Step |
Hyp |
Ref |
Expression |
1 |
|
i1fadd.1 |
⊢ ( 𝜑 → 𝐹 ∈ dom ∫1 ) |
2 |
|
i1fadd.2 |
⊢ ( 𝜑 → 𝐺 ∈ dom ∫1 ) |
3 |
|
i1ff |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 : ℝ ⟶ ℝ ) |
4 |
1 3
|
syl |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ ) |
5 |
4
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ℝ ) |
6 |
|
i1ff |
⊢ ( 𝐺 ∈ dom ∫1 → 𝐺 : ℝ ⟶ ℝ ) |
7 |
2 6
|
syl |
⊢ ( 𝜑 → 𝐺 : ℝ ⟶ ℝ ) |
8 |
7
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn ℝ ) |
9 |
|
reex |
⊢ ℝ ∈ V |
10 |
9
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
11 |
|
inidm |
⊢ ( ℝ ∩ ℝ ) = ℝ |
12 |
5 8 10 10 11
|
offn |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐺 ) Fn ℝ ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( 𝐹 ∘f + 𝐺 ) Fn ℝ ) |
14 |
|
fniniseg |
⊢ ( ( 𝐹 ∘f + 𝐺 ) Fn ℝ → ( 𝑧 ∈ ( ◡ ( 𝐹 ∘f + 𝐺 ) “ { 𝐴 } ) ↔ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) ) |
15 |
13 14
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( 𝑧 ∈ ( ◡ ( 𝐹 ∘f + 𝐺 ) “ { 𝐴 } ) ↔ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) ) |
16 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → 𝐺 Fn ℝ ) |
17 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → 𝑧 ∈ ℝ ) |
18 |
|
fnfvelrn |
⊢ ( ( 𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ ) → ( 𝐺 ‘ 𝑧 ) ∈ ran 𝐺 ) |
19 |
16 17 18
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ ran 𝐺 ) |
20 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) |
21 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
22 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
23 |
5 8 10 10 11 21 22
|
ofval |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑧 ) + ( 𝐺 ‘ 𝑧 ) ) ) |
24 |
23
|
ad2ant2r |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑧 ) + ( 𝐺 ‘ 𝑧 ) ) ) |
25 |
20 24
|
eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → 𝐴 = ( ( 𝐹 ‘ 𝑧 ) + ( 𝐺 ‘ 𝑧 ) ) ) |
26 |
25
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) = ( ( ( 𝐹 ‘ 𝑧 ) + ( 𝐺 ‘ 𝑧 ) ) − ( 𝐺 ‘ 𝑧 ) ) ) |
27 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
28 |
|
fss |
⊢ ( ( 𝐹 : ℝ ⟶ ℝ ∧ ℝ ⊆ ℂ ) → 𝐹 : ℝ ⟶ ℂ ) |
29 |
4 27 28
|
sylancl |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ ) |
30 |
29
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → 𝐹 : ℝ ⟶ ℂ ) |
31 |
30 17
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ ℂ ) |
32 |
|
fss |
⊢ ( ( 𝐺 : ℝ ⟶ ℝ ∧ ℝ ⊆ ℂ ) → 𝐺 : ℝ ⟶ ℂ ) |
33 |
7 27 32
|
sylancl |
⊢ ( 𝜑 → 𝐺 : ℝ ⟶ ℂ ) |
34 |
33
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → 𝐺 : ℝ ⟶ ℂ ) |
35 |
34 17
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ ℂ ) |
36 |
31 35
|
pncand |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( ( ( 𝐹 ‘ 𝑧 ) + ( 𝐺 ‘ 𝑧 ) ) − ( 𝐺 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) |
37 |
26 36
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) ) |
38 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → 𝐹 Fn ℝ ) |
39 |
|
fniniseg |
⊢ ( 𝐹 Fn ℝ → ( 𝑧 ∈ ( ◡ 𝐹 “ { ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) } ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) ) ) ) |
40 |
38 39
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( 𝑧 ∈ ( ◡ 𝐹 “ { ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) } ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) ) ) ) |
41 |
17 37 40
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ { ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) } ) ) |
42 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
43 |
|
fniniseg |
⊢ ( 𝐺 Fn ℝ → ( 𝑧 ∈ ( ◡ 𝐺 “ { ( 𝐺 ‘ 𝑧 ) } ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) ) |
44 |
16 43
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ( 𝑧 ∈ ( ◡ 𝐺 “ { ( 𝐺 ‘ 𝑧 ) } ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) ) |
45 |
17 42 44
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → 𝑧 ∈ ( ◡ 𝐺 “ { ( 𝐺 ‘ 𝑧 ) } ) ) |
46 |
41 45
|
elind |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) } ) ∩ ( ◡ 𝐺 “ { ( 𝐺 ‘ 𝑧 ) } ) ) ) |
47 |
|
oveq2 |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑧 ) → ( 𝐴 − 𝑦 ) = ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) ) |
48 |
47
|
sneqd |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑧 ) → { ( 𝐴 − 𝑦 ) } = { ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) } ) |
49 |
48
|
imaeq2d |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑧 ) → ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) = ( ◡ 𝐹 “ { ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) } ) ) |
50 |
|
sneq |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑧 ) → { 𝑦 } = { ( 𝐺 ‘ 𝑧 ) } ) |
51 |
50
|
imaeq2d |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑧 ) → ( ◡ 𝐺 “ { 𝑦 } ) = ( ◡ 𝐺 “ { ( 𝐺 ‘ 𝑧 ) } ) ) |
52 |
49 51
|
ineq12d |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑧 ) → ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) = ( ( ◡ 𝐹 “ { ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) } ) ∩ ( ◡ 𝐺 “ { ( 𝐺 ‘ 𝑧 ) } ) ) ) |
53 |
52
|
eleq2d |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑧 ) → ( 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ↔ 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) } ) ∩ ( ◡ 𝐺 “ { ( 𝐺 ‘ 𝑧 ) } ) ) ) ) |
54 |
53
|
rspcev |
⊢ ( ( ( 𝐺 ‘ 𝑧 ) ∈ ran 𝐺 ∧ 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − ( 𝐺 ‘ 𝑧 ) ) } ) ∩ ( ◡ 𝐺 “ { ( 𝐺 ‘ 𝑧 ) } ) ) ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ) |
55 |
19 46 54
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ) |
56 |
55
|
ex |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ) ) |
57 |
|
elin |
⊢ ( 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ↔ ( 𝑧 ∈ ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∧ 𝑧 ∈ ( ◡ 𝐺 “ { 𝑦 } ) ) ) |
58 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → 𝐹 Fn ℝ ) |
59 |
|
fniniseg |
⊢ ( 𝐹 Fn ℝ → ( 𝑧 ∈ ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ) ) ) |
60 |
58 59
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( 𝑧 ∈ ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ) ) ) |
61 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → 𝐺 Fn ℝ ) |
62 |
|
fniniseg |
⊢ ( 𝐺 Fn ℝ → ( 𝑧 ∈ ( ◡ 𝐺 “ { 𝑦 } ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) |
63 |
61 62
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( 𝑧 ∈ ( ◡ 𝐺 “ { 𝑦 } ) ↔ ( 𝑧 ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) |
64 |
60 63
|
anbi12d |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( ( 𝑧 ∈ ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∧ 𝑧 ∈ ( ◡ 𝐺 “ { 𝑦 } ) ) ↔ ( ( 𝑧 ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ) ∧ ( 𝑧 ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) ) |
65 |
|
anandi |
⊢ ( ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ↔ ( ( 𝑧 ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ) ∧ ( 𝑧 ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) |
66 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → 𝑧 ∈ ℝ ) |
67 |
23
|
ad2ant2r |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑧 ) + ( 𝐺 ‘ 𝑧 ) ) ) |
68 |
|
simprrl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ) |
69 |
|
simprrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → ( 𝐺 ‘ 𝑧 ) = 𝑦 ) |
70 |
68 69
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → ( ( 𝐹 ‘ 𝑧 ) + ( 𝐺 ‘ 𝑧 ) ) = ( ( 𝐴 − 𝑦 ) + 𝑦 ) ) |
71 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → 𝐴 ∈ ℂ ) |
72 |
33
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → 𝐺 : ℝ ⟶ ℂ ) |
73 |
72 66
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ ℂ ) |
74 |
69 73
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → 𝑦 ∈ ℂ ) |
75 |
71 74
|
npcand |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → ( ( 𝐴 − 𝑦 ) + 𝑦 ) = 𝐴 ) |
76 |
67 70 75
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) |
77 |
66 76
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) ∧ ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) ) → ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) |
78 |
77
|
ex |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) → ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) ) |
79 |
65 78
|
syl5bir |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( ( ( 𝑧 ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝐴 − 𝑦 ) ) ∧ ( 𝑧 ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) ) → ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) ) |
80 |
64 79
|
sylbid |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( ( 𝑧 ∈ ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∧ 𝑧 ∈ ( ◡ 𝐺 “ { 𝑦 } ) ) → ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) ) |
81 |
57 80
|
syl5bi |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) → ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) ) |
82 |
81
|
rexlimdvw |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( ∃ 𝑦 ∈ ran 𝐺 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) → ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ) ) |
83 |
56 82
|
impbid |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( ( 𝑧 ∈ ℝ ∧ ( ( 𝐹 ∘f + 𝐺 ) ‘ 𝑧 ) = 𝐴 ) ↔ ∃ 𝑦 ∈ ran 𝐺 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ) ) |
84 |
15 83
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( 𝑧 ∈ ( ◡ ( 𝐹 ∘f + 𝐺 ) “ { 𝐴 } ) ↔ ∃ 𝑦 ∈ ran 𝐺 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ) ) |
85 |
|
eliun |
⊢ ( 𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺 ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ↔ ∃ 𝑦 ∈ ran 𝐺 𝑧 ∈ ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ) |
86 |
84 85
|
bitr4di |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( 𝑧 ∈ ( ◡ ( 𝐹 ∘f + 𝐺 ) “ { 𝐴 } ) ↔ 𝑧 ∈ ∪ 𝑦 ∈ ran 𝐺 ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ) ) |
87 |
86
|
eqrdv |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℂ ) → ( ◡ ( 𝐹 ∘f + 𝐺 ) “ { 𝐴 } ) = ∪ 𝑦 ∈ ran 𝐺 ( ( ◡ 𝐹 “ { ( 𝐴 − 𝑦 ) } ) ∩ ( ◡ 𝐺 “ { 𝑦 } ) ) ) |