Step |
Hyp |
Ref |
Expression |
1 |
|
itg10a.1 |
⊢ ( 𝜑 → 𝐹 ∈ dom ∫1 ) |
2 |
|
itg10a.2 |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
3 |
|
itg10a.3 |
⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) = 0 ) |
4 |
|
itg10a.4 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ℝ ∖ 𝐴 ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
5 |
|
itg1val |
⊢ ( 𝐹 ∈ dom ∫1 → ( ∫1 ‘ 𝐹 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
6 |
1 5
|
syl |
⊢ ( 𝜑 → ( ∫1 ‘ 𝐹 ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) ) |
7 |
|
i1ff |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 : ℝ ⟶ ℝ ) |
8 |
1 7
|
syl |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ ) |
9 |
8
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ℝ ) |
10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝐹 Fn ℝ ) |
11 |
|
fniniseg |
⊢ ( 𝐹 Fn ℝ → ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑘 } ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) ) |
12 |
10 11
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑘 } ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) ) |
13 |
|
eldifsni |
⊢ ( 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) → 𝑘 ≠ 0 ) |
14 |
13
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) → 𝑘 ≠ 0 ) |
15 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) → 𝑥 ∈ ℝ ) |
16 |
|
eldif |
⊢ ( 𝑥 ∈ ( ℝ ∖ 𝐴 ) ↔ ( 𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴 ) ) |
17 |
|
simplrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) ∧ 𝑥 ∈ ( ℝ ∖ 𝐴 ) ) → ( 𝐹 ‘ 𝑥 ) = 𝑘 ) |
18 |
4
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) ∧ 𝑥 ∈ ( ℝ ∖ 𝐴 ) ) → ( 𝐹 ‘ 𝑥 ) = 0 ) |
19 |
17 18
|
eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) ∧ 𝑥 ∈ ( ℝ ∖ 𝐴 ) ) → 𝑘 = 0 ) |
20 |
19
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) → ( 𝑥 ∈ ( ℝ ∖ 𝐴 ) → 𝑘 = 0 ) ) |
21 |
16 20
|
syl5bir |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) → ( ( 𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴 ) → 𝑘 = 0 ) ) |
22 |
15 21
|
mpand |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) → ( ¬ 𝑥 ∈ 𝐴 → 𝑘 = 0 ) ) |
23 |
22
|
necon1ad |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) → ( 𝑘 ≠ 0 → 𝑥 ∈ 𝐴 ) ) |
24 |
14 23
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) ) → 𝑥 ∈ 𝐴 ) |
25 |
24
|
ex |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ 𝑥 ) = 𝑘 ) → 𝑥 ∈ 𝐴 ) ) |
26 |
12 25
|
sylbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑘 } ) → 𝑥 ∈ 𝐴 ) ) |
27 |
26
|
ssrdv |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ◡ 𝐹 “ { 𝑘 } ) ⊆ 𝐴 ) |
28 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝐴 ⊆ ℝ ) |
29 |
27 28
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ◡ 𝐹 “ { 𝑘 } ) ⊆ ℝ ) |
30 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol* ‘ 𝐴 ) = 0 ) |
31 |
|
ovolssnul |
⊢ ( ( ( ◡ 𝐹 “ { 𝑘 } ) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ ( vol* ‘ 𝐴 ) = 0 ) → ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = 0 ) |
32 |
27 28 30 31
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = 0 ) |
33 |
|
nulmbl |
⊢ ( ( ( ◡ 𝐹 “ { 𝑘 } ) ⊆ ℝ ∧ ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = 0 ) → ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol ) |
34 |
29 32 33
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol ) |
35 |
|
mblvol |
⊢ ( ( ◡ 𝐹 “ { 𝑘 } ) ∈ dom vol → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
36 |
34 35
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = ( vol* ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) |
37 |
36 32
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) = 0 ) |
38 |
37
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) = ( 𝑘 · 0 ) ) |
39 |
8
|
frnd |
⊢ ( 𝜑 → ran 𝐹 ⊆ ℝ ) |
40 |
39
|
ssdifssd |
⊢ ( 𝜑 → ( ran 𝐹 ∖ { 0 } ) ⊆ ℝ ) |
41 |
40
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝑘 ∈ ℝ ) |
42 |
41
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → 𝑘 ∈ ℂ ) |
43 |
42
|
mul01d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑘 · 0 ) = 0 ) |
44 |
38 43
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) = 0 ) |
45 |
44
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) = Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) 0 ) |
46 |
|
i1frn |
⊢ ( 𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin ) |
47 |
1 46
|
syl |
⊢ ( 𝜑 → ran 𝐹 ∈ Fin ) |
48 |
|
difss |
⊢ ( ran 𝐹 ∖ { 0 } ) ⊆ ran 𝐹 |
49 |
|
ssfi |
⊢ ( ( ran 𝐹 ∈ Fin ∧ ( ran 𝐹 ∖ { 0 } ) ⊆ ran 𝐹 ) → ( ran 𝐹 ∖ { 0 } ) ∈ Fin ) |
50 |
47 48 49
|
sylancl |
⊢ ( 𝜑 → ( ran 𝐹 ∖ { 0 } ) ∈ Fin ) |
51 |
50
|
olcd |
⊢ ( 𝜑 → ( ( ran 𝐹 ∖ { 0 } ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( ran 𝐹 ∖ { 0 } ) ∈ Fin ) ) |
52 |
|
sumz |
⊢ ( ( ( ran 𝐹 ∖ { 0 } ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( ran 𝐹 ∖ { 0 } ) ∈ Fin ) → Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) 0 = 0 ) |
53 |
51 52
|
syl |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) 0 = 0 ) |
54 |
45 53
|
eqtrd |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑘 · ( vol ‘ ( ◡ 𝐹 “ { 𝑘 } ) ) ) = 0 ) |
55 |
6 54
|
eqtrd |
⊢ ( 𝜑 → ( ∫1 ‘ 𝐹 ) = 0 ) |