Step |
Hyp |
Ref |
Expression |
1 |
|
dvdivcncf.s |
⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } ) |
2 |
|
dvdivcncf.f |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℂ ) |
3 |
|
dvdivcncf.g |
⊢ ( 𝜑 → 𝐺 : 𝑋 ⟶ ( ℂ ∖ { 0 } ) ) |
4 |
|
dvdivcncf.fdv |
⊢ ( 𝜑 → ( 𝑆 D 𝐹 ) ∈ ( 𝑋 –cn→ ℂ ) ) |
5 |
|
dvdivcncf.gdv |
⊢ ( 𝜑 → ( 𝑆 D 𝐺 ) ∈ ( 𝑋 –cn→ ℂ ) ) |
6 |
|
cncff |
⊢ ( ( 𝑆 D 𝐹 ) ∈ ( 𝑋 –cn→ ℂ ) → ( 𝑆 D 𝐹 ) : 𝑋 ⟶ ℂ ) |
7 |
|
fdm |
⊢ ( ( 𝑆 D 𝐹 ) : 𝑋 ⟶ ℂ → dom ( 𝑆 D 𝐹 ) = 𝑋 ) |
8 |
4 6 7
|
3syl |
⊢ ( 𝜑 → dom ( 𝑆 D 𝐹 ) = 𝑋 ) |
9 |
|
cncff |
⊢ ( ( 𝑆 D 𝐺 ) ∈ ( 𝑋 –cn→ ℂ ) → ( 𝑆 D 𝐺 ) : 𝑋 ⟶ ℂ ) |
10 |
|
fdm |
⊢ ( ( 𝑆 D 𝐺 ) : 𝑋 ⟶ ℂ → dom ( 𝑆 D 𝐺 ) = 𝑋 ) |
11 |
5 9 10
|
3syl |
⊢ ( 𝜑 → dom ( 𝑆 D 𝐺 ) = 𝑋 ) |
12 |
1 2 3 8 11
|
dvdivf |
⊢ ( 𝜑 → ( 𝑆 D ( 𝐹 ∘f / 𝐺 ) ) = ( ( ( ( 𝑆 D 𝐹 ) ∘f · 𝐺 ) ∘f − ( ( 𝑆 D 𝐺 ) ∘f · 𝐹 ) ) ∘f / ( 𝐺 ∘f · 𝐺 ) ) ) |
13 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
14 |
|
sseq1 |
⊢ ( 𝑆 = ℝ → ( 𝑆 ⊆ ℂ ↔ ℝ ⊆ ℂ ) ) |
15 |
13 14
|
mpbiri |
⊢ ( 𝑆 = ℝ → 𝑆 ⊆ ℂ ) |
16 |
|
eqimss |
⊢ ( 𝑆 = ℂ → 𝑆 ⊆ ℂ ) |
17 |
15 16
|
pm3.2i |
⊢ ( ( 𝑆 = ℝ → 𝑆 ⊆ ℂ ) ∧ ( 𝑆 = ℂ → 𝑆 ⊆ ℂ ) ) |
18 |
|
elpri |
⊢ ( 𝑆 ∈ { ℝ , ℂ } → ( 𝑆 = ℝ ∨ 𝑆 = ℂ ) ) |
19 |
1 18
|
syl |
⊢ ( 𝜑 → ( 𝑆 = ℝ ∨ 𝑆 = ℂ ) ) |
20 |
|
pm3.44 |
⊢ ( ( ( 𝑆 = ℝ → 𝑆 ⊆ ℂ ) ∧ ( 𝑆 = ℂ → 𝑆 ⊆ ℂ ) ) → ( ( 𝑆 = ℝ ∨ 𝑆 = ℂ ) → 𝑆 ⊆ ℂ ) ) |
21 |
17 19 20
|
mpsyl |
⊢ ( 𝜑 → 𝑆 ⊆ ℂ ) |
22 |
|
difssd |
⊢ ( 𝜑 → ( ℂ ∖ { 0 } ) ⊆ ℂ ) |
23 |
3 22
|
fssd |
⊢ ( 𝜑 → 𝐺 : 𝑋 ⟶ ℂ ) |
24 |
|
dvbsss |
⊢ dom ( 𝑆 D 𝐹 ) ⊆ 𝑆 |
25 |
8 24
|
eqsstrrdi |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑆 ) |
26 |
|
dvcn |
⊢ ( ( ( 𝑆 ⊆ ℂ ∧ 𝐺 : 𝑋 ⟶ ℂ ∧ 𝑋 ⊆ 𝑆 ) ∧ dom ( 𝑆 D 𝐺 ) = 𝑋 ) → 𝐺 ∈ ( 𝑋 –cn→ ℂ ) ) |
27 |
21 23 25 11 26
|
syl31anc |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝑋 –cn→ ℂ ) ) |
28 |
4 27
|
mulcncff |
⊢ ( 𝜑 → ( ( 𝑆 D 𝐹 ) ∘f · 𝐺 ) ∈ ( 𝑋 –cn→ ℂ ) ) |
29 |
|
dvcn |
⊢ ( ( ( 𝑆 ⊆ ℂ ∧ 𝐹 : 𝑋 ⟶ ℂ ∧ 𝑋 ⊆ 𝑆 ) ∧ dom ( 𝑆 D 𝐹 ) = 𝑋 ) → 𝐹 ∈ ( 𝑋 –cn→ ℂ ) ) |
30 |
21 2 25 8 29
|
syl31anc |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 –cn→ ℂ ) ) |
31 |
5 30
|
mulcncff |
⊢ ( 𝜑 → ( ( 𝑆 D 𝐺 ) ∘f · 𝐹 ) ∈ ( 𝑋 –cn→ ℂ ) ) |
32 |
28 31
|
subcncff |
⊢ ( 𝜑 → ( ( ( 𝑆 D 𝐹 ) ∘f · 𝐺 ) ∘f − ( ( 𝑆 D 𝐺 ) ∘f · 𝐹 ) ) ∈ ( 𝑋 –cn→ ℂ ) ) |
33 |
|
eldifi |
⊢ ( 𝑥 ∈ ( ℂ ∖ { 0 } ) → 𝑥 ∈ ℂ ) |
34 |
33
|
adantr |
⊢ ( ( 𝑥 ∈ ( ℂ ∖ { 0 } ) ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → 𝑥 ∈ ℂ ) |
35 |
|
eldifi |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 0 } ) → 𝑦 ∈ ℂ ) |
36 |
35
|
adantl |
⊢ ( ( 𝑥 ∈ ( ℂ ∖ { 0 } ) ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → 𝑦 ∈ ℂ ) |
37 |
34 36
|
mulcld |
⊢ ( ( 𝑥 ∈ ( ℂ ∖ { 0 } ) ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 · 𝑦 ) ∈ ℂ ) |
38 |
|
eldifsni |
⊢ ( 𝑥 ∈ ( ℂ ∖ { 0 } ) → 𝑥 ≠ 0 ) |
39 |
38
|
adantr |
⊢ ( ( 𝑥 ∈ ( ℂ ∖ { 0 } ) ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → 𝑥 ≠ 0 ) |
40 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 0 } ) → 𝑦 ≠ 0 ) |
41 |
40
|
adantl |
⊢ ( ( 𝑥 ∈ ( ℂ ∖ { 0 } ) ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → 𝑦 ≠ 0 ) |
42 |
34 36 39 41
|
mulne0d |
⊢ ( ( 𝑥 ∈ ( ℂ ∖ { 0 } ) ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 · 𝑦 ) ≠ 0 ) |
43 |
|
eldifsn |
⊢ ( ( 𝑥 · 𝑦 ) ∈ ( ℂ ∖ { 0 } ) ↔ ( ( 𝑥 · 𝑦 ) ∈ ℂ ∧ ( 𝑥 · 𝑦 ) ≠ 0 ) ) |
44 |
37 42 43
|
sylanbrc |
⊢ ( ( 𝑥 ∈ ( ℂ ∖ { 0 } ) ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 · 𝑦 ) ∈ ( ℂ ∖ { 0 } ) ) |
45 |
44
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ℂ ∖ { 0 } ) ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) ) → ( 𝑥 · 𝑦 ) ∈ ( ℂ ∖ { 0 } ) ) |
46 |
1 25
|
ssexd |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
47 |
|
inidm |
⊢ ( 𝑋 ∩ 𝑋 ) = 𝑋 |
48 |
45 3 3 46 46 47
|
off |
⊢ ( 𝜑 → ( 𝐺 ∘f · 𝐺 ) : 𝑋 ⟶ ( ℂ ∖ { 0 } ) ) |
49 |
27 27
|
mulcncff |
⊢ ( 𝜑 → ( 𝐺 ∘f · 𝐺 ) ∈ ( 𝑋 –cn→ ℂ ) ) |
50 |
|
cncffvrn |
⊢ ( ( ( ℂ ∖ { 0 } ) ⊆ ℂ ∧ ( 𝐺 ∘f · 𝐺 ) ∈ ( 𝑋 –cn→ ℂ ) ) → ( ( 𝐺 ∘f · 𝐺 ) ∈ ( 𝑋 –cn→ ( ℂ ∖ { 0 } ) ) ↔ ( 𝐺 ∘f · 𝐺 ) : 𝑋 ⟶ ( ℂ ∖ { 0 } ) ) ) |
51 |
22 49 50
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐺 ∘f · 𝐺 ) ∈ ( 𝑋 –cn→ ( ℂ ∖ { 0 } ) ) ↔ ( 𝐺 ∘f · 𝐺 ) : 𝑋 ⟶ ( ℂ ∖ { 0 } ) ) ) |
52 |
48 51
|
mpbird |
⊢ ( 𝜑 → ( 𝐺 ∘f · 𝐺 ) ∈ ( 𝑋 –cn→ ( ℂ ∖ { 0 } ) ) ) |
53 |
32 52
|
divcncff |
⊢ ( 𝜑 → ( ( ( ( 𝑆 D 𝐹 ) ∘f · 𝐺 ) ∘f − ( ( 𝑆 D 𝐺 ) ∘f · 𝐹 ) ) ∘f / ( 𝐺 ∘f · 𝐺 ) ) ∈ ( 𝑋 –cn→ ℂ ) ) |
54 |
12 53
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑆 D ( 𝐹 ∘f / 𝐺 ) ) ∈ ( 𝑋 –cn→ ℂ ) ) |