Step |
Hyp |
Ref |
Expression |
1 |
|
dvdivf.s |
⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } ) |
2 |
|
dvdivf.f |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℂ ) |
3 |
|
dvdivf.g |
⊢ ( 𝜑 → 𝐺 : 𝑋 ⟶ ( ℂ ∖ { 0 } ) ) |
4 |
|
dvdivf.fdv |
⊢ ( 𝜑 → dom ( 𝑆 D 𝐹 ) = 𝑋 ) |
5 |
|
dvdivf.gdv |
⊢ ( 𝜑 → dom ( 𝑆 D 𝐺 ) = 𝑋 ) |
6 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
7 |
|
dvfg |
⊢ ( 𝑆 ∈ { ℝ , ℂ } → ( 𝑆 D 𝐹 ) : dom ( 𝑆 D 𝐹 ) ⟶ ℂ ) |
8 |
1 7
|
syl |
⊢ ( 𝜑 → ( 𝑆 D 𝐹 ) : dom ( 𝑆 D 𝐹 ) ⟶ ℂ ) |
9 |
4
|
feq2d |
⊢ ( 𝜑 → ( ( 𝑆 D 𝐹 ) : dom ( 𝑆 D 𝐹 ) ⟶ ℂ ↔ ( 𝑆 D 𝐹 ) : 𝑋 ⟶ ℂ ) ) |
10 |
8 9
|
mpbid |
⊢ ( 𝜑 → ( 𝑆 D 𝐹 ) : 𝑋 ⟶ ℂ ) |
11 |
10
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝑆 D 𝐹 ) ‘ 𝑥 ) ∈ ℂ ) |
12 |
2
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
13 |
12
|
oveq2d |
⊢ ( 𝜑 → ( 𝑆 D 𝐹 ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ( 𝐹 ‘ 𝑥 ) ) ) ) |
14 |
10
|
feqmptd |
⊢ ( 𝜑 → ( 𝑆 D 𝐹 ) = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑆 D 𝐹 ) ‘ 𝑥 ) ) ) |
15 |
13 14
|
eqtr3d |
⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ( 𝐹 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑆 D 𝐹 ) ‘ 𝑥 ) ) ) |
16 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝐺 ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) |
17 |
|
dvfg |
⊢ ( 𝑆 ∈ { ℝ , ℂ } → ( 𝑆 D 𝐺 ) : dom ( 𝑆 D 𝐺 ) ⟶ ℂ ) |
18 |
1 17
|
syl |
⊢ ( 𝜑 → ( 𝑆 D 𝐺 ) : dom ( 𝑆 D 𝐺 ) ⟶ ℂ ) |
19 |
5
|
feq2d |
⊢ ( 𝜑 → ( ( 𝑆 D 𝐺 ) : dom ( 𝑆 D 𝐺 ) ⟶ ℂ ↔ ( 𝑆 D 𝐺 ) : 𝑋 ⟶ ℂ ) ) |
20 |
18 19
|
mpbid |
⊢ ( 𝜑 → ( 𝑆 D 𝐺 ) : 𝑋 ⟶ ℂ ) |
21 |
20
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝑆 D 𝐺 ) ‘ 𝑥 ) ∈ ℂ ) |
22 |
3
|
feqmptd |
⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝑋 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
23 |
22
|
oveq2d |
⊢ ( 𝜑 → ( 𝑆 D 𝐺 ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ( 𝐺 ‘ 𝑥 ) ) ) ) |
24 |
20
|
feqmptd |
⊢ ( 𝜑 → ( 𝑆 D 𝐺 ) = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑆 D 𝐺 ) ‘ 𝑥 ) ) ) |
25 |
23 24
|
eqtr3d |
⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ( 𝐺 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑆 D 𝐺 ) ‘ 𝑥 ) ) ) |
26 |
1 6 11 15 16 21 25
|
dvmptdiv |
⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ( ( 𝐹 ‘ 𝑥 ) / ( 𝐺 ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ 𝑋 ↦ ( ( ( ( ( 𝑆 D 𝐹 ) ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) − ( ( ( 𝑆 D 𝐺 ) ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ) / ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) ) ) |
27 |
|
ovex |
⊢ ( 𝑆 D 𝐹 ) ∈ V |
28 |
27
|
dmex |
⊢ dom ( 𝑆 D 𝐹 ) ∈ V |
29 |
4 28
|
eqeltrrdi |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
30 |
29 6 16 12 22
|
offval2 |
⊢ ( 𝜑 → ( 𝐹 ∘f / 𝐺 ) = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝐹 ‘ 𝑥 ) / ( 𝐺 ‘ 𝑥 ) ) ) ) |
31 |
30
|
oveq2d |
⊢ ( 𝜑 → ( 𝑆 D ( 𝐹 ∘f / 𝐺 ) ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ( ( 𝐹 ‘ 𝑥 ) / ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
32 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( ( ( 𝑆 D 𝐹 ) ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) − ( ( ( 𝑆 D 𝐺 ) ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ) ∈ V ) |
33 |
16
|
eldifad |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝐺 ‘ 𝑥 ) ∈ ℂ ) |
34 |
33
|
sqcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ∈ ℂ ) |
35 |
11 33
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( ( 𝑆 D 𝐹 ) ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ∈ ℂ ) |
36 |
21 6
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( ( 𝑆 D 𝐺 ) ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ∈ ℂ ) |
37 |
29 11 33 14 22
|
offval2 |
⊢ ( 𝜑 → ( ( 𝑆 D 𝐹 ) ∘f · 𝐺 ) = ( 𝑥 ∈ 𝑋 ↦ ( ( ( 𝑆 D 𝐹 ) ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) |
38 |
29 21 6 24 12
|
offval2 |
⊢ ( 𝜑 → ( ( 𝑆 D 𝐺 ) ∘f · 𝐹 ) = ( 𝑥 ∈ 𝑋 ↦ ( ( ( 𝑆 D 𝐺 ) ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ) ) |
39 |
29 35 36 37 38
|
offval2 |
⊢ ( 𝜑 → ( ( ( 𝑆 D 𝐹 ) ∘f · 𝐺 ) ∘f − ( ( 𝑆 D 𝐺 ) ∘f · 𝐹 ) ) = ( 𝑥 ∈ 𝑋 ↦ ( ( ( ( 𝑆 D 𝐹 ) ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) − ( ( ( 𝑆 D 𝐺 ) ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
40 |
29 16 16 22 22
|
offval2 |
⊢ ( 𝜑 → ( 𝐺 ∘f · 𝐺 ) = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) |
41 |
33
|
sqvald |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) = ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) |
42 |
41
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝐺 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) |
43 |
40 42
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐺 ∘f · 𝐺 ) = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) ) |
44 |
29 32 34 39 43
|
offval2 |
⊢ ( 𝜑 → ( ( ( ( 𝑆 D 𝐹 ) ∘f · 𝐺 ) ∘f − ( ( 𝑆 D 𝐺 ) ∘f · 𝐹 ) ) ∘f / ( 𝐺 ∘f · 𝐺 ) ) = ( 𝑥 ∈ 𝑋 ↦ ( ( ( ( ( 𝑆 D 𝐹 ) ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) − ( ( ( 𝑆 D 𝐺 ) ‘ 𝑥 ) · ( 𝐹 ‘ 𝑥 ) ) ) / ( ( 𝐺 ‘ 𝑥 ) ↑ 2 ) ) ) ) |
45 |
26 31 44
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝑆 D ( 𝐹 ∘f / 𝐺 ) ) = ( ( ( ( 𝑆 D 𝐹 ) ∘f · 𝐺 ) ∘f − ( ( 𝑆 D 𝐺 ) ∘f · 𝐹 ) ) ∘f / ( 𝐺 ∘f · 𝐺 ) ) ) |