Step |
Hyp |
Ref |
Expression |
1 |
|
sitgval.b |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
2 |
|
sitgval.j |
⊢ 𝐽 = ( TopOpen ‘ 𝑊 ) |
3 |
|
sitgval.s |
⊢ 𝑆 = ( sigaGen ‘ 𝐽 ) |
4 |
|
sitgval.0 |
⊢ 0 = ( 0g ‘ 𝑊 ) |
5 |
|
sitgval.x |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
6 |
|
sitgval.h |
⊢ 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) ) |
7 |
|
sitgval.1 |
⊢ ( 𝜑 → 𝑊 ∈ 𝑉 ) |
8 |
|
sitgval.2 |
⊢ ( 𝜑 → 𝑀 ∈ ∪ ran measures ) |
9 |
|
sitgadd.1 |
⊢ ( 𝜑 → 𝑊 ∈ TopSp ) |
10 |
|
sitgadd.2 |
⊢ ( 𝜑 → ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ∈ SLMod ) |
11 |
|
sitgadd.3 |
⊢ ( 𝜑 → 𝐽 ∈ Fre ) |
12 |
|
sitgadd.4 |
⊢ ( 𝜑 → 𝐹 ∈ dom ( 𝑊 sitg 𝑀 ) ) |
13 |
|
sitgadd.5 |
⊢ ( 𝜑 → 𝐺 ∈ dom ( 𝑊 sitg 𝑀 ) ) |
14 |
|
sitgadd.6 |
⊢ ( 𝜑 → ( Scalar ‘ 𝑊 ) ∈ ℝExt ) |
15 |
|
sitgadd.7 |
⊢ + = ( +g ‘ 𝑊 ) |
16 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ∈ SLMod ) |
17 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → 𝜑 ) |
18 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
19 |
18
|
rrhfe |
⊢ ( ( Scalar ‘ 𝑊 ) ∈ ℝExt → ( ℝHom ‘ ( Scalar ‘ 𝑊 ) ) : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
20 |
14 19
|
syl |
⊢ ( 𝜑 → ( ℝHom ‘ ( Scalar ‘ 𝑊 ) ) : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
21 |
6
|
feq1i |
⊢ ( 𝐻 : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ↔ ( ℝHom ‘ ( Scalar ‘ 𝑊 ) ) : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
22 |
20 21
|
sylibr |
⊢ ( 𝜑 → 𝐻 : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
23 |
22
|
ffnd |
⊢ ( 𝜑 → 𝐻 Fn ℝ ) |
24 |
17 23
|
syl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → 𝐻 Fn ℝ ) |
25 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
26 |
25
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( 0 [,) +∞ ) ⊆ ℝ ) |
27 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) |
28 |
27
|
eldifad |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) ) |
29 |
|
xp1st |
⊢ ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) → ( 1st ‘ 𝑝 ) ∈ ran 𝐹 ) |
30 |
28 29
|
syl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( 1st ‘ 𝑝 ) ∈ ran 𝐹 ) |
31 |
|
xp2nd |
⊢ ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) → ( 2nd ‘ 𝑝 ) ∈ ran 𝐺 ) |
32 |
28 31
|
syl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( 2nd ‘ 𝑝 ) ∈ ran 𝐺 ) |
33 |
27
|
eldifbd |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ¬ 𝑝 ∈ { 〈 0 , 0 〉 } ) |
34 |
|
velsn |
⊢ ( 𝑝 ∈ { 〈 0 , 0 〉 } ↔ 𝑝 = 〈 0 , 0 〉 ) |
35 |
34
|
notbii |
⊢ ( ¬ 𝑝 ∈ { 〈 0 , 0 〉 } ↔ ¬ 𝑝 = 〈 0 , 0 〉 ) |
36 |
33 35
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ¬ 𝑝 = 〈 0 , 0 〉 ) |
37 |
|
eqopi |
⊢ ( ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) ∧ ( ( 1st ‘ 𝑝 ) = 0 ∧ ( 2nd ‘ 𝑝 ) = 0 ) ) → 𝑝 = 〈 0 , 0 〉 ) |
38 |
37
|
ex |
⊢ ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) → ( ( ( 1st ‘ 𝑝 ) = 0 ∧ ( 2nd ‘ 𝑝 ) = 0 ) → 𝑝 = 〈 0 , 0 〉 ) ) |
39 |
38
|
con3d |
⊢ ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) → ( ¬ 𝑝 = 〈 0 , 0 〉 → ¬ ( ( 1st ‘ 𝑝 ) = 0 ∧ ( 2nd ‘ 𝑝 ) = 0 ) ) ) |
40 |
39
|
imp |
⊢ ( ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) ∧ ¬ 𝑝 = 〈 0 , 0 〉 ) → ¬ ( ( 1st ‘ 𝑝 ) = 0 ∧ ( 2nd ‘ 𝑝 ) = 0 ) ) |
41 |
28 36 40
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ¬ ( ( 1st ‘ 𝑝 ) = 0 ∧ ( 2nd ‘ 𝑝 ) = 0 ) ) |
42 |
|
ianor |
⊢ ( ¬ ( ( 1st ‘ 𝑝 ) = 0 ∧ ( 2nd ‘ 𝑝 ) = 0 ) ↔ ( ¬ ( 1st ‘ 𝑝 ) = 0 ∨ ¬ ( 2nd ‘ 𝑝 ) = 0 ) ) |
43 |
|
df-ne |
⊢ ( ( 1st ‘ 𝑝 ) ≠ 0 ↔ ¬ ( 1st ‘ 𝑝 ) = 0 ) |
44 |
|
df-ne |
⊢ ( ( 2nd ‘ 𝑝 ) ≠ 0 ↔ ¬ ( 2nd ‘ 𝑝 ) = 0 ) |
45 |
43 44
|
orbi12i |
⊢ ( ( ( 1st ‘ 𝑝 ) ≠ 0 ∨ ( 2nd ‘ 𝑝 ) ≠ 0 ) ↔ ( ¬ ( 1st ‘ 𝑝 ) = 0 ∨ ¬ ( 2nd ‘ 𝑝 ) = 0 ) ) |
46 |
42 45
|
bitr4i |
⊢ ( ¬ ( ( 1st ‘ 𝑝 ) = 0 ∧ ( 2nd ‘ 𝑝 ) = 0 ) ↔ ( ( 1st ‘ 𝑝 ) ≠ 0 ∨ ( 2nd ‘ 𝑝 ) ≠ 0 ) ) |
47 |
41 46
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( ( 1st ‘ 𝑝 ) ≠ 0 ∨ ( 2nd ‘ 𝑝 ) ≠ 0 ) ) |
48 |
1 2 3 4 5 6 7 8 12 13 9 11
|
sibfinima |
⊢ ( ( ( 𝜑 ∧ ( 1st ‘ 𝑝 ) ∈ ran 𝐹 ∧ ( 2nd ‘ 𝑝 ) ∈ ran 𝐺 ) ∧ ( ( 1st ‘ 𝑝 ) ≠ 0 ∨ ( 2nd ‘ 𝑝 ) ≠ 0 ) ) → ( 𝑀 ‘ ( ( ◡ 𝐹 “ { ( 1st ‘ 𝑝 ) } ) ∩ ( ◡ 𝐺 “ { ( 2nd ‘ 𝑝 ) } ) ) ) ∈ ( 0 [,) +∞ ) ) |
49 |
17 30 32 47 48
|
syl31anc |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( 𝑀 ‘ ( ( ◡ 𝐹 “ { ( 1st ‘ 𝑝 ) } ) ∩ ( ◡ 𝐺 “ { ( 2nd ‘ 𝑝 ) } ) ) ) ∈ ( 0 [,) +∞ ) ) |
50 |
|
fnfvima |
⊢ ( ( 𝐻 Fn ℝ ∧ ( 0 [,) +∞ ) ⊆ ℝ ∧ ( 𝑀 ‘ ( ( ◡ 𝐹 “ { ( 1st ‘ 𝑝 ) } ) ∩ ( ◡ 𝐺 “ { ( 2nd ‘ 𝑝 ) } ) ) ) ∈ ( 0 [,) +∞ ) ) → ( 𝐻 ‘ ( 𝑀 ‘ ( ( ◡ 𝐹 “ { ( 1st ‘ 𝑝 ) } ) ∩ ( ◡ 𝐺 “ { ( 2nd ‘ 𝑝 ) } ) ) ) ) ∈ ( 𝐻 “ ( 0 [,) +∞ ) ) ) |
51 |
24 26 49 50
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( 𝐻 ‘ ( 𝑀 ‘ ( ( ◡ 𝐹 “ { ( 1st ‘ 𝑝 ) } ) ∩ ( ◡ 𝐺 “ { ( 2nd ‘ 𝑝 ) } ) ) ) ) ∈ ( 𝐻 “ ( 0 [,) +∞ ) ) ) |
52 |
|
imassrn |
⊢ ( 𝐻 “ ( 0 [,) +∞ ) ) ⊆ ran 𝐻 |
53 |
22
|
frnd |
⊢ ( 𝜑 → ran 𝐻 ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
54 |
52 53
|
sstrid |
⊢ ( 𝜑 → ( 𝐻 “ ( 0 [,) +∞ ) ) ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
55 |
|
eqid |
⊢ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) = ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) |
56 |
55 18
|
ressbas2 |
⊢ ( ( 𝐻 “ ( 0 [,) +∞ ) ) ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) → ( 𝐻 “ ( 0 [,) +∞ ) ) = ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) ) |
57 |
54 56
|
syl |
⊢ ( 𝜑 → ( 𝐻 “ ( 0 [,) +∞ ) ) = ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) ) |
58 |
17 57
|
syl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( 𝐻 “ ( 0 [,) +∞ ) ) = ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) ) |
59 |
51 58
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( 𝐻 ‘ ( 𝑀 ‘ ( ( ◡ 𝐹 “ { ( 1st ‘ 𝑝 ) } ) ∩ ( ◡ 𝐺 “ { ( 2nd ‘ 𝑝 ) } ) ) ) ) ∈ ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) ) |
60 |
1 2 3 4 5 6 7 8 13
|
sibff |
⊢ ( 𝜑 → 𝐺 : ∪ dom 𝑀 ⟶ ∪ 𝐽 ) |
61 |
1 2
|
tpsuni |
⊢ ( 𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽 ) |
62 |
|
feq3 |
⊢ ( 𝐵 = ∪ 𝐽 → ( 𝐺 : ∪ dom 𝑀 ⟶ 𝐵 ↔ 𝐺 : ∪ dom 𝑀 ⟶ ∪ 𝐽 ) ) |
63 |
9 61 62
|
3syl |
⊢ ( 𝜑 → ( 𝐺 : ∪ dom 𝑀 ⟶ 𝐵 ↔ 𝐺 : ∪ dom 𝑀 ⟶ ∪ 𝐽 ) ) |
64 |
60 63
|
mpbird |
⊢ ( 𝜑 → 𝐺 : ∪ dom 𝑀 ⟶ 𝐵 ) |
65 |
64
|
frnd |
⊢ ( 𝜑 → ran 𝐺 ⊆ 𝐵 ) |
66 |
65
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ran 𝐺 ⊆ 𝐵 ) |
67 |
66 32
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( 2nd ‘ 𝑝 ) ∈ 𝐵 ) |
68 |
6
|
fvexi |
⊢ 𝐻 ∈ V |
69 |
|
imaexg |
⊢ ( 𝐻 ∈ V → ( 𝐻 “ ( 0 [,) +∞ ) ) ∈ V ) |
70 |
|
eqid |
⊢ ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) = ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) |
71 |
70 1
|
resvbas |
⊢ ( ( 𝐻 “ ( 0 [,) +∞ ) ) ∈ V → 𝐵 = ( Base ‘ ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) ) |
72 |
68 69 71
|
mp2b |
⊢ 𝐵 = ( Base ‘ ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) |
73 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
74 |
70 73 18
|
resvsca |
⊢ ( ( 𝐻 “ ( 0 [,) +∞ ) ) ∈ V → ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) = ( Scalar ‘ ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) ) |
75 |
68 69 74
|
mp2b |
⊢ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) = ( Scalar ‘ ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) |
76 |
70 5
|
resvvsca |
⊢ ( ( 𝐻 “ ( 0 [,) +∞ ) ) ∈ V → · = ( ·𝑠 ‘ ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) ) |
77 |
68 69 76
|
mp2b |
⊢ · = ( ·𝑠 ‘ ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) |
78 |
|
eqid |
⊢ ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) = ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) |
79 |
72 75 77 78
|
slmdvscl |
⊢ ( ( ( 𝑊 ↾v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ∈ SLMod ∧ ( 𝐻 ‘ ( 𝑀 ‘ ( ( ◡ 𝐹 “ { ( 1st ‘ 𝑝 ) } ) ∩ ( ◡ 𝐺 “ { ( 2nd ‘ 𝑝 ) } ) ) ) ) ∈ ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) ∧ ( 2nd ‘ 𝑝 ) ∈ 𝐵 ) → ( ( 𝐻 ‘ ( 𝑀 ‘ ( ( ◡ 𝐹 “ { ( 1st ‘ 𝑝 ) } ) ∩ ( ◡ 𝐺 “ { ( 2nd ‘ 𝑝 ) } ) ) ) ) · ( 2nd ‘ 𝑝 ) ) ∈ 𝐵 ) |
80 |
16 59 67 79
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { 〈 0 , 0 〉 } ) ) → ( ( 𝐻 ‘ ( 𝑀 ‘ ( ( ◡ 𝐹 “ { ( 1st ‘ 𝑝 ) } ) ∩ ( ◡ 𝐺 “ { ( 2nd ‘ 𝑝 ) } ) ) ) ) · ( 2nd ‘ 𝑝 ) ) ∈ 𝐵 ) |