑡 → 𝑡 = ( 𝐹 ‘ 𝐵 ) ) ) |
330 |
291 329
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → 𝑡 = ( 𝐹 ‘ 𝐵 ) ) |
331 |
|
velsn |
⊢ ( 𝑡 ∈ { ( 𝐹 ‘ 𝐵 ) } ↔ 𝑡 = ( 𝐹 ‘ 𝐵 ) ) |
332 |
330 331
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐵 ) } ) |
333 |
332
|
ex |
⊢ ( 𝜑 → ( 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐵 ) } ) ) |
334 |
333
|
ssrdv |
⊢ ( 𝜑 → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ { ( 𝐹 ‘ 𝐵 ) } ) |
335 |
334
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ { ( 𝐹 ‘ 𝐵 ) } ) |
336 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
337 |
336
|
snssd |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → { ( 𝐹 ‘ 𝐵 ) } ⊆ 𝑢 ) |
338 |
335 337
|
sstrd |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ 𝑢 ) |
339 |
338
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ 𝑢 ) |
340 |
289 339
|
unssd |
⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) |
341 |
243 340
|
eqsstrid |
⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) |
342 |
159 163 164 341
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) |
343 |
242 342
|
unssd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ 𝑤 ) ∪ ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) |
344 |
158 343
|
eqsstrid |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) |
345 |
|
eleq2 |
⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ) |
346 |
|
imaeq2 |
⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ) |
347 |
346
|
sseq1d |
⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) ) |
348 |
345 347
|
anbi12d |
⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) ) ) |
349 |
348
|
rspcev |
⊢ ( ( ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
350 |
103 157 344 349
|
syl12anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
351 |
79
|
a1i |
⊢ ( 𝑤 ∈ 𝐽 → 𝐽 ∈ Top ) |
352 |
|
iooretop |
⊢ ( -∞ (,) 𝐵 ) ∈ ( topGen ‘ ran (,) ) |
353 |
352 2
|
eleqtrri |
⊢ ( -∞ (,) 𝐵 ) ∈ 𝐽 |
354 |
|
inopn |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( -∞ (,) 𝐵 ) ∈ 𝐽 ) → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∈ 𝐽 ) |
355 |
79 353 354
|
mp3an13 |
⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∈ 𝐽 ) |
356 |
96
|
a1i |
⊢ ( 𝑤 ∈ 𝐽 → ( -∞ (,) 𝐴 ) ∈ 𝐽 ) |
357 |
|
unopn |
⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∈ 𝐽 ∧ ( -∞ (,) 𝐴 ) ∈ 𝐽 ) → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 ) |
358 |
351 355 356 357
|
syl3anc |
⊢ ( 𝑤 ∈ 𝐽 → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 ) |
359 |
358
|
3ad2ant2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 ) |
360 |
359
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 ) |
361 |
|
simpll1 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ) |
362 |
|
simpll3 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
363 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
364 |
|
simpll |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) |
365 |
262
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
366 |
|
eqimss |
⊢ ( ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) = ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ⊆ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) |
367 |
109 366
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ⊆ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) |
368 |
|
difcom |
⊢ ( ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ⊆ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
369 |
367 368
|
sylib |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
370 |
365 369
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
371 |
370
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
372 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ℝ ) |
373 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) |
374 |
|
elioore |
⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) → 𝑦 ∈ ℝ ) |
375 |
374
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ℝ ) |
376 |
|
elioo3g |
⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) ↔ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
377 |
376
|
biimpi |
⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) → ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
378 |
377
|
simprld |
⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) → 𝐵 < 𝑦 ) |
379 |
378
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 < 𝑦 ) |
380 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 ∈ ℝ ) |
381 |
380 375
|
ltnled |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵 ) ) |
382 |
379 381
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑦 ≤ 𝐵 ) |
383 |
382
|
intn3an3d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) |
384 |
262
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
385 |
|
elicc2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
386 |
384 385
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
387 |
383 386
|
mtbird |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
388 |
387
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) |
389 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ∈ ℝ ) |
390 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ≤ 𝐵 ) |
391 |
389 380 375 390 379
|
lelttrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 < 𝑦 ) |
392 |
389 375 391
|
ltnsymd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑦 < 𝐴 ) |
393 |
392
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐵 ) ) |
394 |
388 393
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐵 ) ) |
395 |
129
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 ) |
396 |
394 395
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 ) |
397 |
375 396 133
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) |
398 |
397 394
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
399 |
398
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) ) |
400 |
399
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) ) |
401 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
402 |
400 401
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
403 |
402
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
404 |
403
|
stoic1a |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) |
405 |
404
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) |
406 |
|
ioran |
⊢ ( ¬ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) ↔ ( ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) ) |
407 |
373 405 406
|
sylanbrc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) ) |
408 |
|
elun |
⊢ ( 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) ) |
409 |
407 408
|
sylnibr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) |
410 |
372 409
|
eldifd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) |
411 |
371 410
|
sseldd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
412 |
411
|
adantllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
413 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝜑 ) |
414 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
415 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
416 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
417 |
138
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
418 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
419 |
417 418
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) |
420 |
20
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
421 |
420 143
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) ) |
422 |
416 419 421
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
423 |
413 414 415 422
|
syl21anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
424 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
425 |
423 424
|
eleqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
426 |
|
elinel1 |
⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ 𝑤 ) |
427 |
425 426
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ 𝑤 ) |
428 |
364 412 427
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ 𝑤 ) |
429 |
|
simp-4l |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) ) |
430 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
431 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
432 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝜑 ) |
433 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 ) |
434 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
435 |
433 434
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
436 |
432 435 137
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
437 |
433
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
438 |
436 437
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
439 |
438
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
440 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝜑 ) |
441 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐵 ∈ ℝ* ) |
442 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
443 |
442
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → +∞ ∈ ℝ* ) |
444 |
|
rexr |
⊢ ( 𝑦 ∈ ℝ → 𝑦 ∈ ℝ* ) |
445 |
444
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ* ) |
446 |
441 443 445
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
447 |
446
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
448 |
|
mnflt |
⊢ ( 𝑦 ∈ ℝ → -∞ < 𝑦 ) |
449 |
448
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → -∞ < 𝑦 ) |
450 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
451 |
450
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → -∞ ∈ ℝ* ) |
452 |
451 441 445
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
453 |
|
elioo3g |
⊢ ( 𝑦 ∈ ( -∞ (,) 𝐵 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ) |
454 |
453
|
notbii |
⊢ ( ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ↔ ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ) |
455 |
454
|
biimpi |
⊢ ( ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ) |
456 |
455
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ) |
457 |
|
nan |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ) |
458 |
456 457
|
mpbi |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) |
459 |
452 458
|
mpidan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) |
460 |
|
nan |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ -∞ < 𝑦 ) → ¬ 𝑦 < 𝐵 ) ) |
461 |
459 460
|
mpbi |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ -∞ < 𝑦 ) → ¬ 𝑦 < 𝐵 ) |
462 |
449 461
|
mpdan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ 𝑦 < 𝐵 ) |
463 |
462
|
anim1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( ¬ 𝑦 < 𝐵 ∧ ¬ 𝑦 = 𝐵 ) ) |
464 |
|
pm4.56 |
⊢ ( ( ¬ 𝑦 < 𝐵 ∧ ¬ 𝑦 = 𝐵 ) ↔ ¬ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) ) |
465 |
463 464
|
sylib |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ¬ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) ) |
466 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ ) |
467 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐵 ∈ ℝ ) |
468 |
466 467
|
jca |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
469 |
468
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
470 |
|
leloe |
⊢ ( ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ≤ 𝐵 ↔ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) ) ) |
471 |
469 470
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝑦 ≤ 𝐵 ↔ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) ) ) |
472 |
465 471
|
mtbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ¬ 𝑦 ≤ 𝐵 ) |
473 |
6
|
anim1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) |
474 |
473
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) |
475 |
|
ltnle |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵 ) ) |
476 |
474 475
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵 ) ) |
477 |
472 476
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝐵 < 𝑦 ) |
478 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝑦 ∈ ℝ ) |
479 |
478
|
ltpnfd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝑦 < +∞ ) |
480 |
477 479
|
jca |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) |
481 |
447 480 376
|
sylanbrc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝑦 ∈ ( 𝐵 (,) +∞ ) ) |
482 |
440 481 398
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
483 |
439 482
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
484 |
483
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) ) |
485 |
484
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) ) |
486 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
487 |
485 486
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
488 |
487
|
stoic1a |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) |
489 |
488
|
notnotrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( -∞ (,) 𝐵 ) ) |
490 |
429 430 431 489
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( -∞ (,) 𝐵 ) ) |
491 |
428 490
|
elind |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) |
492 |
491
|
ex |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) → 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) |
493 |
492
|
orrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) |
494 |
493
|
orcomd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∨ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) ) |
495 |
|
elun |
⊢ ( 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ↔ ( 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∨ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) ) |
496 |
494 495
|
sylibr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) |
497 |
361 362 363 496
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) |
498 |
104
|
simpld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝜑 ) |
499 |
498
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 ) |
500 |
|
simpll2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 ) |
501 |
11 500 160
|
sylancr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ⊆ ℝ ) |
502 |
499 501
|
jca |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ) |
503 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
504 |
65
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn ℝ ) |
505 |
504
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝐺 Fn ℝ ) |
506 |
|
ssinss1 |
⊢ ( 𝑤 ⊆ ℝ → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ⊆ ℝ ) |
507 |
506
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ⊆ ℝ ) |
508 |
|
ioossre |
⊢ ( -∞ (,) 𝐴 ) ⊆ ℝ |
509 |
508
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( -∞ (,) 𝐴 ) ⊆ ℝ ) |
510 |
|
unima |
⊢ ( ( 𝐺 Fn ℝ ∧ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ⊆ ℝ ∧ ( -∞ (,) 𝐴 ) ⊆ ℝ ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∪ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ) |
511 |
505 507 509 510
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∪ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ) |
512 |
165
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → Fun 𝐺 ) |
513 |
|
fvelima |
⊢ ( ( Fun 𝐺 ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → ∃ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ( 𝐺 ‘ 𝑧 ) = 𝑦 ) |
514 |
512 513
|
sylancom |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → ∃ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ( 𝐺 ‘ 𝑧 ) = 𝑦 ) |
515 |
171
|
3ad2ant3 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 = ( 𝐺 ‘ 𝑧 ) ) |
516 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝜑 ) |
517 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
518 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( -∞ (,) 𝐴 ) ) |
519 |
273 267 269
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) ) |
520 |
519
|
3adant2 |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) ) |
521 |
|
simp2 |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
522 |
520 521
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
523 |
516 517 518 522
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
524 |
|
simplll |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) |
525 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝜑 ) |
526 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) |
527 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) |
528 |
|
elinel1 |
⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ 𝑤 ) |
529 |
528
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ 𝑤 ) |
530 |
|
elinel2 |
⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ ( -∞ (,) 𝐵 ) ) |
531 |
|
elioore |
⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) → 𝑧 ∈ ℝ ) |
532 |
530 531
|
syl |
⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ ℝ ) |
533 |
532
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ℝ ) |
534 |
26
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝐴 ∈ ℝ* ) |
535 |
533
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ℝ* ) |
536 |
|
mnflt |
⊢ ( 𝑧 ∈ ℝ → -∞ < 𝑧 ) |
537 |
533 536
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → -∞ < 𝑧 ) |
538 |
450
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → -∞ ∈ ℝ* ) |
539 |
538 534 535
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ) |
540 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) |
541 |
540 254
|
sylnib |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ) |
542 |
|
nan |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ) ↔ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ) |
543 |
541 542
|
mpbi |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) |
544 |
539 543
|
mpdan |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) |
545 |
|
nan |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ↔ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ -∞ < 𝑧 ) → ¬ 𝑧 < 𝐴 ) ) |
546 |
544 545
|
mpbi |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ -∞ < 𝑧 ) → ¬ 𝑧 < 𝐴 ) |
547 |
537 546
|
mpdan |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 < 𝐴 ) |
548 |
534 535 547
|
xrnltled |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝐴 ≤ 𝑧 ) |
549 |
|
simp1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝜑 ) |
550 |
530
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( -∞ (,) 𝐵 ) ) |
551 |
531
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ ℝ ) |
552 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝐵 ∈ ℝ ) |
553 |
|
elioo3g |
⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐵 ) ) ) |
554 |
553
|
biimpi |
⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) → ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐵 ) ) ) |
555 |
554
|
simprrd |
⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) → 𝑧 < 𝐵 ) |
556 |
555
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝑧 < 𝐵 ) |
557 |
551 552 556
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝑧 ≤ 𝐵 ) |
558 |
549 550 557
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ≤ 𝐵 ) |
559 |
262
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
560 |
559 264
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) ) |
561 |
533 548 558 560
|
mpbir3and |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) |
562 |
529 561
|
elind |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
563 |
525 526 527 562
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
564 |
|
elinel2 |
⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) |
565 |
564
|
anim2i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
566 |
565
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
567 |
566 186
|
syl |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
568 |
20
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
569 |
|
simpr |
⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
570 |
195
|
adantr |
⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) ) |
571 |
569 570
|
eleqtrd |
⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
572 |
571
|
adantll |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
573 |
201
|
simplbda |
⊢ ( ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) |
574 |
568 572 573
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) |
575 |
567 574
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
576 |
575
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
577 |
524 563 576
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
578 |
523 577
|
pm2.61dan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
579 |
578
|
3adant3 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 ) |
580 |
515 579
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 ∈ 𝑢 ) |
581 |
580
|
3adant1r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 ∈ 𝑢 ) |
582 |
581
|
rexlimdv3a |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → ( ∃ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ( 𝐺 ‘ 𝑧 ) = 𝑦 → 𝑦 ∈ 𝑢 ) ) |
583 |
514 582
|
mpd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → 𝑦 ∈ 𝑢 ) |
584 |
583
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) → 𝑦 ∈ 𝑢 ) ) |
585 |
584
|
ssrdv |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ⊆ 𝑢 ) |
586 |
288
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ 𝑢 ) |
587 |
585 586
|
unssd |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∪ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) |
588 |
511 587
|
eqsstrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) |
589 |
502 362 503 588
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) |
590 |
|
eleq2 |
⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ) |
591 |
|
imaeq2 |
⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ) |
592 |
591
|
sseq1d |
⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) ) |
593 |
590 592
|
anbi12d |
⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) ) ) |
594 |
593
|
rspcev |
⊢ ( ( ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
595 |
360 497 589 594
|
syl12anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
596 |
350 595
|
pm2.61dan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
597 |
|
simpll2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 ) |
598 |
|
iooretop |
⊢ ( 𝐴 (,) +∞ ) ∈ ( topGen ‘ ran (,) ) |
599 |
598 2
|
eleqtrri |
⊢ ( 𝐴 (,) +∞ ) ∈ 𝐽 |
600 |
|
inopn |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( 𝐴 (,) +∞ ) ∈ 𝐽 ) → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∈ 𝐽 ) |
601 |
79 599 600
|
mp3an13 |
⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∈ 𝐽 ) |
602 |
98
|
a1i |
⊢ ( 𝑤 ∈ 𝐽 → ( 𝐵 (,) +∞ ) ∈ 𝐽 ) |
603 |
|
unopn |
⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∈ 𝐽 ∧ ( 𝐵 (,) +∞ ) ∈ 𝐽 ) → ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ) |
604 |
351 601 602 603
|
syl3anc |
⊢ ( 𝑤 ∈ 𝐽 → ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ) |
605 |
597 604
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ) |
606 |
|
simplll |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ) |
607 |
606
|
simpld |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) ) |
608 |
607
|
simpld |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝜑 ) |
609 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
610 |
|
simp-5r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ℝ ) |
611 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
612 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → 𝜑 ) |
613 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐴 ∈ ℝ* ) |
614 |
451 613 445
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
615 |
614
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
616 |
448
|
anim1i |
⊢ ( ( 𝑦 ∈ ℝ ∧ 𝑦 < 𝐴 ) → ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) ) |
617 |
616
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) ) |
618 |
|
elioo3g |
⊢ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) ) ) |
619 |
615 617 618
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → 𝑦 ∈ ( -∞ (,) 𝐴 ) ) |
620 |
|
eleq1 |
⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ ( -∞ (,) 𝐴 ) ↔ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) ) |
621 |
620
|
anbi2d |
⊢ ( 𝑧 = 𝑦 → ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ↔ ( 𝜑 ∧ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) ) ) |
622 |
|
fveq2 |
⊢ ( 𝑧 = 𝑦 → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑦 ) ) |
623 |
622
|
eqeq1d |
⊢ ( 𝑧 = 𝑦 → ( ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) ↔ ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) ) |
624 |
621 623
|
imbi12d |
⊢ ( 𝑧 = 𝑦 → ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) ) ) |
625 |
624 519
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) |
626 |
612 619 625
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) |
627 |
626
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) ) |
628 |
627
|
ad4ant14 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) ) |
629 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
630 |
628 629
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
631 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
632 |
630 631
|
pm2.65da |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ¬ 𝑦 < 𝐴 ) |
633 |
5
|
anim1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) |
634 |
633
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) |
635 |
|
lenlt |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝐴 ≤ 𝑦 ↔ ¬ 𝑦 < 𝐴 ) ) |
636 |
634 635
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐴 ≤ 𝑦 ↔ ¬ 𝑦 < 𝐴 ) ) |
637 |
632 636
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝐴 ≤ 𝑦 ) |
638 |
606 611 637
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ≤ 𝑦 ) |
639 |
|
ltpnf |
⊢ ( 𝑦 ∈ ℝ → 𝑦 < +∞ ) |
640 |
639
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 < +∞ ) |
641 |
446
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
642 |
376
|
notbii |
⊢ ( ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ↔ ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
643 |
642
|
biimpi |
⊢ ( ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) → ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
644 |
643
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
645 |
|
imnan |
⊢ ( ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ¬ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ↔ ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
646 |
644 645
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ¬ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
647 |
641 646
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) |
648 |
|
ancom |
⊢ ( ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ↔ ( 𝑦 < +∞ ∧ 𝐵 < 𝑦 ) ) |
649 |
647 648
|
sylnib |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝑦 < +∞ ∧ 𝐵 < 𝑦 ) ) |
650 |
|
imnan |
⊢ ( ( 𝑦 < +∞ → ¬ 𝐵 < 𝑦 ) ↔ ¬ ( 𝑦 < +∞ ∧ 𝐵 < 𝑦 ) ) |
651 |
649 650
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 < +∞ → ¬ 𝐵 < 𝑦 ) ) |
652 |
640 651
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝐵 < 𝑦 ) |
653 |
468
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
654 |
|
lenlt |
⊢ ( ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑦 ) ) |
655 |
653 654
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑦 ) ) |
656 |
652 655
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ≤ 𝐵 ) |
657 |
607 656
|
sylancom |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ≤ 𝐵 ) |
658 |
262
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
659 |
658 385
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
660 |
610 638 657 659
|
mpbir3and |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
661 |
608 609 660 422
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
662 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
663 |
661 662
|
eleqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
664 |
663 426
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ 𝑤 ) |
665 |
|
fveq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝐴 ) ) |
666 |
29
|
ancli |
⊢ ( 𝜑 → ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
667 |
|
eleq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
668 |
667
|
anbi2d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) ) ) |
669 |
|
fveq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) |
670 |
665 669
|
eqeq12d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) ) |
671 |
668 670
|
imbi12d |
⊢ ( 𝑦 = 𝐴 → ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) ) ) |
672 |
671 137
|
vtoclg |
⊢ ( 𝐴 ∈ ℝ → ( ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) ) |
673 |
5 666 672
|
sylc |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) |
674 |
665 673
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) |
675 |
674
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ 𝑦 = 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) |
676 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝜑 ) |
677 |
614
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
678 |
448
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → -∞ < 𝑦 ) |
679 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 ∈ ℝ ) |
680 |
676 5
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝐴 ∈ ℝ ) |
681 |
445
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ ℝ* ) |
682 |
26
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝐴 ∈ ℝ* ) |
683 |
639
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝑦 < +∞ ) |
684 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) |
685 |
442
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → +∞ ∈ ℝ* ) |
686 |
682 685 681
|
3jca |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
687 |
|
elioo3g |
⊢ ( 𝑦 ∈ ( 𝐴 (,) +∞ ) ↔ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
688 |
687
|
notbii |
⊢ ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ↔ ¬ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
689 |
688
|
biimpi |
⊢ ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) → ¬ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
690 |
|
nan |
⊢ ( ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) → ¬ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) ) ↔ ( ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ∧ ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
691 |
689 690
|
mpbi |
⊢ ( ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ∧ ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) |
692 |
684 686 691
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) |
693 |
|
ancom |
⊢ ( ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ↔ ( 𝑦 < +∞ ∧ 𝐴 < 𝑦 ) ) |
694 |
692 693
|
sylnib |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝑦 < +∞ ∧ 𝐴 < 𝑦 ) ) |
695 |
|
nan |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝑦 < +∞ ∧ 𝐴 < 𝑦 ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ 𝑦 < +∞ ) → ¬ 𝐴 < 𝑦 ) ) |
696 |
694 695
|
mpbi |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ 𝑦 < +∞ ) → ¬ 𝐴 < 𝑦 ) |
697 |
683 696
|
mpdan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ 𝐴 < 𝑦 ) |
698 |
681 682 697
|
xrnltled |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝑦 ≤ 𝐴 ) |
699 |
698
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 ≤ 𝐴 ) |
700 |
|
neqne |
⊢ ( ¬ 𝑦 = 𝐴 → 𝑦 ≠ 𝐴 ) |
701 |
700
|
necomd |
⊢ ( ¬ 𝑦 = 𝐴 → 𝐴 ≠ 𝑦 ) |
702 |
701
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝐴 ≠ 𝑦 ) |
703 |
679 680 699 702
|
leneltd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 < 𝐴 ) |
704 |
678 703
|
jca |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) ) |
705 |
677 704 618
|
sylanbrc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 ∈ ( -∞ (,) 𝐴 ) ) |
706 |
676 705 625
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) |
707 |
675 706
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) |
708 |
707
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) ) |
709 |
708
|
ad4ant14 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) ) |
710 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
711 |
709 710
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
712 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
713 |
711 712
|
condan |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 (,) +∞ ) ) |
714 |
606 611 713
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝐴 (,) +∞ ) ) |
715 |
664 714
|
elind |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) |
716 |
715
|
adantlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) |
717 |
|
pm5.6 |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ↔ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐵 (,) +∞ ) ∨ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ) ) |
718 |
716 717
|
mpbi |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐵 (,) +∞ ) ∨ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ) |
719 |
718
|
orcomd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) ) |
720 |
|
elun |
⊢ ( 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) ) |
721 |
719 720
|
sylibr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) |
722 |
721
|
3adantll2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) |
723 |
|
simp1ll |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝜑 ) |
724 |
723
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 ) |
725 |
|
simpll3 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
726 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
727 |
504
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝐺 Fn ℝ ) |
728 |
|
ioossre |
⊢ ( 𝐴 (,) +∞ ) ⊆ ℝ |
729 |
728
|
olci |
⊢ ( 𝑤 ⊆ ℝ ∨ ( 𝐴 (,) +∞ ) ⊆ ℝ ) |
730 |
|
inss |
⊢ ( ( 𝑤 ⊆ ℝ ∨ ( 𝐴 (,) +∞ ) ⊆ ℝ ) → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ ) |
731 |
729 730
|
ax-mp |
⊢ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ |
732 |
731
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ ) |
733 |
|
ioossre |
⊢ ( 𝐵 (,) +∞ ) ⊆ ℝ |
734 |
733
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐵 (,) +∞ ) ⊆ ℝ ) |
735 |
|
unima |
⊢ ( ( 𝐺 Fn ℝ ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ ∧ ( 𝐵 (,) +∞ ) ⊆ ℝ ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ) |
736 |
727 732 734 735
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ) |
737 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → 𝜑 ) |
738 |
731
|
sseli |
⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ ℝ ) |
739 |
738
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → 𝑦 ∈ ℝ ) |
740 |
737 739 446
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
741 |
|
simpr |
⊢ ( ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∧ 𝐵 < 𝑦 ) → 𝐵 < 𝑦 ) |
742 |
738
|
ltpnfd |
⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 < +∞ ) |
743 |
742
|
adantr |
⊢ ( ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∧ 𝐵 < 𝑦 ) → 𝑦 < +∞ ) |
744 |
741 743
|
jca |
⊢ ( ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) |
745 |
744
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) |
746 |
740 745 376
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝐵 (,) +∞ ) ) |
747 |
737 746 398
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
748 |
747
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
749 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
750 |
748 749
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
751 |
750
|
adantl3r |
⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
752 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝜑 ) |
753 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) |
754 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ¬ 𝐵 < 𝑦 ) |
755 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝜑 ) |
756 |
738
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝑦 ∈ ℝ ) |
757 |
756
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ℝ ) |
758 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝐴 ∈ ℝ ) |
759 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ ( 𝐴 (,) +∞ ) ) |
760 |
687
|
biimpi |
⊢ ( 𝑦 ∈ ( 𝐴 (,) +∞ ) → ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) ) |
761 |
760
|
simprld |
⊢ ( 𝑦 ∈ ( 𝐴 (,) +∞ ) → 𝐴 < 𝑦 ) |
762 |
759 761
|
syl |
⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝐴 < 𝑦 ) |
763 |
762
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝐴 < 𝑦 ) |
764 |
758 756 763
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝐴 ≤ 𝑦 ) |
765 |
764
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝐴 ≤ 𝑦 ) |
766 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ¬ 𝐵 < 𝑦 ) |
767 |
755 757 468
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
768 |
767 654
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑦 ) ) |
769 |
766 768
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ≤ 𝐵 ) |
770 |
262
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
771 |
770 385
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
772 |
757 765 769 771
|
mpbir3and |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
773 |
755 772 137
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
774 |
752 753 754 773
|
syl21anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
775 |
|
elinel1 |
⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ 𝑤 ) |
776 |
775
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ 𝑤 ) |
777 |
776 772
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
778 |
777
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
779 |
778 149
|
sylibr |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
780 |
195
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) ) |
781 |
779 780
|
eleqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
782 |
20
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
783 |
782 143
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) ) |
784 |
781 783
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) |
785 |
784
|
simprd |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) |
786 |
785
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) |
787 |
774 786
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
788 |
751 787
|
pm2.61dan |
⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
789 |
788
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∀ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
790 |
504
|
fndmd |
⊢ ( 𝜑 → dom 𝐺 = ℝ ) |
791 |
731 790
|
sseqtrrid |
⊢ ( 𝜑 → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) |
792 |
166 791
|
jca |
⊢ ( 𝜑 → ( Fun 𝐺 ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) ) |
793 |
792
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( Fun 𝐺 ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) ) |
794 |
|
funimass4 |
⊢ ( ( Fun 𝐺 ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ⊆ 𝑢 ↔ ∀ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ) |
795 |
793 794
|
syl |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ⊆ 𝑢 ↔ ∀ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ) |
796 |
789 795
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ⊆ 𝑢 ) |
797 |
338
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ 𝑢 ) |
798 |
796 797
|
unssd |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) |
799 |
736 798
|
eqsstrd |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) |
800 |
724 725 726 799
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) |
801 |
|
eleq2 |
⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ) |
802 |
|
imaeq2 |
⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ) |
803 |
802
|
sseq1d |
⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) ) |
804 |
801 803
|
anbi12d |
⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) ) ) |
805 |
804
|
rspcev |
⊢ ( ( ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
806 |
605 722 800 805
|
syl12anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
807 |
|
simpll2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 ) |
808 |
|
iooretop |
⊢ ( 𝐴 (,) 𝐵 ) ∈ ( topGen ‘ ran (,) ) |
809 |
808 2
|
eleqtrri |
⊢ ( 𝐴 (,) 𝐵 ) ∈ 𝐽 |
810 |
|
inopn |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( 𝐴 (,) 𝐵 ) ∈ 𝐽 ) → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 ) |
811 |
79 809 810
|
mp3an13 |
⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 ) |
812 |
807 811
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 ) |
813 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ℝ ) |
814 |
637
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝐴 ≤ 𝑦 ) |
815 |
|
simpll |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) ) |
816 |
815 404 656
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≤ 𝐵 ) |
817 |
816
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≤ 𝐵 ) |
818 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 ) |
819 |
818 262
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
820 |
819 385
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
821 |
813 814 817 820
|
mpbir3and |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
822 |
821
|
adantllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
823 |
818 821 137
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
824 |
823
|
adantllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
825 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
826 |
824 825
|
eqeltrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) |
827 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 ) |
828 |
827 20
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
829 |
828 143
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) ) |
830 |
822 826 829
|
mpbir2and |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
831 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
832 |
830 831
|
eleqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
833 |
832 426
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ 𝑤 ) |
834 |
|
simp-5r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ℝ ) |
835 |
827 834 822
|
jca31 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
836 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) |
837 |
826 836
|
jca |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ) |
838 |
|
nelneq |
⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) |
839 |
669
|
necon3bi |
⊢ ( ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) → 𝑦 ≠ 𝐴 ) |
840 |
837 838 839
|
3syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≠ 𝐴 ) |
841 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) |
842 |
826 841
|
jca |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ) |
843 |
|
nelneq |
⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
844 |
|
fveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) ) |
845 |
844
|
necon3bi |
⊢ ( ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) → 𝑦 ≠ 𝐵 ) |
846 |
842 843 845
|
3syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≠ 𝐵 ) |
847 |
613
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝐴 ∈ ℝ* ) |
848 |
441
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝐵 ∈ ℝ* ) |
849 |
444
|
ad4antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 ∈ ℝ* ) |
850 |
847 848 849
|
3jca |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
851 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → 𝑦 ≠ 𝐴 ) |
852 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ∈ ℝ ) |
853 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ℝ ) |
854 |
262
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) |
855 |
854 385
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) ) |
856 |
135 855
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) |
857 |
856
|
simp2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝑦 ) |
858 |
857
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝑦 ) |
859 |
852 853 858
|
3jca |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ) ) |
860 |
859
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ) ) |
861 |
|
leltne |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ) → ( 𝐴 < 𝑦 ↔ 𝑦 ≠ 𝐴 ) ) |
862 |
860 861
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → ( 𝐴 < 𝑦 ↔ 𝑦 ≠ 𝐴 ) ) |
863 |
851 862
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → 𝐴 < 𝑦 ) |
864 |
863
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝐴 < 𝑦 ) |
865 |
|
necom |
⊢ ( 𝑦 ≠ 𝐵 ↔ 𝐵 ≠ 𝑦 ) |
866 |
865
|
biimpi |
⊢ ( 𝑦 ≠ 𝐵 → 𝐵 ≠ 𝑦 ) |
867 |
866
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → 𝐵 ≠ 𝑦 ) |
868 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ ) |
869 |
856
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ≤ 𝐵 ) |
870 |
869
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ≤ 𝐵 ) |
871 |
853 868 870
|
3jca |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ≤ 𝐵 ) ) |
872 |
871
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ≤ 𝐵 ) ) |
873 |
|
leltne |
⊢ ( ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ≤ 𝐵 ) → ( 𝑦 < 𝐵 ↔ 𝐵 ≠ 𝑦 ) ) |
874 |
872 873
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝑦 < 𝐵 ↔ 𝐵 ≠ 𝑦 ) ) |
875 |
867 874
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 < 𝐵 ) |
876 |
875
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 < 𝐵 ) |
877 |
864 876
|
jca |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝐴 < 𝑦 ∧ 𝑦 < 𝐵 ) ) |
878 |
|
elioo3g |
⊢ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < 𝐵 ) ) ) |
879 |
850 877 878
|
sylanbrc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ) |
880 |
835 840 846 879
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ) |
881 |
833 880
|
elind |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) |
882 |
881
|
3adantll2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) |
883 |
165
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → Fun 𝐺 ) |
884 |
|
fvelima |
⊢ ( ( Fun 𝐺 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ∃ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ( 𝐺 ‘ 𝑦 ) = 𝑡 ) |
885 |
883 884
|
sylancom |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ∃ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ( 𝐺 ‘ 𝑦 ) = 𝑡 ) |
886 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐺 ‘ 𝑦 ) = 𝑡 ) |
887 |
|
simp1l |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → 𝜑 ) |
888 |
|
inss2 |
⊢ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 (,) 𝐵 ) |
889 |
|
ioossicc |
⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
890 |
888 889
|
sstri |
⊢ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) |
891 |
890
|
sseli |
⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
892 |
891
|
3ad2ant2 |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
893 |
887 892 137
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
894 |
|
sslin |
⊢ ( ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
895 |
889 894
|
ax-mp |
⊢ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) |
896 |
895
|
sseli |
⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
897 |
896
|
adantl |
⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
898 |
195
|
adantr |
⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) ) |
899 |
897 898
|
eleqtrd |
⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
900 |
899
|
adantll |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ) |
901 |
20
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
902 |
901 143
|
syl |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) ) |
903 |
900 902
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) |
904 |
903
|
simprd |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) |
905 |
904
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) |
906 |
893 905
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) |
907 |
886 906
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → 𝑡 ∈ 𝑢 ) |
908 |
907
|
3exp |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐺 ‘ 𝑦 ) = 𝑡 → 𝑡 ∈ 𝑢 ) ) ) |
909 |
908
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐺 ‘ 𝑦 ) = 𝑡 → 𝑡 ∈ 𝑢 ) ) ) |
910 |
909
|
rexlimdv |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ( ∃ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ( 𝐺 ‘ 𝑦 ) = 𝑡 → 𝑡 ∈ 𝑢 ) ) |
911 |
885 910
|
mpd |
⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → 𝑡 ∈ 𝑢 ) |
912 |
911
|
ralrimiva |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ∀ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) 𝑡 ∈ 𝑢 ) |
913 |
|
dfss3 |
⊢ ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ↔ ∀ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) 𝑡 ∈ 𝑢 ) |
914 |
912 913
|
sylibr |
⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) |
915 |
914
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) |
916 |
915
|
3adant2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) |
917 |
916
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) |
918 |
|
eleq2 |
⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) |
919 |
|
imaeq2 |
⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) |
920 |
919
|
sseq1d |
⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) ) |
921 |
918 920
|
anbi12d |
⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) ) ) |
922 |
921
|
rspcev |
⊢ ( ( ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
923 |
812 882 917 922
|
syl12anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
924 |
806 923
|
pm2.61dan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
925 |
596 924
|
pm2.61dan |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
926 |
93 925
|
syld3an1 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
927 |
926
|
rexlimdv3a |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) ) |
928 |
88 927
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) |
929 |
928
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) ) |
930 |
929
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ∀ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) ) |
931 |
11
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐽 ∈ ( TopOn ‘ ℝ ) ) |
932 |
|
resttopon |
⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ ran 𝐹 ⊆ 𝑌 ) → ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) ) |
933 |
17 71 932
|
syl2anc |
⊢ ( 𝜑 → ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) ) |
934 |
933
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) ) |
935 |
|
iscnp |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) ∧ 𝑦 ∈ ℝ ) → ( 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) ) ) ) |
936 |
931 934 466 935
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) ) ) ) |
937 |
66 930 936
|
mpbir2and |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ) |
938 |
937
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ ℝ 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ) |
939 |
|
cncnp |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) ) → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑦 ∈ ℝ 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ) ) ) |
940 |
11 933 939
|
sylancr |
⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑦 ∈ ℝ 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ) ) ) |
941 |
65 938 940
|
mpbir2and |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ) |
942 |
|
fnssres |
⊢ ( ( 𝐺 Fn ℝ ∧ ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) → ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) Fn ( 𝐴 [,] 𝐵 ) ) |
943 |
504 12 942
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) Fn ( 𝐴 [,] 𝐵 ) ) |
944 |
|
fvres |
⊢ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) → ( ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
945 |
944
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) ) |
946 |
945 137
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
947 |
943 20 946
|
eqfnfvd |
⊢ ( 𝜑 → ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) = 𝐹 ) |
948 |
941 947
|
jca |
⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) = 𝐹 ) ) |