| Step | Hyp | Ref
| Expression |
| 1 | | dvcnv.f |
. . . . 5
⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌) |
| 2 | | f1of 6848 |
. . . . 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 | ffvelcdmd 7105 |
. . 3
⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌) |
| 6 | | dvcnv.k |
. . . . . 6
⊢ 𝐾 = (𝐽 ↾t 𝑆) |
| 7 | | dvcnv.j |
. . . . . . . 8
⊢ 𝐽 =
(TopOpen‘ℂfld) |
| 8 | 7 | cnfldtopon 24803 |
. . . . . . 7
⊢ 𝐽 ∈
(TopOn‘ℂ) |
| 9 | | dvcnv.s |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
| 10 | | recnprss 25939 |
. . . . . . . 8
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
| 11 | 9, 10 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 12 | | resttopon 23169 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝑆 ⊆ ℂ)
→ (𝐽
↾t 𝑆)
∈ (TopOn‘𝑆)) |
| 13 | 8, 11, 12 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
| 14 | 6, 13 | eqeltrid 2845 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆)) |
| 15 | | topontop 22919 |
. . . . 5
⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝐾 ∈ Top) |
| 16 | 14, 15 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ Top) |
| 17 | | dvcnv.y |
. . . 4
⊢ (𝜑 → 𝑌 ∈ 𝐾) |
| 18 | | isopn3i 23090 |
. . . 4
⊢ ((𝐾 ∈ Top ∧ 𝑌 ∈ 𝐾) → ((int‘𝐾)‘𝑌) = 𝑌) |
| 19 | 16, 17, 18 | syl2anc 584 |
. . 3
⊢ (𝜑 → ((int‘𝐾)‘𝑌) = 𝑌) |
| 20 | 5, 19 | eleqtrrd 2844 |
. 2
⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝐾)‘𝑌)) |
| 21 | | f1ocnv 6860 |
. . . . . . . . 9
⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋) |
| 22 | | f1of 6848 |
. . . . . . . . 9
⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋) |
| 23 | 1, 21, 22 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → ◡𝐹:𝑌⟶𝑋) |
| 24 | | eldifi 4131 |
. . . . . . . 8
⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ∈ 𝑌) |
| 25 | | ffvelcdm 7101 |
. . . . . . . 8
⊢ ((◡𝐹:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (◡𝐹‘𝑧) ∈ 𝑋) |
| 26 | 23, 24, 25 | syl2an 596 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (◡𝐹‘𝑧) ∈ 𝑋) |
| 27 | 26 | anim1i 615 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (◡𝐹‘𝑧) ≠ 𝐶) → ((◡𝐹‘𝑧) ∈ 𝑋 ∧ (◡𝐹‘𝑧) ≠ 𝐶)) |
| 28 | | eldifsn 4786 |
. . . . . 6
⊢ ((◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶}) ↔ ((◡𝐹‘𝑧) ∈ 𝑋 ∧ (◡𝐹‘𝑧) ≠ 𝐶)) |
| 29 | 27, 28 | sylibr 234 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) ∧ (◡𝐹‘𝑧) ≠ 𝐶) → (◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶})) |
| 30 | 29 | anasss 466 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ∧ (◡𝐹‘𝑧) ≠ 𝐶)) → (◡𝐹‘𝑧) ∈ (𝑋 ∖ {𝐶})) |
| 31 | | eldifi 4131 |
. . . . . . 7
⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ∈ 𝑋) |
| 32 | | dvcnv.d |
. . . . . . . . . 10
⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) |
| 33 | | dvbsss 25937 |
. . . . . . . . . 10
⊢ dom
(𝑆 D 𝐹) ⊆ 𝑆 |
| 34 | 32, 33 | eqsstrrdi 4029 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
| 35 | 34, 11 | sstrd 3994 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ⊆ ℂ) |
| 36 | 35 | sselda 3983 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ℂ) |
| 37 | 31, 36 | sylan2 593 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ∈ ℂ) |
| 38 | 34, 4 | sseldd 3984 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ 𝑆) |
| 39 | 11, 38 | sseldd 3984 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 40 | 39 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐶 ∈ ℂ) |
| 41 | 37, 40 | subcld 11620 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ∈ ℂ) |
| 42 | | toponss 22933 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆) |
| 43 | 14, 17, 42 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
| 44 | 43, 11 | sstrd 3994 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ⊆ ℂ) |
| 45 | 3, 44 | fssd 6753 |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
| 46 | | ffvelcdm 7101 |
. . . . . . 7
⊢ ((𝐹:𝑋⟶ℂ ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ℂ) |
| 47 | 45, 31, 46 | syl2an 596 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝑦) ∈ ℂ) |
| 48 | 44, 5 | sseldd 3984 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
| 49 | 48 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝐹‘𝐶) ∈ ℂ) |
| 50 | 47, 49 | subcld 11620 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ∈ ℂ) |
| 51 | | eldifsni 4790 |
. . . . . . 7
⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) → 𝑦 ≠ 𝐶) |
| 52 | 51 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ≠ 𝐶) |
| 53 | 47, 49 | subeq0ad 11630 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ (𝐹‘𝑦) = (𝐹‘𝐶))) |
| 54 | | f1of1 6847 |
. . . . . . . . . . 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 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐹:𝑋–1-1→𝑌) |
| 57 | 31 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝑦 ∈ 𝑋) |
| 58 | 4 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → 𝐶 ∈ 𝑋) |
| 59 | | f1fveq 7282 |
. . . . . . . . 9
⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑦 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶)) |
| 60 | 56, 57, 58, 59 | syl12anc 837 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) = (𝐹‘𝐶) ↔ 𝑦 = 𝐶)) |
| 61 | 53, 60 | bitrd 279 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) = 0 ↔ 𝑦 = 𝐶)) |
| 62 | 61 | necon3bid 2985 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0 ↔ 𝑦 ≠ 𝐶)) |
| 63 | 52, 62 | mpbird 257 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝐹‘𝑦) − (𝐹‘𝐶)) ≠ 0) |
| 64 | 41, 50, 63 | divcld 12043 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) ∈ ℂ) |
| 65 | | limcresi 25920 |
. . . . . 6
⊢ (◡𝐹 limℂ (𝐹‘𝐶)) ⊆ ((◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) limℂ (𝐹‘𝐶)) |
| 66 | 23 | feqmptd 6977 |
. . . . . . . . 9
⊢ (𝜑 → ◡𝐹 = (𝑧 ∈ 𝑌 ↦ (◡𝐹‘𝑧))) |
| 67 | 66 | reseq1d 5996 |
. . . . . . . 8
⊢ (𝜑 → (◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = ((𝑧 ∈ 𝑌 ↦ (◡𝐹‘𝑧)) ↾ (𝑌 ∖ {(𝐹‘𝐶)}))) |
| 68 | | difss 4136 |
. . . . . . . . 9
⊢ (𝑌 ∖ {(𝐹‘𝐶)}) ⊆ 𝑌 |
| 69 | | resmpt 6055 |
. . . . . . . . 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 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝜑 → (◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) = (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧))) |
| 72 | 71 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → ((◡𝐹 ↾ (𝑌 ∖ {(𝐹‘𝐶)})) limℂ (𝐹‘𝐶)) = ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧)) limℂ (𝐹‘𝐶))) |
| 73 | 65, 72 | sseqtrid 4026 |
. . . . 5
⊢ (𝜑 → (◡𝐹 limℂ (𝐹‘𝐶)) ⊆ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧)) limℂ (𝐹‘𝐶))) |
| 74 | | f1ocnvfv1 7296 |
. . . . . . 7
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) |
| 75 | 1, 4, 74 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) |
| 76 | | dvcnv.i |
. . . . . . 7
⊢ (𝜑 → ◡𝐹 ∈ (𝑌–cn→𝑋)) |
| 77 | 76, 5 | cnlimci 25924 |
. . . . . 6
⊢ (𝜑 → (◡𝐹‘(𝐹‘𝐶)) ∈ (◡𝐹 limℂ (𝐹‘𝐶))) |
| 78 | 75, 77 | eqeltrrd 2842 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ (◡𝐹 limℂ (𝐹‘𝐶))) |
| 79 | 73, 78 | sseldd 3984 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (◡𝐹‘𝑧)) limℂ (𝐹‘𝐶))) |
| 80 | 45, 35, 4 | dvlem 25931 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ ℂ) |
| 81 | 37, 40, 52 | subne0d 11629 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (𝑦 − 𝐶) ≠ 0) |
| 82 | 50, 41, 63, 81 | divne0d 12059 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ≠ 0) |
| 83 | | eldifsn 4786 |
. . . . . . . 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 583 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) ∈ (ℂ ∖
{0})) |
| 85 | 84 | fmpttd 7135 |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))):(𝑋 ∖ {𝐶})⟶(ℂ ∖
{0})) |
| 86 | | difss 4136 |
. . . . . . 7
⊢ (ℂ
∖ {0}) ⊆ ℂ |
| 87 | 86 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (ℂ ∖ {0})
⊆ ℂ) |
| 88 | | eqid 2737 |
. . . . . 6
⊢ (𝐽 ↾t (ℂ
∖ {0})) = (𝐽
↾t (ℂ ∖ {0})) |
| 89 | 4, 32 | eleqtrrd 2844 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) |
| 90 | | dvfg 25941 |
. . . . . . . . . 10
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
| 91 | | ffun 6739 |
. . . . . . . . . 10
⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) |
| 92 | | funfvbrb 7071 |
. . . . . . . . . 10
⊢ (Fun
(𝑆 D 𝐹) → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))) |
| 93 | 9, 90, 91, 92 | 4syl 19 |
. . . . . . . . 9
⊢ (𝜑 → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))) |
| 94 | 89, 93 | mpbid 232 |
. . . . . . . 8
⊢ (𝜑 → 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶)) |
| 95 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) |
| 96 | 6, 7, 95, 11, 45, 34 | eldv 25933 |
. . . . . . . 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 | 94, 96 | mpbid 232 |
. . . . . . 7
⊢ (𝜑 → (𝐶 ∈ ((int‘𝐾)‘𝑋) ∧ ((𝑆 D 𝐹)‘𝐶) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) limℂ 𝐶))) |
| 98 | 97 | simprd 495 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) limℂ 𝐶)) |
| 99 | | resttopon 23169 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ (ℂ ∖ {0}) ⊆ ℂ) → (𝐽 ↾t (ℂ ∖ {0}))
∈ (TopOn‘(ℂ ∖ {0}))) |
| 100 | 8, 86, 99 | mp2an 692 |
. . . . . . . . 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 11256 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈
ℂ) |
| 104 | 101, 102,
103 | cnmptc 23670 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 1)
∈ ((𝐽
↾t (ℂ ∖ {0})) Cn 𝐽)) |
| 105 | 101 | cnmptid 23669 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ 𝑥) ∈ ((𝐽 ↾t (ℂ ∖ {0}))
Cn (𝐽 ↾t
(ℂ ∖ {0})))) |
| 106 | 7, 88 | divcn 24892 |
. . . . . . . . 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 23674 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) ∈ ((𝐽 ↾t (ℂ
∖ {0})) Cn 𝐽)) |
| 109 | 9, 90 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
| 110 | 32 | feq2d 6722 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ)) |
| 111 | 109, 110 | mpbid 232 |
. . . . . . . . 9
⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ) |
| 112 | 111, 4 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ℂ) |
| 113 | 109 | ffnd 6737 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 D 𝐹) Fn dom (𝑆 D 𝐹)) |
| 114 | | fnfvelrn 7100 |
. . . . . . . . . 10
⊢ (((𝑆 D 𝐹) Fn dom (𝑆 D 𝐹) ∧ 𝐶 ∈ dom (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹)) |
| 115 | 113, 89, 114 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹)) |
| 116 | | dvcnv.z |
. . . . . . . . 9
⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹)) |
| 117 | | nelne2 3040 |
. . . . . . . . 9
⊢ ((((𝑆 D 𝐹)‘𝐶) ∈ ran (𝑆 D 𝐹) ∧ ¬ 0 ∈ ran (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐶) ≠ 0) |
| 118 | 115, 116,
117 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ≠ 0) |
| 119 | | eldifsn 4786 |
. . . . . . . 8
⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) ↔
(((𝑆 D 𝐹)‘𝐶) ∈ ℂ ∧ ((𝑆 D 𝐹)‘𝐶) ≠ 0)) |
| 120 | 112, 118,
119 | sylanbrc 583 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖
{0})) |
| 121 | 100 | toponunii 22922 |
. . . . . . . 8
⊢ (ℂ
∖ {0}) = ∪ (𝐽 ↾t (ℂ ∖
{0})) |
| 122 | 121 | cncnpi 23286 |
. . . . . . 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))) |
| 123 | 108, 120,
122 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) ∈ (((𝐽 ↾t (ℂ
∖ {0})) CnP 𝐽)‘((𝑆 D 𝐹)‘𝐶))) |
| 124 | 85, 87, 7, 88, 98, 123 | limccnp 25926 |
. . . . 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)) |
| 125 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = ((𝑆 D 𝐹)‘𝐶) → (1 / 𝑥) = (1 / ((𝑆 D 𝐹)‘𝐶))) |
| 126 | | eqid 2737 |
. . . . . . 7
⊢ (𝑥 ∈ (ℂ ∖ {0})
↦ (1 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0})
↦ (1 / 𝑥)) |
| 127 | | ovex 7464 |
. . . . . . 7
⊢ (1 /
((𝑆 D 𝐹)‘𝐶)) ∈ V |
| 128 | 125, 126,
127 | fvmpt 7016 |
. . . . . 6
⊢ (((𝑆 D 𝐹)‘𝐶) ∈ (ℂ ∖ {0}) →
((𝑥 ∈ (ℂ ∖
{0}) ↦ (1 / 𝑥))‘((𝑆 D 𝐹)‘𝐶)) = (1 / ((𝑆 D 𝐹)‘𝐶))) |
| 129 | 120, 128 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥))‘((𝑆 D 𝐹)‘𝐶)) = (1 / ((𝑆 D 𝐹)‘𝐶))) |
| 130 | | eqidd 2738 |
. . . . . . . 8
⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) |
| 131 | | eqidd 2738 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (1 /
𝑥))) |
| 132 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑥 = (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)) → (1 / 𝑥) = (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) |
| 133 | 84, 130, 131, 132 | fmptco 7149 |
. . . . . . 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))))) |
| 134 | 50, 41, 63, 81 | recdivd 12060 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {𝐶})) → (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶))) = ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) |
| 135 | 134 | mpteq2dva 5242 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ (1 / (((𝐹‘𝑦) − (𝐹‘𝐶)) / (𝑦 − 𝐶)))) = (𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))))) |
| 136 | 133, 135 | eqtrd 2777 |
. . . . . 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))))) |
| 137 | 136 | oveq1d 7446 |
. . . . 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)) |
| 138 | 124, 129,
137 | 3eltr3d 2855 |
. . . 4
⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑦 ∈ (𝑋 ∖ {𝐶}) ↦ ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶)))) limℂ 𝐶)) |
| 139 | | oveq1 7438 |
. . . . 5
⊢ (𝑦 = (◡𝐹‘𝑧) → (𝑦 − 𝐶) = ((◡𝐹‘𝑧) − 𝐶)) |
| 140 | | fveq2 6906 |
. . . . . 6
⊢ (𝑦 = (◡𝐹‘𝑧) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑧))) |
| 141 | 140 | oveq1d 7446 |
. . . . 5
⊢ (𝑦 = (◡𝐹‘𝑧) → ((𝐹‘𝑦) − (𝐹‘𝐶)) = ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶))) |
| 142 | 139, 141 | oveq12d 7449 |
. . . 4
⊢ (𝑦 = (◡𝐹‘𝑧) → ((𝑦 − 𝐶) / ((𝐹‘𝑦) − (𝐹‘𝐶))) = (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶)))) |
| 143 | | eldifsni 4790 |
. . . . . . . . 9
⊢ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) → 𝑧 ≠ (𝐹‘𝐶)) |
| 144 | 143 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝑧 ≠ (𝐹‘𝐶)) |
| 145 | 144 | necomd 2996 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘𝐶) ≠ 𝑧) |
| 146 | | f1ocnvfvb 7299 |
. . . . . . . . 9
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝐶) = 𝑧 ↔ (◡𝐹‘𝑧) = 𝐶)) |
| 147 | 1, 4, 24, 146 | syl2an3an 1424 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) = 𝑧 ↔ (◡𝐹‘𝑧) = 𝐶)) |
| 148 | 147 | necon3abid 2977 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘𝐶) ≠ 𝑧 ↔ ¬ (◡𝐹‘𝑧) = 𝐶)) |
| 149 | 145, 148 | mpbid 232 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ¬ (◡𝐹‘𝑧) = 𝐶) |
| 150 | 149 | pm2.21d 121 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((◡𝐹‘𝑧) = 𝐶 → (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (1 / ((𝑆 D 𝐹)‘𝐶)))) |
| 151 | 150 | impr 454 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ∧ (◡𝐹‘𝑧) = 𝐶)) → (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (1 / ((𝑆 D 𝐹)‘𝐶))) |
| 152 | 30, 64, 79, 138, 142, 151 | limcco 25928 |
. . 3
⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶)))) limℂ (𝐹‘𝐶))) |
| 153 | 75 | eqcomd 2743 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 = (◡𝐹‘(𝐹‘𝐶))) |
| 154 | 153 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → 𝐶 = (◡𝐹‘(𝐹‘𝐶))) |
| 155 | 154 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((◡𝐹‘𝑧) − 𝐶) = ((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶)))) |
| 156 | | f1ocnvfv2 7297 |
. . . . . . . 8
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧) |
| 157 | 1, 24, 156 | syl2an 596 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧) |
| 158 | 157 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶)) = (𝑧 − (𝐹‘𝐶))) |
| 159 | 155, 158 | oveq12d 7449 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)})) → (((◡𝐹‘𝑧) − 𝐶) / ((𝐹‘(◡𝐹‘𝑧)) − (𝐹‘𝐶))) = (((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) |
| 160 | 159 | mpteq2dva 5242 |
. . . 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))))) |
| 161 | 160 | oveq1d 7446 |
. . 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))) |
| 162 | 152, 161 | eleqtrd 2843 |
. 2
⊢ (𝜑 → (1 / ((𝑆 D 𝐹)‘𝐶)) ∈ ((𝑧 ∈ (𝑌 ∖ {(𝐹‘𝐶)}) ↦ (((◡𝐹‘𝑧) − (◡𝐹‘(𝐹‘𝐶))) / (𝑧 − (𝐹‘𝐶)))) limℂ (𝐹‘𝐶))) |
| 163 | | eqid 2737 |
. . 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)))) |
| 164 | 23, 35 | fssd 6753 |
. . 3
⊢ (𝜑 → ◡𝐹:𝑌⟶ℂ) |
| 165 | 6, 7, 163, 11, 164, 43 | eldv 25933 |
. 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))))) |
| 166 | 20, 162, 165 | mpbir2and 713 |
1
⊢ (𝜑 → (𝐹‘𝐶)(𝑆 D ◡𝐹)(1 / ((𝑆 D 𝐹)‘𝐶))) |