Step | Hyp | Ref
| Expression |
1 | | i1fadd.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ dom
∫1) |
2 | | i1ff 24573 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ 𝐹:ℝ⟶ℝ) |
3 | 1, 2 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
4 | 3 | ffnd 6546 |
. . . . . . 7
⊢ (𝜑 → 𝐹 Fn ℝ) |
5 | | i1fadd.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ dom
∫1) |
6 | | i1ff 24573 |
. . . . . . . . 9
⊢ (𝐺 ∈ dom ∫1
→ 𝐺:ℝ⟶ℝ) |
7 | 5, 6 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐺:ℝ⟶ℝ) |
8 | 7 | ffnd 6546 |
. . . . . . 7
⊢ (𝜑 → 𝐺 Fn ℝ) |
9 | | reex 10820 |
. . . . . . . 8
⊢ ℝ
∈ V |
10 | 9 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ℝ ∈
V) |
11 | | inidm 4133 |
. . . . . . 7
⊢ (ℝ
∩ ℝ) = ℝ |
12 | 4, 8, 10, 10, 11 | offn 7481 |
. . . . . 6
⊢ (𝜑 → (𝐹 ∘f · 𝐺) Fn ℝ) |
13 | 12 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐹 ∘f ·
𝐺) Fn
ℝ) |
14 | | fniniseg 6880 |
. . . . 5
⊢ ((𝐹 ∘f ·
𝐺) Fn ℝ → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f · 𝐺)‘𝑧) = 𝐴))) |
15 | 13, 14 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ (𝑧 ∈ ℝ ∧ ((𝐹 ∘f · 𝐺)‘𝑧) = 𝐴))) |
16 | 4 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → 𝐹 Fn ℝ) |
17 | 8 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → 𝐺 Fn ℝ) |
18 | 9 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) →
ℝ ∈ V) |
19 | | eqidd 2738 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧)) |
20 | | eqidd 2738 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧)) |
21 | 16, 17, 18, 18, 11, 19, 20 | ofval 7479 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → ((𝐹 ∘f ·
𝐺)‘𝑧) = ((𝐹‘𝑧) · (𝐺‘𝑧))) |
22 | 21 | eqeq1d 2739 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹 ∘f ·
𝐺)‘𝑧) = 𝐴 ↔ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) |
23 | 22 | pm5.32da 582 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) →
((𝑧 ∈ ℝ ∧
((𝐹 ∘f
· 𝐺)‘𝑧) = 𝐴) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))) |
24 | 8 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐺 Fn ℝ) |
25 | | simprl 771 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ ℝ) |
26 | | fnfvelrn 6901 |
. . . . . . . . 9
⊢ ((𝐺 Fn ℝ ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) ∈ ran 𝐺) |
27 | 24, 25, 26 | syl2anc 587 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ran 𝐺) |
28 | | eldifsni 4703 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ (ℂ ∖ {0})
→ 𝐴 ≠
0) |
29 | 28 | ad2antlr 727 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐴 ≠ 0) |
30 | | simprr 773 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴) |
31 | 3 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐹:ℝ⟶ℝ) |
32 | 31, 25 | ffvelrnd 6905 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℝ) |
33 | 32 | recnd 10861 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) ∈ ℂ) |
34 | 33 | mul01d 11031 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · 0) = 0) |
35 | 29, 30, 34 | 3netr4d 3018 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) ≠ ((𝐹‘𝑧) · 0)) |
36 | | oveq2 7221 |
. . . . . . . . . 10
⊢ ((𝐺‘𝑧) = 0 → ((𝐹‘𝑧) · (𝐺‘𝑧)) = ((𝐹‘𝑧) · 0)) |
37 | 36 | necon3i 2973 |
. . . . . . . . 9
⊢ (((𝐹‘𝑧) · (𝐺‘𝑧)) ≠ ((𝐹‘𝑧) · 0) → (𝐺‘𝑧) ≠ 0) |
38 | 35, 37 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ≠ 0) |
39 | | eldifsn 4700 |
. . . . . . . 8
⊢ ((𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0}) ↔ ((𝐺‘𝑧) ∈ ran 𝐺 ∧ (𝐺‘𝑧) ≠ 0)) |
40 | 27, 38, 39 | sylanbrc 586 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0})) |
41 | 7 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐺:ℝ⟶ℝ) |
42 | 41, 25 | ffvelrnd 6905 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℝ) |
43 | 42 | recnd 10861 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) ∈ ℂ) |
44 | 33, 43, 38 | divcan4d 11614 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (((𝐹‘𝑧) · (𝐺‘𝑧)) / (𝐺‘𝑧)) = (𝐹‘𝑧)) |
45 | 30 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (((𝐹‘𝑧) · (𝐺‘𝑧)) / (𝐺‘𝑧)) = (𝐴 / (𝐺‘𝑧))) |
46 | 44, 45 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐹‘𝑧) = (𝐴 / (𝐺‘𝑧))) |
47 | 31 | ffnd 6546 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝐹 Fn ℝ) |
48 | | fniniseg 6880 |
. . . . . . . . . 10
⊢ (𝐹 Fn ℝ → (𝑧 ∈ (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / (𝐺‘𝑧))))) |
49 | 47, 48 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝑧 ∈ (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / (𝐺‘𝑧))))) |
50 | 25, 46, 49 | mpbir2and 713 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))})) |
51 | | eqidd 2738 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝐺‘𝑧) = (𝐺‘𝑧)) |
52 | | fniniseg 6880 |
. . . . . . . . . 10
⊢ (𝐺 Fn ℝ → (𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧)))) |
53 | 24, 52 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → (𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = (𝐺‘𝑧)))) |
54 | 25, 51, 53 | mpbir2and 713 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ (◡𝐺 “ {(𝐺‘𝑧)})) |
55 | 50, 54 | elind 4108 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → 𝑧 ∈ ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)}))) |
56 | | oveq2 7221 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐺‘𝑧) → (𝐴 / 𝑦) = (𝐴 / (𝐺‘𝑧))) |
57 | 56 | sneqd 4553 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐺‘𝑧) → {(𝐴 / 𝑦)} = {(𝐴 / (𝐺‘𝑧))}) |
58 | 57 | imaeq2d 5929 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐹 “ {(𝐴 / 𝑦)}) = (◡𝐹 “ {(𝐴 / (𝐺‘𝑧))})) |
59 | | sneq 4551 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐺‘𝑧) → {𝑦} = {(𝐺‘𝑧)}) |
60 | 59 | imaeq2d 5929 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐺‘𝑧) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(𝐺‘𝑧)})) |
61 | 58, 60 | ineq12d 4128 |
. . . . . . . . 9
⊢ (𝑦 = (𝐺‘𝑧) → ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)}))) |
62 | 61 | eleq2d 2823 |
. . . . . . . 8
⊢ (𝑦 = (𝐺‘𝑧) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ 𝑧 ∈ ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)})))) |
63 | 62 | rspcev 3537 |
. . . . . . 7
⊢ (((𝐺‘𝑧) ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ((◡𝐹 “ {(𝐴 / (𝐺‘𝑧))}) ∩ (◡𝐺 “ {(𝐺‘𝑧)}))) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))) |
64 | 40, 55, 63 | syl2anc 587 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))) |
65 | 64 | ex 416 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) →
((𝑧 ∈ ℝ ∧
((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴) → ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))) |
66 | | fniniseg 6880 |
. . . . . . . . . . 11
⊢ (𝐹 Fn ℝ → (𝑧 ∈ (◡𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦)))) |
67 | 16, 66 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡𝐹 “ {(𝐴 / 𝑦)}) ↔ (𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦)))) |
68 | | fniniseg 6880 |
. . . . . . . . . . 11
⊢ (𝐺 Fn ℝ → (𝑧 ∈ (◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦))) |
69 | 17, 68 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡𝐺 “ {𝑦}) ↔ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦))) |
70 | 67, 69 | anbi12d 634 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) →
((𝑧 ∈ (◡𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦})) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦)))) |
71 | | elin 3882 |
. . . . . . . . 9
⊢ (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ (◡𝐹 “ {(𝐴 / 𝑦)}) ∧ 𝑧 ∈ (◡𝐺 “ {𝑦}))) |
72 | | anandi 676 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) ↔ ((𝑧 ∈ ℝ ∧ (𝐹‘𝑧) = (𝐴 / 𝑦)) ∧ (𝑧 ∈ ℝ ∧ (𝐺‘𝑧) = 𝑦))) |
73 | 70, 71, 72 | 3bitr4g 317 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦)))) |
74 | 73 | adantr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦)))) |
75 | | eldifi 4041 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ (ℂ ∖ {0})
→ 𝐴 ∈
ℂ) |
76 | 75 | ad2antlr 727 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐴 ∈ ℂ) |
77 | 7 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝐺:ℝ⟶ℝ) |
78 | 77 | frnd 6553 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ran 𝐺 ⊆
ℝ) |
79 | | simprl 771 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ (ran 𝐺 ∖ {0})) |
80 | | eldifsn 4700 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0)) |
81 | 79, 80 | sylib 221 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (𝑦 ∈ ran 𝐺 ∧ 𝑦 ≠ 0)) |
82 | 81 | simpld 498 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ran 𝐺) |
83 | 78, 82 | sseldd 3902 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℝ) |
84 | 83 | recnd 10861 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ∈ ℂ) |
85 | 81 | simprd 499 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → 𝑦 ≠ 0) |
86 | 76, 84, 85 | divcan1d 11609 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → ((𝐴 / 𝑦) · 𝑦) = 𝐴) |
87 | | oveq12 7222 |
. . . . . . . . . . 11
⊢ (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = ((𝐴 / 𝑦) · 𝑦)) |
88 | 87 | eqeq1d 2739 |
. . . . . . . . . 10
⊢ (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → (((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴 ↔ ((𝐴 / 𝑦) · 𝑦) = 𝐴)) |
89 | 86, 88 | syl5ibrcom 250 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ (𝑦 ∈ (ran 𝐺 ∖ {0}) ∧ 𝑧 ∈ ℝ)) → (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) |
90 | 89 | anassrs 471 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑧 ∈ ℝ) → (((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦) → ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴)) |
91 | 90 | imdistanda 575 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → ((𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) = (𝐴 / 𝑦) ∧ (𝐺‘𝑧) = 𝑦)) → (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))) |
92 | 74, 91 | sylbid 243 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) ∧ 𝑦 ∈ (ran 𝐺 ∖ {0})) → (𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))) |
93 | 92 | rexlimdva 3203 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) →
(∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) → (𝑧 ∈ ℝ ∧ ((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴))) |
94 | 65, 93 | impbid 215 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) →
((𝑧 ∈ ℝ ∧
((𝐹‘𝑧) · (𝐺‘𝑧)) = 𝐴) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))) |
95 | 15, 23, 94 | 3bitrd 308 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))) |
96 | | eliun 4908 |
. . 3
⊢ (𝑧 ∈ ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})) ↔ ∃𝑦 ∈ (ran 𝐺 ∖ {0})𝑧 ∈ ((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))) |
97 | 95, 96 | bitr4di 292 |
. 2
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝑧 ∈ (◡(𝐹 ∘f · 𝐺) “ {𝐴}) ↔ 𝑧 ∈ ∪
𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))) |
98 | 97 | eqrdv 2735 |
1
⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝐴}) = ∪
𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))) |