Step |
Hyp |
Ref |
Expression |
1 |
|
i1fadd.1 |
⊢ ( 𝜑 → 𝐹 ∈ dom ∫1 ) |
2 |
|
i1fadd.2 |
⊢ ( 𝜑 → 𝐺 ∈ dom ∫1 ) |
3 |
|
itg1add.3 |
⊢ 𝐼 = ( 𝑖 ∈ ℝ , 𝑗 ∈ ℝ ↦ if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) ) |
4 |
|
iffalse |
⊢ ( ¬ ( 𝑖 = 0 ∧ 𝑗 = 0 ) → if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) = ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) |
5 |
4
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ ¬ ( 𝑖 = 0 ∧ 𝑗 = 0 ) ) → if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) = ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) |
6 |
|
i1fima |
⊢ ( 𝐹 ∈ dom ∫1 → ( ◡ 𝐹 “ { 𝑖 } ) ∈ dom vol ) |
7 |
1 6
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ { 𝑖 } ) ∈ dom vol ) |
8 |
|
i1fima |
⊢ ( 𝐺 ∈ dom ∫1 → ( ◡ 𝐺 “ { 𝑗 } ) ∈ dom vol ) |
9 |
2 8
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐺 “ { 𝑗 } ) ∈ dom vol ) |
10 |
|
inmbl |
⊢ ( ( ( ◡ 𝐹 “ { 𝑖 } ) ∈ dom vol ∧ ( ◡ 𝐺 “ { 𝑗 } ) ∈ dom vol ) → ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ∈ dom vol ) |
11 |
7 9 10
|
syl2anc |
⊢ ( 𝜑 → ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ∈ dom vol ) |
12 |
11
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ ¬ ( 𝑖 = 0 ∧ 𝑗 = 0 ) ) → ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ∈ dom vol ) |
13 |
|
mblvol |
⊢ ( ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ∈ dom vol → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) = ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) |
14 |
12 13
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ ¬ ( 𝑖 = 0 ∧ 𝑗 = 0 ) ) → ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) = ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) |
15 |
5 14
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ ¬ ( 𝑖 = 0 ∧ 𝑗 = 0 ) ) → if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) = ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) |
16 |
|
neorian |
⊢ ( ( 𝑖 ≠ 0 ∨ 𝑗 ≠ 0 ) ↔ ¬ ( 𝑖 = 0 ∧ 𝑗 = 0 ) ) |
17 |
|
inss1 |
⊢ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ⊆ ( ◡ 𝐹 “ { 𝑖 } ) |
18 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → ( ◡ 𝐹 “ { 𝑖 } ) ∈ dom vol ) |
19 |
|
mblss |
⊢ ( ( ◡ 𝐹 “ { 𝑖 } ) ∈ dom vol → ( ◡ 𝐹 “ { 𝑖 } ) ⊆ ℝ ) |
20 |
18 19
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → ( ◡ 𝐹 “ { 𝑖 } ) ⊆ ℝ ) |
21 |
|
mblvol |
⊢ ( ( ◡ 𝐹 “ { 𝑖 } ) ∈ dom vol → ( vol ‘ ( ◡ 𝐹 “ { 𝑖 } ) ) = ( vol* ‘ ( ◡ 𝐹 “ { 𝑖 } ) ) ) |
22 |
18 21
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑖 } ) ) = ( vol* ‘ ( ◡ 𝐹 “ { 𝑖 } ) ) ) |
23 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → 𝐹 ∈ dom ∫1 ) |
24 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → 𝑖 ∈ ℝ ) |
25 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → 𝑖 ≠ 0 ) |
26 |
|
eldifsn |
⊢ ( 𝑖 ∈ ( ℝ ∖ { 0 } ) ↔ ( 𝑖 ∈ ℝ ∧ 𝑖 ≠ 0 ) ) |
27 |
24 25 26
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → 𝑖 ∈ ( ℝ ∖ { 0 } ) ) |
28 |
|
i1fima2sn |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑖 ∈ ( ℝ ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑖 } ) ) ∈ ℝ ) |
29 |
23 27 28
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → ( vol ‘ ( ◡ 𝐹 “ { 𝑖 } ) ) ∈ ℝ ) |
30 |
22 29
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → ( vol* ‘ ( ◡ 𝐹 “ { 𝑖 } ) ) ∈ ℝ ) |
31 |
|
ovolsscl |
⊢ ( ( ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ⊆ ( ◡ 𝐹 “ { 𝑖 } ) ∧ ( ◡ 𝐹 “ { 𝑖 } ) ⊆ ℝ ∧ ( vol* ‘ ( ◡ 𝐹 “ { 𝑖 } ) ) ∈ ℝ ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ∈ ℝ ) |
32 |
17 20 30 31
|
mp3an2i |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑖 ≠ 0 ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ∈ ℝ ) |
33 |
|
inss2 |
⊢ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ⊆ ( ◡ 𝐺 “ { 𝑗 } ) |
34 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) → 𝐺 ∈ dom ∫1 ) |
35 |
34 8
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) → ( ◡ 𝐺 “ { 𝑗 } ) ∈ dom vol ) |
36 |
35
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → ( ◡ 𝐺 “ { 𝑗 } ) ∈ dom vol ) |
37 |
|
mblss |
⊢ ( ( ◡ 𝐺 “ { 𝑗 } ) ∈ dom vol → ( ◡ 𝐺 “ { 𝑗 } ) ⊆ ℝ ) |
38 |
36 37
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → ( ◡ 𝐺 “ { 𝑗 } ) ⊆ ℝ ) |
39 |
|
mblvol |
⊢ ( ( ◡ 𝐺 “ { 𝑗 } ) ∈ dom vol → ( vol ‘ ( ◡ 𝐺 “ { 𝑗 } ) ) = ( vol* ‘ ( ◡ 𝐺 “ { 𝑗 } ) ) ) |
40 |
36 39
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → ( vol ‘ ( ◡ 𝐺 “ { 𝑗 } ) ) = ( vol* ‘ ( ◡ 𝐺 “ { 𝑗 } ) ) ) |
41 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → 𝐺 ∈ dom ∫1 ) |
42 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → 𝑗 ∈ ℝ ) |
43 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → 𝑗 ≠ 0 ) |
44 |
|
eldifsn |
⊢ ( 𝑗 ∈ ( ℝ ∖ { 0 } ) ↔ ( 𝑗 ∈ ℝ ∧ 𝑗 ≠ 0 ) ) |
45 |
42 43 44
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → 𝑗 ∈ ( ℝ ∖ { 0 } ) ) |
46 |
|
i1fima2sn |
⊢ ( ( 𝐺 ∈ dom ∫1 ∧ 𝑗 ∈ ( ℝ ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐺 “ { 𝑗 } ) ) ∈ ℝ ) |
47 |
41 45 46
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → ( vol ‘ ( ◡ 𝐺 “ { 𝑗 } ) ) ∈ ℝ ) |
48 |
40 47
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → ( vol* ‘ ( ◡ 𝐺 “ { 𝑗 } ) ) ∈ ℝ ) |
49 |
|
ovolsscl |
⊢ ( ( ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ⊆ ( ◡ 𝐺 “ { 𝑗 } ) ∧ ( ◡ 𝐺 “ { 𝑗 } ) ⊆ ℝ ∧ ( vol* ‘ ( ◡ 𝐺 “ { 𝑗 } ) ) ∈ ℝ ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ∈ ℝ ) |
50 |
33 38 48 49
|
mp3an2i |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ 𝑗 ≠ 0 ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ∈ ℝ ) |
51 |
32 50
|
jaodan |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ ( 𝑖 ≠ 0 ∨ 𝑗 ≠ 0 ) ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ∈ ℝ ) |
52 |
16 51
|
sylan2br |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ ¬ ( 𝑖 = 0 ∧ 𝑗 = 0 ) ) → ( vol* ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ∈ ℝ ) |
53 |
15 52
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) ∧ ¬ ( 𝑖 = 0 ∧ 𝑗 = 0 ) ) → if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) ∈ ℝ ) |
54 |
53
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) → ( ¬ ( 𝑖 = 0 ∧ 𝑗 = 0 ) → if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) ∈ ℝ ) ) |
55 |
|
iftrue |
⊢ ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) → if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) = 0 ) |
56 |
|
0re |
⊢ 0 ∈ ℝ |
57 |
55 56
|
eqeltrdi |
⊢ ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) → if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) ∈ ℝ ) |
58 |
54 57
|
pm2.61d2 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ ) ) → if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) ∈ ℝ ) |
59 |
58
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ℝ ∀ 𝑗 ∈ ℝ if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) ∈ ℝ ) |
60 |
3
|
fmpo |
⊢ ( ∀ 𝑖 ∈ ℝ ∀ 𝑗 ∈ ℝ if ( ( 𝑖 = 0 ∧ 𝑗 = 0 ) , 0 , ( vol ‘ ( ( ◡ 𝐹 “ { 𝑖 } ) ∩ ( ◡ 𝐺 “ { 𝑗 } ) ) ) ) ∈ ℝ ↔ 𝐼 : ( ℝ × ℝ ) ⟶ ℝ ) |
61 |
59 60
|
sylib |
⊢ ( 𝜑 → 𝐼 : ( ℝ × ℝ ) ⟶ ℝ ) |