Step |
Hyp |
Ref |
Expression |
1 |
|
smfco.s |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
2 |
|
smfco.f |
⊢ ( 𝜑 → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) |
3 |
|
smfco.j |
⊢ 𝐽 = ( topGen ‘ ran (,) ) |
4 |
|
smfco.b |
⊢ 𝐵 = ( SalGen ‘ 𝐽 ) |
5 |
|
smfco.h |
⊢ ( 𝜑 → 𝐻 ∈ ( SMblFn ‘ 𝐵 ) ) |
6 |
|
nfv |
⊢ Ⅎ 𝑎 𝜑 |
7 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ dom 𝐻 ) ⊆ dom 𝐹 |
8 |
7
|
a1i |
⊢ ( 𝜑 → ( ◡ 𝐹 “ dom 𝐻 ) ⊆ dom 𝐹 ) |
9 |
|
eqid |
⊢ dom 𝐹 = dom 𝐹 |
10 |
1 2 9
|
smfdmss |
⊢ ( 𝜑 → dom 𝐹 ⊆ ∪ 𝑆 ) |
11 |
8 10
|
sstrd |
⊢ ( 𝜑 → ( ◡ 𝐹 “ dom 𝐻 ) ⊆ ∪ 𝑆 ) |
12 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
13 |
3 12
|
eqeltri |
⊢ 𝐽 ∈ Top |
14 |
13
|
a1i |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
15 |
14 4
|
salgencld |
⊢ ( 𝜑 → 𝐵 ∈ SAlg ) |
16 |
|
eqid |
⊢ dom 𝐻 = dom 𝐻 |
17 |
15 5 16
|
smff |
⊢ ( 𝜑 → 𝐻 : dom 𝐻 ⟶ ℝ ) |
18 |
17
|
ffund |
⊢ ( 𝜑 → Fun 𝐻 ) |
19 |
1 2 9
|
smff |
⊢ ( 𝜑 → 𝐹 : dom 𝐹 ⟶ ℝ ) |
20 |
19
|
ffund |
⊢ ( 𝜑 → Fun 𝐹 ) |
21 |
18 20
|
fco3 |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝐹 ) : ( ◡ 𝐹 “ dom 𝐻 ) ⟶ ran 𝐻 ) |
22 |
17
|
frnd |
⊢ ( 𝜑 → ran 𝐻 ⊆ ℝ ) |
23 |
21 22
|
fssd |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝐹 ) : ( ◡ 𝐹 “ dom 𝐻 ) ⟶ ℝ ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( 𝐻 ∘ 𝐹 ) : ( ◡ 𝐹 “ dom 𝐻 ) ⟶ ℝ ) |
25 |
|
rexr |
⊢ ( 𝑎 ∈ ℝ → 𝑎 ∈ ℝ* ) |
26 |
25
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝑎 ∈ ℝ* ) |
27 |
24 26
|
preimaioomnf |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( ◡ ( 𝐻 ∘ 𝐹 ) “ ( -∞ (,) 𝑎 ) ) = { 𝑥 ∈ ( ◡ 𝐹 “ dom 𝐻 ) ∣ ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑥 ) < 𝑎 } ) |
28 |
27
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → { 𝑥 ∈ ( ◡ 𝐹 “ dom 𝐻 ) ∣ ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑥 ) < 𝑎 } = ( ◡ ( 𝐻 ∘ 𝐹 ) “ ( -∞ (,) 𝑎 ) ) ) |
29 |
|
cnvco |
⊢ ◡ ( 𝐻 ∘ 𝐹 ) = ( ◡ 𝐹 ∘ ◡ 𝐻 ) |
30 |
29
|
imaeq1i |
⊢ ( ◡ ( 𝐻 ∘ 𝐹 ) “ ( -∞ (,) 𝑎 ) ) = ( ( ◡ 𝐹 ∘ ◡ 𝐻 ) “ ( -∞ (,) 𝑎 ) ) |
31 |
30
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( ◡ ( 𝐻 ∘ 𝐹 ) “ ( -∞ (,) 𝑎 ) ) = ( ( ◡ 𝐹 ∘ ◡ 𝐻 ) “ ( -∞ (,) 𝑎 ) ) ) |
32 |
|
imaco |
⊢ ( ( ◡ 𝐹 ∘ ◡ 𝐻 ) “ ( -∞ (,) 𝑎 ) ) = ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) |
33 |
32
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( ( ◡ 𝐹 ∘ ◡ 𝐻 ) “ ( -∞ (,) 𝑎 ) ) = ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) ) |
34 |
28 31 33
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → { 𝑥 ∈ ( ◡ 𝐹 “ dom 𝐻 ) ∣ ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑥 ) < 𝑎 } = ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) ) |
35 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝐻 : dom 𝐻 ⟶ ℝ ) |
36 |
35 26
|
preimaioomnf |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = { 𝑥 ∈ dom 𝐻 ∣ ( 𝐻 ‘ 𝑥 ) < 𝑎 } ) |
37 |
36
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → { 𝑥 ∈ dom 𝐻 ∣ ( 𝐻 ‘ 𝑥 ) < 𝑎 } = ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) |
38 |
37
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = { 𝑥 ∈ dom 𝐻 ∣ ( 𝐻 ‘ 𝑥 ) < 𝑎 } ) |
39 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝐵 ∈ SAlg ) |
40 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝐻 ∈ ( SMblFn ‘ 𝐵 ) ) |
41 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → 𝑎 ∈ ℝ ) |
42 |
39 40 16 41
|
smfpreimalt |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → { 𝑥 ∈ dom 𝐻 ∣ ( 𝐻 ‘ 𝑥 ) < 𝑎 } ∈ ( 𝐵 ↾t dom 𝐻 ) ) |
43 |
38 42
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ∈ ( 𝐵 ↾t dom 𝐻 ) ) |
44 |
15
|
elexd |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
45 |
5
|
dmexd |
⊢ ( 𝜑 → dom 𝐻 ∈ V ) |
46 |
|
elrest |
⊢ ( ( 𝐵 ∈ V ∧ dom 𝐻 ∈ V ) → ( ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ∈ ( 𝐵 ↾t dom 𝐻 ) ↔ ∃ 𝑒 ∈ 𝐵 ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) ) ) |
47 |
44 45 46
|
syl2anc |
⊢ ( 𝜑 → ( ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ∈ ( 𝐵 ↾t dom 𝐻 ) ↔ ∃ 𝑒 ∈ 𝐵 ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) ) ) |
48 |
47
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ∈ ( 𝐵 ↾t dom 𝐻 ) ↔ ∃ 𝑒 ∈ 𝐵 ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) ) ) |
49 |
43 48
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ∃ 𝑒 ∈ 𝐵 ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) ) |
50 |
|
imaeq2 |
⊢ ( ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) → ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) = ( ◡ 𝐹 “ ( 𝑒 ∩ dom 𝐻 ) ) ) |
51 |
50
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ∧ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) ) → ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) = ( ◡ 𝐹 “ ( 𝑒 ∩ dom 𝐻 ) ) ) |
52 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → ( 𝑆 ↾t dom 𝐹 ) ∈ V ) |
53 |
2
|
elexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
54 |
|
cnvexg |
⊢ ( 𝐹 ∈ V → ◡ 𝐹 ∈ V ) |
55 |
53 54
|
syl |
⊢ ( 𝜑 → ◡ 𝐹 ∈ V ) |
56 |
|
imaexg |
⊢ ( ◡ 𝐹 ∈ V → ( ◡ 𝐹 “ dom 𝐻 ) ∈ V ) |
57 |
55 56
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ dom 𝐻 ) ∈ V ) |
58 |
57
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → ( ◡ 𝐹 “ dom 𝐻 ) ∈ V ) |
59 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → 𝑆 ∈ SAlg ) |
60 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) |
61 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → 𝑒 ∈ 𝐵 ) |
62 |
|
eqid |
⊢ ( ◡ 𝐹 “ 𝑒 ) = ( ◡ 𝐹 “ 𝑒 ) |
63 |
59 60 9 3 4 61 62
|
smfpimbor1 |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t dom 𝐹 ) ) |
64 |
|
eqid |
⊢ ( ( ◡ 𝐹 “ 𝑒 ) ∩ ( ◡ 𝐹 “ dom 𝐻 ) ) = ( ( ◡ 𝐹 “ 𝑒 ) ∩ ( ◡ 𝐹 “ dom 𝐻 ) ) |
65 |
52 58 63 64
|
elrestd |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → ( ( ◡ 𝐹 “ 𝑒 ) ∩ ( ◡ 𝐹 “ dom 𝐻 ) ) ∈ ( ( 𝑆 ↾t dom 𝐹 ) ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
66 |
|
inpreima |
⊢ ( Fun 𝐹 → ( ◡ 𝐹 “ ( 𝑒 ∩ dom 𝐻 ) ) = ( ( ◡ 𝐹 “ 𝑒 ) ∩ ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
67 |
20 66
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( 𝑒 ∩ dom 𝐻 ) ) = ( ( ◡ 𝐹 “ 𝑒 ) ∩ ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
68 |
67
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → ( ◡ 𝐹 “ ( 𝑒 ∩ dom 𝐻 ) ) = ( ( ◡ 𝐹 “ 𝑒 ) ∩ ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
69 |
2
|
dmexd |
⊢ ( 𝜑 → dom 𝐹 ∈ V ) |
70 |
|
restabs |
⊢ ( ( 𝑆 ∈ SAlg ∧ ( ◡ 𝐹 “ dom 𝐻 ) ⊆ dom 𝐹 ∧ dom 𝐹 ∈ V ) → ( ( 𝑆 ↾t dom 𝐹 ) ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) = ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
71 |
1 8 69 70
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑆 ↾t dom 𝐹 ) ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) = ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
72 |
71
|
eqcomd |
⊢ ( 𝜑 → ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) = ( ( 𝑆 ↾t dom 𝐹 ) ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
73 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) = ( ( 𝑆 ↾t dom 𝐹 ) ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
74 |
68 73
|
eleq12d |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → ( ( ◡ 𝐹 “ ( 𝑒 ∩ dom 𝐻 ) ) ∈ ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ↔ ( ( ◡ 𝐹 “ 𝑒 ) ∩ ( ◡ 𝐹 “ dom 𝐻 ) ) ∈ ( ( 𝑆 ↾t dom 𝐹 ) ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) ) |
75 |
65 74
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ) → ( ◡ 𝐹 “ ( 𝑒 ∩ dom 𝐻 ) ) ∈ ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
76 |
75
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ∧ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) ) → ( ◡ 𝐹 “ ( 𝑒 ∩ dom 𝐻 ) ) ∈ ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
77 |
51 76
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝐵 ∧ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) ) → ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) ∈ ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
78 |
77
|
3exp |
⊢ ( 𝜑 → ( 𝑒 ∈ 𝐵 → ( ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) → ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) ∈ ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) ) ) |
79 |
78
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( 𝑒 ∈ 𝐵 → ( ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) → ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) ∈ ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) ) ) |
80 |
79
|
rexlimdv |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( ∃ 𝑒 ∈ 𝐵 ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) = ( 𝑒 ∩ dom 𝐻 ) → ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) ∈ ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) ) |
81 |
49 80
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → ( ◡ 𝐹 “ ( ◡ 𝐻 “ ( -∞ (,) 𝑎 ) ) ) ∈ ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
82 |
34 81
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ℝ ) → { 𝑥 ∈ ( ◡ 𝐹 “ dom 𝐻 ) ∣ ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑥 ) < 𝑎 } ∈ ( 𝑆 ↾t ( ◡ 𝐹 “ dom 𝐻 ) ) ) |
83 |
6 1 11 23 82
|
issmfd |
⊢ ( 𝜑 → ( 𝐻 ∘ 𝐹 ) ∈ ( SMblFn ‘ 𝑆 ) ) |