Step |
Hyp |
Ref |
Expression |
1 |
|
i1fima |
⊢ ( 𝐹 ∈ dom ∫1 → ( ◡ 𝐹 “ 𝐴 ) ∈ dom vol ) |
2 |
1
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( ◡ 𝐹 “ 𝐴 ) ∈ dom vol ) |
3 |
|
mblvol |
⊢ ( ( ◡ 𝐹 “ 𝐴 ) ∈ dom vol → ( vol ‘ ( ◡ 𝐹 “ 𝐴 ) ) = ( vol* ‘ ( ◡ 𝐹 “ 𝐴 ) ) ) |
4 |
2 3
|
syl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( vol ‘ ( ◡ 𝐹 “ 𝐴 ) ) = ( vol* ‘ ( ◡ 𝐹 “ 𝐴 ) ) ) |
5 |
|
i1ff |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 : ℝ ⟶ ℝ ) |
6 |
5
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → 𝐹 : ℝ ⟶ ℝ ) |
7 |
|
ffun |
⊢ ( 𝐹 : ℝ ⟶ ℝ → Fun 𝐹 ) |
8 |
|
inpreima |
⊢ ( Fun 𝐹 → ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) = ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) ) |
9 |
6 7 8
|
3syl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) = ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) ) |
10 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ 𝐴 ) ⊆ dom 𝐹 |
11 |
|
cnvimarndm |
⊢ ( ◡ 𝐹 “ ran 𝐹 ) = dom 𝐹 |
12 |
10 11
|
sseqtrri |
⊢ ( ◡ 𝐹 “ 𝐴 ) ⊆ ( ◡ 𝐹 “ ran 𝐹 ) |
13 |
|
df-ss |
⊢ ( ( ◡ 𝐹 “ 𝐴 ) ⊆ ( ◡ 𝐹 “ ran 𝐹 ) ↔ ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) = ( ◡ 𝐹 “ 𝐴 ) ) |
14 |
12 13
|
mpbi |
⊢ ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) = ( ◡ 𝐹 “ 𝐴 ) |
15 |
9 14
|
eqtr2di |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( ◡ 𝐹 “ 𝐴 ) = ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) ) |
16 |
|
elinel1 |
⊢ ( 0 ∈ ( 𝐴 ∩ ran 𝐹 ) → 0 ∈ 𝐴 ) |
17 |
16
|
con3i |
⊢ ( ¬ 0 ∈ 𝐴 → ¬ 0 ∈ ( 𝐴 ∩ ran 𝐹 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ¬ 0 ∈ ( 𝐴 ∩ ran 𝐹 ) ) |
19 |
|
disjsn |
⊢ ( ( ( 𝐴 ∩ ran 𝐹 ) ∩ { 0 } ) = ∅ ↔ ¬ 0 ∈ ( 𝐴 ∩ ran 𝐹 ) ) |
20 |
18 19
|
sylibr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( ( 𝐴 ∩ ran 𝐹 ) ∩ { 0 } ) = ∅ ) |
21 |
|
inss2 |
⊢ ( 𝐴 ∩ ran 𝐹 ) ⊆ ran 𝐹 |
22 |
5
|
frnd |
⊢ ( 𝐹 ∈ dom ∫1 → ran 𝐹 ⊆ ℝ ) |
23 |
21 22
|
sstrid |
⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐴 ∩ ran 𝐹 ) ⊆ ℝ ) |
24 |
23
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( 𝐴 ∩ ran 𝐹 ) ⊆ ℝ ) |
25 |
|
reldisj |
⊢ ( ( 𝐴 ∩ ran 𝐹 ) ⊆ ℝ → ( ( ( 𝐴 ∩ ran 𝐹 ) ∩ { 0 } ) = ∅ ↔ ( 𝐴 ∩ ran 𝐹 ) ⊆ ( ℝ ∖ { 0 } ) ) ) |
26 |
24 25
|
syl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( ( ( 𝐴 ∩ ran 𝐹 ) ∩ { 0 } ) = ∅ ↔ ( 𝐴 ∩ ran 𝐹 ) ⊆ ( ℝ ∖ { 0 } ) ) ) |
27 |
20 26
|
mpbid |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( 𝐴 ∩ ran 𝐹 ) ⊆ ( ℝ ∖ { 0 } ) ) |
28 |
|
imass2 |
⊢ ( ( 𝐴 ∩ ran 𝐹 ) ⊆ ( ℝ ∖ { 0 } ) → ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) ⊆ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) |
29 |
27 28
|
syl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) ⊆ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) |
30 |
15 29
|
eqsstrd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( ◡ 𝐹 “ 𝐴 ) ⊆ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) |
31 |
|
i1fima |
⊢ ( 𝐹 ∈ dom ∫1 → ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ∈ dom vol ) |
32 |
31
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ∈ dom vol ) |
33 |
|
mblss |
⊢ ( ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ∈ dom vol → ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ⊆ ℝ ) |
34 |
32 33
|
syl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ⊆ ℝ ) |
35 |
|
mblvol |
⊢ ( ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ∈ dom vol → ( vol ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) = ( vol* ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ) |
36 |
32 35
|
syl |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( vol ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) = ( vol* ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ) |
37 |
|
isi1f |
⊢ ( 𝐹 ∈ dom ∫1 ↔ ( 𝐹 ∈ MblFn ∧ ( 𝐹 : ℝ ⟶ ℝ ∧ ran 𝐹 ∈ Fin ∧ ( vol ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ∈ ℝ ) ) ) |
38 |
37
|
simprbi |
⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐹 : ℝ ⟶ ℝ ∧ ran 𝐹 ∈ Fin ∧ ( vol ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ∈ ℝ ) ) |
39 |
38
|
simp3d |
⊢ ( 𝐹 ∈ dom ∫1 → ( vol ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ∈ ℝ ) |
40 |
39
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( vol ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ∈ ℝ ) |
41 |
36 40
|
eqeltrrd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( vol* ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ∈ ℝ ) |
42 |
|
ovolsscl |
⊢ ( ( ( ◡ 𝐹 “ 𝐴 ) ⊆ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ∧ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ⊆ ℝ ∧ ( vol* ‘ ( ◡ 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ∈ ℝ ) → ( vol* ‘ ( ◡ 𝐹 “ 𝐴 ) ) ∈ ℝ ) |
43 |
30 34 41 42
|
syl3anc |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( vol* ‘ ( ◡ 𝐹 “ 𝐴 ) ) ∈ ℝ ) |
44 |
4 43
|
eqeltrd |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴 ) → ( vol ‘ ( ◡ 𝐹 “ 𝐴 ) ) ∈ ℝ ) |