Step |
Hyp |
Ref |
Expression |
1 |
|
i1fmulc.2 |
⊢ ( 𝜑 → 𝐹 ∈ dom ∫1 ) |
2 |
|
i1fmulc.3 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
3 |
|
reex |
⊢ ℝ ∈ V |
4 |
3
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ℝ ∈ V ) |
5 |
|
i1ff |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 : ℝ ⟶ ℝ ) |
6 |
1 5
|
syl |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ ) |
7 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → 𝐹 : ℝ ⟶ ℝ ) |
8 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → 𝐴 ∈ ℝ ) |
9 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → 0 ∈ ℝ ) |
10 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 0 ) ∧ 𝑥 ∈ ℝ ) → 𝐴 = 0 ) |
11 |
10
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 0 ) ∧ 𝑥 ∈ ℝ ) → ( 𝐴 · 𝑥 ) = ( 0 · 𝑥 ) ) |
12 |
|
mul02lem2 |
⊢ ( 𝑥 ∈ ℝ → ( 0 · 𝑥 ) = 0 ) |
13 |
12
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 0 ) ∧ 𝑥 ∈ ℝ ) → ( 0 · 𝑥 ) = 0 ) |
14 |
11 13
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 0 ) ∧ 𝑥 ∈ ℝ ) → ( 𝐴 · 𝑥 ) = 0 ) |
15 |
4 7 8 9 14
|
caofid2 |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) = ( ℝ × { 0 } ) ) |
16 |
|
i1f0 |
⊢ ( ℝ × { 0 } ) ∈ dom ∫1 |
17 |
15 16
|
eqeltrdi |
⊢ ( ( 𝜑 ∧ 𝐴 = 0 ) → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ dom ∫1 ) |
18 |
|
remulcl |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 · 𝑦 ) ∈ ℝ ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ ) |
20 |
|
fconst6g |
⊢ ( 𝐴 ∈ ℝ → ( ℝ × { 𝐴 } ) : ℝ ⟶ ℝ ) |
21 |
2 20
|
syl |
⊢ ( 𝜑 → ( ℝ × { 𝐴 } ) : ℝ ⟶ ℝ ) |
22 |
3
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
23 |
|
inidm |
⊢ ( ℝ ∩ ℝ ) = ℝ |
24 |
19 21 6 22 22 23
|
off |
⊢ ( 𝜑 → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) : ℝ ⟶ ℝ ) |
25 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) : ℝ ⟶ ℝ ) |
26 |
|
i1frn |
⊢ ( 𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin ) |
27 |
1 26
|
syl |
⊢ ( 𝜑 → ran 𝐹 ∈ Fin ) |
28 |
|
ovex |
⊢ ( 𝐴 · 𝑦 ) ∈ V |
29 |
|
eqid |
⊢ ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) = ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) |
30 |
28 29
|
fnmpti |
⊢ ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) Fn ran 𝐹 |
31 |
|
dffn4 |
⊢ ( ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) Fn ran 𝐹 ↔ ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) : ran 𝐹 –onto→ ran ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) ) |
32 |
30 31
|
mpbi |
⊢ ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) : ran 𝐹 –onto→ ran ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) |
33 |
|
fofi |
⊢ ( ( ran 𝐹 ∈ Fin ∧ ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) : ran 𝐹 –onto→ ran ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) ) → ran ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) ∈ Fin ) |
34 |
27 32 33
|
sylancl |
⊢ ( 𝜑 → ran ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) ∈ Fin ) |
35 |
|
id |
⊢ ( 𝑤 ∈ ran 𝐹 → 𝑤 ∈ ran 𝐹 ) |
36 |
|
elsni |
⊢ ( 𝑥 ∈ { 𝐴 } → 𝑥 = 𝐴 ) |
37 |
36
|
oveq1d |
⊢ ( 𝑥 ∈ { 𝐴 } → ( 𝑥 · 𝑤 ) = ( 𝐴 · 𝑤 ) ) |
38 |
|
oveq2 |
⊢ ( 𝑦 = 𝑤 → ( 𝐴 · 𝑦 ) = ( 𝐴 · 𝑤 ) ) |
39 |
38
|
rspceeqv |
⊢ ( ( 𝑤 ∈ ran 𝐹 ∧ ( 𝑥 · 𝑤 ) = ( 𝐴 · 𝑤 ) ) → ∃ 𝑦 ∈ ran 𝐹 ( 𝑥 · 𝑤 ) = ( 𝐴 · 𝑦 ) ) |
40 |
35 37 39
|
syl2anr |
⊢ ( ( 𝑥 ∈ { 𝐴 } ∧ 𝑤 ∈ ran 𝐹 ) → ∃ 𝑦 ∈ ran 𝐹 ( 𝑥 · 𝑤 ) = ( 𝐴 · 𝑦 ) ) |
41 |
|
ovex |
⊢ ( 𝑥 · 𝑤 ) ∈ V |
42 |
|
eqeq1 |
⊢ ( 𝑧 = ( 𝑥 · 𝑤 ) → ( 𝑧 = ( 𝐴 · 𝑦 ) ↔ ( 𝑥 · 𝑤 ) = ( 𝐴 · 𝑦 ) ) ) |
43 |
42
|
rexbidv |
⊢ ( 𝑧 = ( 𝑥 · 𝑤 ) → ( ∃ 𝑦 ∈ ran 𝐹 𝑧 = ( 𝐴 · 𝑦 ) ↔ ∃ 𝑦 ∈ ran 𝐹 ( 𝑥 · 𝑤 ) = ( 𝐴 · 𝑦 ) ) ) |
44 |
41 43
|
elab |
⊢ ( ( 𝑥 · 𝑤 ) ∈ { 𝑧 ∣ ∃ 𝑦 ∈ ran 𝐹 𝑧 = ( 𝐴 · 𝑦 ) } ↔ ∃ 𝑦 ∈ ran 𝐹 ( 𝑥 · 𝑤 ) = ( 𝐴 · 𝑦 ) ) |
45 |
40 44
|
sylibr |
⊢ ( ( 𝑥 ∈ { 𝐴 } ∧ 𝑤 ∈ ran 𝐹 ) → ( 𝑥 · 𝑤 ) ∈ { 𝑧 ∣ ∃ 𝑦 ∈ ran 𝐹 𝑧 = ( 𝐴 · 𝑦 ) } ) |
46 |
45
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ { 𝐴 } ∧ 𝑤 ∈ ran 𝐹 ) ) → ( 𝑥 · 𝑤 ) ∈ { 𝑧 ∣ ∃ 𝑦 ∈ ran 𝐹 𝑧 = ( 𝐴 · 𝑦 ) } ) |
47 |
|
fconstg |
⊢ ( 𝐴 ∈ ℝ → ( ℝ × { 𝐴 } ) : ℝ ⟶ { 𝐴 } ) |
48 |
2 47
|
syl |
⊢ ( 𝜑 → ( ℝ × { 𝐴 } ) : ℝ ⟶ { 𝐴 } ) |
49 |
6
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ℝ ) |
50 |
|
dffn3 |
⊢ ( 𝐹 Fn ℝ ↔ 𝐹 : ℝ ⟶ ran 𝐹 ) |
51 |
49 50
|
sylib |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ran 𝐹 ) |
52 |
46 48 51 22 22 23
|
off |
⊢ ( 𝜑 → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) : ℝ ⟶ { 𝑧 ∣ ∃ 𝑦 ∈ ran 𝐹 𝑧 = ( 𝐴 · 𝑦 ) } ) |
53 |
52
|
frnd |
⊢ ( 𝜑 → ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ⊆ { 𝑧 ∣ ∃ 𝑦 ∈ ran 𝐹 𝑧 = ( 𝐴 · 𝑦 ) } ) |
54 |
29
|
rnmpt |
⊢ ran ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) = { 𝑧 ∣ ∃ 𝑦 ∈ ran 𝐹 𝑧 = ( 𝐴 · 𝑦 ) } |
55 |
53 54
|
sseqtrrdi |
⊢ ( 𝜑 → ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ⊆ ran ( 𝑦 ∈ ran 𝐹 ↦ ( 𝐴 · 𝑦 ) ) ) |
56 |
34 55
|
ssfid |
⊢ ( 𝜑 → ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ Fin ) |
57 |
56
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ Fin ) |
58 |
24
|
frnd |
⊢ ( 𝜑 → ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ⊆ ℝ ) |
59 |
58
|
ssdifssd |
⊢ ( 𝜑 → ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ⊆ ℝ ) |
60 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ⊆ ℝ ) |
61 |
60
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑦 ∈ ℝ ) |
62 |
1 2
|
i1fmulclem |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ℝ ) → ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑦 } ) = ( ◡ 𝐹 “ { ( 𝑦 / 𝐴 ) } ) ) |
63 |
61 62
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑦 } ) = ( ◡ 𝐹 “ { ( 𝑦 / 𝐴 ) } ) ) |
64 |
|
i1fima |
⊢ ( 𝐹 ∈ dom ∫1 → ( ◡ 𝐹 “ { ( 𝑦 / 𝐴 ) } ) ∈ dom vol ) |
65 |
1 64
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ { ( 𝑦 / 𝐴 ) } ) ∈ dom vol ) |
66 |
65
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( ◡ 𝐹 “ { ( 𝑦 / 𝐴 ) } ) ∈ dom vol ) |
67 |
63 66
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑦 } ) ∈ dom vol ) |
68 |
63
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑦 } ) ) = ( vol ‘ ( ◡ 𝐹 “ { ( 𝑦 / 𝐴 ) } ) ) ) |
69 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐹 ∈ dom ∫1 ) |
70 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐴 ∈ ℝ ) |
71 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐴 ≠ 0 ) |
72 |
61 70 71
|
redivcld |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑦 / 𝐴 ) ∈ ℝ ) |
73 |
61
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑦 ∈ ℂ ) |
74 |
70
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝐴 ∈ ℂ ) |
75 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) → 𝑦 ≠ 0 ) |
76 |
75
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → 𝑦 ≠ 0 ) |
77 |
73 74 76 71
|
divne0d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑦 / 𝐴 ) ≠ 0 ) |
78 |
|
eldifsn |
⊢ ( ( 𝑦 / 𝐴 ) ∈ ( ℝ ∖ { 0 } ) ↔ ( ( 𝑦 / 𝐴 ) ∈ ℝ ∧ ( 𝑦 / 𝐴 ) ≠ 0 ) ) |
79 |
72 77 78
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( 𝑦 / 𝐴 ) ∈ ( ℝ ∖ { 0 } ) ) |
80 |
|
i1fima2sn |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ ( 𝑦 / 𝐴 ) ∈ ( ℝ ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { ( 𝑦 / 𝐴 ) } ) ) ∈ ℝ ) |
81 |
69 79 80
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( vol ‘ ( ◡ 𝐹 “ { ( 𝑦 / 𝐴 ) } ) ) ∈ ℝ ) |
82 |
68 81
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 0 ) ∧ 𝑦 ∈ ( ran ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∖ { 0 } ) ) → ( vol ‘ ( ◡ ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) “ { 𝑦 } ) ) ∈ ℝ ) |
83 |
25 57 67 82
|
i1fd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ dom ∫1 ) |
84 |
17 83
|
pm2.61dane |
⊢ ( 𝜑 → ( ( ℝ × { 𝐴 } ) ∘f · 𝐹 ) ∈ dom ∫1 ) |