Step | Hyp | Ref
| Expression |
1 | | dvcnv.f |
. . . . 5
⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌) |
2 | | f1of 6137 |
. . . . 5
⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
4 | | dvcnv.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ 𝑋) |
5 | 3, 4 | ffvelrnd 6360 |
. . 3
⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌) |
6 | | dvcnv.k |
. . . . . 6
⊢ 𝐾 = (𝐽 ↾t 𝑆) |
7 | | dvcnv.j |
. . . . . . . 8
⊢ 𝐽 =
(TopOpen‘ℂfld) |
8 | 7 | cnfldtopon 22586 |
. . . . . . 7
⊢ 𝐽 ∈
(TopOn‘ℂ) |
9 | | dvcnv.s |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
10 | | recnprss 23668 |
. . . . . . . 8
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
11 | 9, 10 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
12 | | resttopon 20965 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝑆 ⊆ ℂ)
→ (𝐽
↾t 𝑆)
∈ (TopOn‘𝑆)) |
13 | 8, 11, 12 | sylancr 695 |
. . . . . 6
⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
14 | 6, 13 | syl5eqel 2705 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆)) |
15 | | topontop 20718 |
. . . . 5
⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝐾 ∈ Top) |
16 | 14, 15 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ Top) |
17 | | dvcnv.y |
. . . 4
⊢ (𝜑 → 𝑌 ∈ 𝐾) |
18 | | isopn3i 20886 |
. . . 4
⊢ ((𝐾 ∈ Top ∧ 𝑌 ∈ 𝐾) → ((int‘𝐾)‘𝑌) = 𝑌) |
19 | 16, 17, 18 | syl2anc 693 |
. . 3
⊢ (𝜑 → ((int‘𝐾)‘𝑌) = 𝑌) |
20 | 5, 19 | eleqtrrd 2704 |
. 2
⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝐾)‘𝑌)) |
21 | | f1ocnv 6149 |
. . . . . . . . 9
⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋) |
22 | | f1of 6137 |
. . . . . . . . 9
⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋) |
23 | 1, 21, 22 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → ◡𝐹:𝑌⟶𝑋) |
24 | | eldifi 3732 |
. . . . . . . 8
⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ∈ 𝑌) |
25 | | ffvelrn 6357 |
. . . . . . . 8
⊢ ((◡𝐹:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (◡𝐹‘𝑧) ∈ 𝑋) |
26 | 23, 24, 25 | syl2an 494 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (◡𝐹‘𝑧) ∈ 𝑋) |
27 | 26 | anim1i 592 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (◡𝐹‘𝑧) ≠ 𝐶) → ((◡𝐹‘𝑧) ∈ 𝑋 ∧ (◡𝐹‘𝑧) ≠ 𝐶)) |
28 | | eldifsn 4317 |
. . . . . 6
⊢ ((◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶}) ↔ ((◡𝐹‘𝑧) ∈ 𝑋 ∧ (◡𝐹‘𝑧) ≠ 𝐶)) |
29 | 27, 28 | sylibr 224 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (◡𝐹‘𝑧) ≠ 𝐶) → (◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶})) |
30 | 29 | anasss 679 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ∧ (◡𝐹‘𝑧) ≠ 𝐶)) → (◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶})) |
31 | | eldifi 3732 |
. . . . . . 7
⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ∈ 𝑋) |
32 | | dvcnv.d |
. . . . . . . . . 10
⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) |
33 | | dvbsss 23666 |
. . . . . . . . . 10
⊢ dom
(𝑆 D 𝐹) ⊆ 𝑆 |
34 | 32, 33 | syl6eqssr 3656 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
35 | 34, 11 | sstrd 3613 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ⊆ ℂ) |
36 | 35 | sselda 3603 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ℂ) |
37 | 31, 36 | sylan2 491 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ∈ ℂ) |
38 | 34, 4 | sseldd 3604 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ 𝑆) |
39 | 11, 38 | sseldd 3604 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ ℂ) |
40 | 39 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐶 ∈ ℂ) |
41 | 37, 40 | subcld 10392 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ∈ ℂ) |
42 | | toponss 20731 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆) |
43 | 14, 17, 42 | syl2anc 693 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
44 | 43, 11 | sstrd 3613 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ⊆ ℂ) |
45 | 3, 44 | fssd 6057 |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
46 | | ffvelrn 6357 |
. . . . . . 7
⊢ ((𝐹:𝑋⟶ℂ ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ℂ) |
47 | 45, 31, 46 | syl2an 494 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝑦) ∈ ℂ) |
48 | 44, 5 | sseldd 3604 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
49 | 48 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝐶) ∈ ℂ) |
50 | 47, 49 | subcld 10392 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ∈ ℂ) |
51 | | eldifsni 4320 |
. . . . . . 7
⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ≠ 𝐶) |
52 | 51 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ≠ 𝐶) |
53 | 47, 49 | subeq0ad 10402 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ (𝐹‘𝑦) = (𝐹‘𝐶))) |
54 | | f1of1 6136 |
. . . . . . . . . . 11
⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌) |
55 | 1, 54 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌) |
56 | 55 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐹:𝑋–1-1→𝑌) |
57 | 31 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ∈ 𝑋) |
58 | 4 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐶 ∈ 𝑋) |
59 | | f1fveq 6519 |
. . . . . . . . 9
⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑦 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶)) |
60 | 56, 57, 58, 59 | syl12anc 1324 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶)) |
61 | 53, 60 | bitrd 268 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ 𝑦 = 𝐶)) |
62 | 61 | necon3bid 2838 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0 ↔ 𝑦 ≠ 𝐶)) |
63 | 52, 62 | mpbird 247 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0) |
64 | 41, 50, 63 | divcld 10801 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) ∈ ℂ) |
65 | | limcresi 23649 |
. . . . . 6
⊢ (◡𝐹 limℂ (𝐹‘𝐶)) ⊆ ((◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) limℂ (𝐹‘𝐶)) |
66 | 23 | feqmptd 6249 |
. . . . . . . . 9
⊢ (𝜑 → ◡𝐹 = (𝑧 ∈ 𝑌 ↦ (◡𝐹‘𝑧))) |
67 | 66 | reseq1d 5395 |
. . . . . . . 8
⊢ (𝜑 → (◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = ((𝑧 ∈ 𝑌 ↦ (◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)}))) |
68 | | difss 3737 |
. . . . . . . . 9
⊢ (𝑌 ∖ {(𝐹‘𝐶)}) ⊆ 𝑌 |
69 | | resmpt 5449 |
. . . . . . . . 9
⊢ ((𝑌 ∖ {(𝐹‘𝐶)}) ⊆ 𝑌 → ((𝑧 ∈ 𝑌 ↦ (◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧))) |
70 | 68, 69 | ax-mp 5 |
. . . . . . . 8
⊢ ((𝑧 ∈ 𝑌 ↦ (◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧)) |
71 | 67, 70 | syl6eq 2672 |
. . . . . . 7
⊢ (𝜑 → (◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧))) |
72 | 71 | oveq1d 6665 |
. . . . . 6
⊢ (𝜑 → ((◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) limℂ (𝐹‘𝐶)) = ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧)) limℂ (𝐹‘𝐶))) |
73 | 65, 72 | syl5sseq 3653 |
. . . . 5
⊢ (𝜑 → (◡𝐹 limℂ (𝐹‘𝐶)) ⊆ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧)) limℂ (𝐹‘𝐶))) |
74 | | f1ocnvfv1 6532 |
. . . . . . 7
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) |
75 | 1, 4, 74 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) |
76 | | dvcnv.i |
. . . . . . 7
⊢ (𝜑 → ◡𝐹 ∈ (𝑌–cn→𝑋)) |
77 | 76, 5 | cnlimci 23653 |
. . . . . 6
⊢ (𝜑 → (◡𝐹‘(𝐹‘𝐶)) ∈ (◡𝐹 limℂ (𝐹‘𝐶))) |
78 | 75, 77 | eqeltrrd 2702 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ (◡𝐹 limℂ (𝐹‘𝐶))) |
79 | 73, 78 | sseldd 3604 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧)) limℂ (𝐹‘𝐶))) |
80 | 45, 35, 4 | dvlem 23660 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ ℂ) |
81 | 37, 40, 52 | subne0d 10401 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ≠ 0) |
82 | 50, 41, 63, 81 | divne0d 10817 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ≠ 0) |
83 | | eldifsn 4317 |
. . . . . . . 8
⊢ ((((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ (ℂ ∖ {0}) ↔
((((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ ℂ ∧ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ≠ 0)) |
84 | 80, 82, 83 | sylanbrc 698 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ (ℂ ∖
{0})) |
85 | | eqid 2622 |
. . . . . . 7
⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) |
86 | 84, 85 | fmptd 6385 |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))):(𝑋 ∖ {𝐶})⟶(ℂ ∖
{0})) |
87 | | difss 3737 |
. . . . . . 7
⊢ (ℂ
∖ {0}) ⊆ ℂ |
88 | 87 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (ℂ ∖ {0})
⊆ ℂ) |
89 | | eqid 2622 |
. . . . . 6
⊢ (𝐽 ↾t (ℂ
∖ {0})) = (𝐽
↾t (ℂ ∖ {0})) |
90 | 4, 32 | eleqtrrd 2704 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) |
91 | | dvfg 23670 |
. . . . . . . . . 10
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
92 | | ffun 6048 |
. . . . . . . . . 10
⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) |
93 | | funfvbrb 6330 |
. . . . . . . . . 10
⊢ (Fun
(𝑆 D 𝐹) → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))) |
94 | 9, 91, 92, 93 | 4syl 19 |
. . . . . . . . 9
⊢ (𝜑 → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))) |
95 | 90, 94 | mpbid 222 |
. . . . . . . 8
⊢ (𝜑 → 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶)) |
96 | 6, 7, 85, 11, 45, 34 | eldv 23662 |
. . . . . . . 8
⊢ (𝜑 → (𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶) ↔ (𝐶 ∈ ((int‘𝐾)‘𝑋) ∧ ((𝑆 D 𝐹)‘𝐶) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) limℂ 𝐶)))) |
97 | 95, 96 | mpbid 222 |
. . . . . . 7
⊢ (𝜑 → (𝐶 ∈ ((int‘𝐾)‘𝑋) ∧ ((𝑆 D 𝐹)‘𝐶) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) limℂ 𝐶))) |
98 | 97 | simprd 479 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) limℂ 𝐶)) |
99 | | resttopon 20965 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾t (ℂ ∖ {0}))
∈ (TopOn‘(ℂ ∖ {0}))) |
100 | 8, 87, 99 | mp2an 708 |
. . . . . . . . 9
⊢ (𝐽 ↾t (ℂ
∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})) |
101 | 100 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (𝐽 ↾t (ℂ ∖ {0}))
∈ (TopOn‘(ℂ ∖ {0}))) |
102 | 8 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐽 ∈
(TopOn‘ℂ)) |
103 | | 1cnd 10056 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈
ℂ) |
104 | 101, 102,
103 | cnmptc 21465 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 1)
∈ ((𝐽
↾t (ℂ ∖ {0})) Cn 𝐽)) |
105 | 101 | cnmptid 21464 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 𝑥) ∈ ((𝐽 ↾t (ℂ ∖ {0}))
Cn (𝐽 ↾t
(ℂ ∖ {0})))) |
106 | 7, 89 | divcn 22671 |
. . . . . . . . 9
⊢ / ∈
((𝐽 ×t
(𝐽 ↾t
(ℂ ∖ {0}))) Cn 𝐽) |
107 | 106 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → / ∈ ((𝐽 ×t (𝐽 ↾t (ℂ
∖ {0}))) Cn 𝐽)) |
108 | 101, 104,
105, 107 | cnmpt12f 21469 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) ∈ ((𝐽 ↾t (ℂ
∖ {0})) Cn 𝐽)) |
109 | 9, 91 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
110 | 32 | feq2d 6031 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ)) |
111 | 109, 110 | mpbid 222 |
. . . . . . . . 9
⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ) |
112 | 111, 4 | ffvelrnd 6360 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ) |
113 | | ffn 6045 |
. . . . . . . . . . 11
⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹)) |
114 | 109, 113 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹)) |
115 | | fnfvelrn 6356 |
. . . . . . . . . 10
⊢ (((𝑆 D 𝐹) Fn dom (𝑆 D 𝐹) ∧ 𝐶 ∈ dom (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹)) |
116 | 114, 90, 115 | syl2anc 693 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹)) |
117 | | dvcnv.z |
. . . . . . . . 9
⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹)) |
118 | | nelne2 2891 |
. . . . . . . . 9
⊢ ((((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹) ∧ ¬ 0 ∈ ran (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ≠ 0) |
119 | 116, 117,
118 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ≠ 0) |
120 | | eldifsn 4317 |
. . . . . . . 8
⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) ↔
(((𝑆 D 𝐹)‘𝐶) ∈ ℂ ∧ ((𝑆 D 𝐹)‘𝐶) ≠ 0)) |
121 | 112, 119,
120 | sylanbrc 698 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖
{0})) |
122 | 100 | toponunii 20721 |
. . . . . . . 8
⊢ (ℂ
∖ {0}) = ∪ (𝐽 ↾t (ℂ ∖
{0})) |
123 | 122 | cncnpi 21082 |
. . . . . . 7
⊢ (((𝑥 ∈ (ℂ ∖ {0})
↦ (1 / 𝑥)) ∈
((𝐽 ↾t
(ℂ ∖ {0})) Cn 𝐽) ∧ ((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0})) →
(𝑥 ∈ (ℂ ∖
{0}) ↦ (1 / 𝑥))
∈ (((𝐽
↾t (ℂ ∖ {0})) CnP 𝐽)‘((𝑆 D 𝐹)‘𝐶))) |
124 | 108, 121,
123 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) ∈ (((𝐽 ↾t (ℂ
∖ {0})) CnP 𝐽)‘((𝑆 D 𝐹)‘𝐶))) |
125 | 86, 88, 7, 89, 98, 124 | limccnp 23655 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥))‘((𝑆 D 𝐹)‘𝐶)) ∈ (((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) ∘ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) limℂ 𝐶)) |
126 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑥 = ((𝑆 D 𝐹)‘𝐶) → (1 / 𝑥) = (1 / ((𝑆 D 𝐹)‘𝐶))) |
127 | | eqid 2622 |
. . . . . . 7
⊢ (𝑥 ∈ (ℂ ∖ {0})
↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0})
↦ (1 / 𝑥)) |
128 | | ovex 6678 |
. . . . . . 7
⊢ (1 /
((𝑆 D 𝐹)‘𝐶)) ∈ V |
129 | 126, 127,
128 | fvmpt 6282 |
. . . . . 6
⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) →
((𝑥 ∈ (ℂ ∖
{0}) ↦ (1 / 𝑥))‘((𝑆 D 𝐹)‘𝐶)) = (1 / ((𝑆 D 𝐹)‘𝐶))) |
130 | 121, 129 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥))‘((𝑆 D 𝐹)‘𝐶)) = (1 / ((𝑆 D 𝐹)‘𝐶))) |
131 | | eqidd 2623 |
. . . . . . . 8
⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) |
132 | | eqidd 2623 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥))) |
133 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝑥 = (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) → (1 / 𝑥) = (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) |
134 | 84, 131, 132, 133 | fmptco 6396 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) ∘ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))))) |
135 | 50, 41, 63, 81 | recdivd 10818 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) |
136 | 135 | mpteq2dva 4744 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))))) |
137 | 134, 136 | eqtrd 2656 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) ∘ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))))) |
138 | 137 | oveq1d 6665 |
. . . . 5
⊢ (𝜑 → (((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) ∘ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) limℂ 𝐶) = ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) limℂ 𝐶)) |
139 | 125, 130,
138 | 3eltr3d 2715 |
. . . 4
⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) limℂ 𝐶)) |
140 | | oveq1 6657 |
. . . . 5
⊢ (𝑦 = (◡𝐹‘𝑧) → (𝑦 − 𝐶) = ((◡𝐹‘𝑧) − 𝐶)) |
141 | | fveq2 6191 |
. . . . . 6
⊢ (𝑦 = (◡𝐹‘𝑧) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑧))) |
142 | 141 | oveq1d 6665 |
. . . . 5
⊢ (𝑦 = (◡𝐹‘𝑧) → ((𝐹‘𝑦) − (𝐹‘𝐶)) = ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶))) |
143 | 140, 142 | oveq12d 6668 |
. . . 4
⊢ (𝑦 = (◡𝐹‘𝑧) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) = (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶)))) |
144 | | eldifsni 4320 |
. . . . . . . . 9
⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ≠ (𝐹‘𝐶)) |
145 | 144 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝑧 ≠ (𝐹‘𝐶)) |
146 | 145 | necomd 2849 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘𝐶) ≠ 𝑧) |
147 | 1 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐹:𝑋–1-1-onto→𝑌) |
148 | 4 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐶 ∈ 𝑋) |
149 | 24 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝑧 ∈ 𝑌) |
150 | | f1ocnvfvb 6535 |
. . . . . . . . 9
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝐶) = 𝑧 ↔ (◡𝐹‘𝑧) = 𝐶)) |
151 | 147, 148,
149, 150 | syl3anc 1326 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) = 𝑧 ↔ (◡𝐹‘𝑧) = 𝐶)) |
152 | 151 | necon3abid 2830 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) ≠ 𝑧 ↔ ¬ (◡𝐹‘𝑧) = 𝐶)) |
153 | 146, 152 | mpbid 222 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ¬ (◡𝐹‘𝑧) = 𝐶) |
154 | 153 | pm2.21d 118 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((◡𝐹‘𝑧) = 𝐶 → (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (1 / ((𝑆 D 𝐹)‘𝐶)))) |
155 | 154 | impr 649 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ∧ (◡𝐹‘𝑧) = 𝐶)) → (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (1 / ((𝑆 D 𝐹)‘𝐶))) |
156 | 30, 64, 79, 139, 143, 155 | limcco 23657 |
. . 3
⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶)))) limℂ (𝐹‘𝐶))) |
157 | 75 | eqcomd 2628 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 = (◡𝐹‘(𝐹‘𝐶))) |
158 | 157 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐶 = (◡𝐹‘(𝐹‘𝐶))) |
159 | 158 | oveq2d 6666 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((◡𝐹‘𝑧) − 𝐶) = ((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶)))) |
160 | | f1ocnvfv2 6533 |
. . . . . . . 8
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧) |
161 | 1, 24, 160 | syl2an 494 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧) |
162 | 161 | oveq1d 6665 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶)) = (𝑧 − (𝐹‘𝐶))) |
163 | 159, 162 | oveq12d 6668 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) |
164 | 163 | mpteq2dva 4744 |
. . . 4
⊢ (𝜑 → (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶)))) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶))))) |
165 | 164 | oveq1d 6665 |
. . 3
⊢ (𝜑 → ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶)))) limℂ (𝐹‘𝐶)) = ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) limℂ (𝐹‘𝐶))) |
166 | 156, 165 | eleqtrd 2703 |
. 2
⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) limℂ (𝐹‘𝐶))) |
167 | | eqid 2622 |
. . 3
⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) |
168 | 23, 35 | fssd 6057 |
. . 3
⊢ (𝜑 → ◡𝐹:𝑌⟶ℂ) |
169 | 6, 7, 167, 11, 168, 43 | eldv 23662 |
. 2
⊢ (𝜑 → ((𝐹‘𝐶)(𝑆 D ◡𝐹)(1 / ((𝑆 D 𝐹)‘𝐶)) ↔ ((𝐹‘𝐶) ∈ ((int‘𝐾)‘𝑌) ∧ (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) limℂ (𝐹‘𝐶))))) |
170 | 20, 166, 169 | mpbir2and 957 |
1
⊢ (𝜑 → (𝐹‘𝐶)(𝑆 D ◡𝐹)(1 / ((𝑆 D 𝐹)‘𝐶))) |