Proof of Theorem dvreslem
| Step | Hyp | Ref
| Expression |
| 1 | | difss 4136 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ⊆ (𝐴 ∩ 𝐵) |
| 2 | | inss2 4238 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 |
| 3 | 1, 2 | sstri 3993 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ⊆ 𝐵 |
| 4 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) ∧ 𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥})) → 𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥})) |
| 5 | 3, 4 | sselid 3981 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) ∧ 𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥})) → 𝑧 ∈ 𝐵) |
| 6 | 5 | fvresd 6926 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) ∧ 𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥})) → ((𝐹 ↾ 𝐵)‘𝑧) = (𝐹‘𝑧)) |
| 7 | | dvres.t |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑇 = (𝐾 ↾t 𝑆) |
| 8 | | dvres.k |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐾 =
(TopOpen‘ℂfld) |
| 9 | 8 | cnfldtop 24804 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐾 ∈ Top |
| 10 | | dvres.s |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 11 | | cnex 11236 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ℂ
∈ V |
| 12 | | ssexg 5323 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑆 ⊆ ℂ ∧ ℂ
∈ V) → 𝑆 ∈
V) |
| 13 | 10, 11, 12 | sylancl 586 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑆 ∈ V) |
| 14 | | resttop 23168 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ Top ∧ 𝑆 ∈ V) → (𝐾 ↾t 𝑆) ∈ Top) |
| 15 | 9, 13, 14 | sylancr 587 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐾 ↾t 𝑆) ∈ Top) |
| 16 | 7, 15 | eqeltrid 2845 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑇 ∈ Top) |
| 17 | | inss1 4237 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 |
| 18 | | dvres.a |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
| 19 | 17, 18 | sstrid 3995 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝑆) |
| 20 | 8 | cnfldtopon 24803 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝐾 ∈
(TopOn‘ℂ) |
| 21 | | resttopon 23169 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐾 ∈ (TopOn‘ℂ)
∧ 𝑆 ⊆ ℂ)
→ (𝐾
↾t 𝑆)
∈ (TopOn‘𝑆)) |
| 22 | 20, 10, 21 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
| 23 | 7, 22 | eqeltrid 2845 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑇 ∈ (TopOn‘𝑆)) |
| 24 | | toponuni 22920 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑇 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝑇) |
| 25 | 23, 24 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑆 = ∪ 𝑇) |
| 26 | 19, 25 | sseqtrd 4020 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ ∪ 𝑇) |
| 27 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ 𝑇 =
∪ 𝑇 |
| 28 | 27 | ntrss2 23065 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑇 ∈ Top ∧ (𝐴 ∩ 𝐵) ⊆ ∪ 𝑇) → ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ⊆ (𝐴 ∩ 𝐵)) |
| 29 | 16, 26, 28 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ⊆ (𝐴 ∩ 𝐵)) |
| 30 | 29, 2 | sstrdi 3996 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ⊆ 𝐵) |
| 31 | 30 | sselda 3983 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → 𝑥 ∈ 𝐵) |
| 32 | 31 | fvresd 6926 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) |
| 33 | 32 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) ∧ 𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥})) → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) |
| 34 | 6, 33 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) ∧ 𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥})) → (((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) = ((𝐹‘𝑧) − (𝐹‘𝑥))) |
| 35 | 34 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) ∧ 𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥})) → ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥)) = (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) |
| 36 | 35 | mpteq2dva 5242 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → (𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)))) |
| 37 | | dvres.g |
. . . . . . . . . . 11
⊢ 𝐺 = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) |
| 38 | 37 | reseq1i 5993 |
. . . . . . . . . 10
⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∖ {𝑥})) = ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) ↾ ((𝐴 ∩ 𝐵) ∖ {𝑥})) |
| 39 | | ssdif 4144 |
. . . . . . . . . . 11
⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → ((𝐴 ∩ 𝐵) ∖ {𝑥}) ⊆ (𝐴 ∖ {𝑥})) |
| 40 | | resmpt 6055 |
. . . . . . . . . . 11
⊢ (((𝐴 ∩ 𝐵) ∖ {𝑥}) ⊆ (𝐴 ∖ {𝑥}) → ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) ↾ ((𝐴 ∩ 𝐵) ∖ {𝑥})) = (𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)))) |
| 41 | 17, 39, 40 | mp2b 10 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) ↾ ((𝐴 ∩ 𝐵) ∖ {𝑥})) = (𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) |
| 42 | 38, 41 | eqtri 2765 |
. . . . . . . . 9
⊢ (𝐺 ↾ ((𝐴 ∩ 𝐵) ∖ {𝑥})) = (𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) |
| 43 | 36, 42 | eqtr4di 2795 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → (𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) = (𝐺 ↾ ((𝐴 ∩ 𝐵) ∖ {𝑥}))) |
| 44 | 43 | oveq1d 7446 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → ((𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥) = ((𝐺 ↾ ((𝐴 ∩ 𝐵) ∖ {𝑥})) limℂ 𝑥)) |
| 45 | | dvres.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
| 46 | 45 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → 𝐹:𝐴⟶ℂ) |
| 47 | 18, 10 | sstrd 3994 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ⊆ ℂ) |
| 48 | 47 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → 𝐴 ⊆ ℂ) |
| 49 | 29, 17 | sstrdi 3996 |
. . . . . . . . . . 11
⊢ (𝜑 → ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ⊆ 𝐴) |
| 50 | 49 | sselda 3983 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → 𝑥 ∈ 𝐴) |
| 51 | 46, 48, 50 | dvlem 25931 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) ∧ 𝑧 ∈ (𝐴 ∖ {𝑥})) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) ∈ ℂ) |
| 52 | 51, 37 | fmptd 7134 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → 𝐺:(𝐴 ∖ {𝑥})⟶ℂ) |
| 53 | 17, 39 | mp1i 13 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → ((𝐴 ∩ 𝐵) ∖ {𝑥}) ⊆ (𝐴 ∖ {𝑥})) |
| 54 | | difss 4136 |
. . . . . . . . 9
⊢ (𝐴 ∖ {𝑥}) ⊆ 𝐴 |
| 55 | 54, 48 | sstrid 3995 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → (𝐴 ∖ {𝑥}) ⊆ ℂ) |
| 56 | | eqid 2737 |
. . . . . . . 8
⊢ (𝐾 ↾t ((𝐴 ∖ {𝑥}) ∪ {𝑥})) = (𝐾 ↾t ((𝐴 ∖ {𝑥}) ∪ {𝑥})) |
| 57 | | difssd 4137 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (∪ 𝑇
∖ 𝐴) ⊆ ∪ 𝑇) |
| 58 | 26, 57 | unssd 4192 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴)) ⊆ ∪
𝑇) |
| 59 | | ssun1 4178 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∩ 𝐵) ⊆ ((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴)) |
| 60 | 59 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ ((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴))) |
| 61 | 27 | ntrss 23063 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ Top ∧ ((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴)) ⊆ ∪
𝑇 ∧ (𝐴 ∩ 𝐵) ⊆ ((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴))) → ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝑇)‘((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴)))) |
| 62 | 16, 58, 60, 61 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝑇)‘((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴)))) |
| 63 | 62, 49 | ssind 4241 |
. . . . . . . . . . 11
⊢ (𝜑 → ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ⊆ (((int‘𝑇)‘((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴))) ∩ 𝐴)) |
| 64 | 18, 25 | sseqtrd 4020 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑇) |
| 65 | 17 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴) |
| 66 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ (𝑇 ↾t 𝐴) = (𝑇 ↾t 𝐴) |
| 67 | 27, 66 | restntr 23190 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑇
∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → ((int‘(𝑇 ↾t 𝐴))‘(𝐴 ∩ 𝐵)) = (((int‘𝑇)‘((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴))) ∩ 𝐴)) |
| 68 | 16, 64, 65, 67 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((int‘(𝑇 ↾t 𝐴))‘(𝐴 ∩ 𝐵)) = (((int‘𝑇)‘((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴))) ∩ 𝐴)) |
| 69 | 7 | oveq1i 7441 |
. . . . . . . . . . . . . . 15
⊢ (𝑇 ↾t 𝐴) = ((𝐾 ↾t 𝑆) ↾t 𝐴) |
| 70 | 9 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐾 ∈ Top) |
| 71 | | restabs 23173 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑆 ∧ 𝑆 ∈ V) → ((𝐾 ↾t 𝑆) ↾t 𝐴) = (𝐾 ↾t 𝐴)) |
| 72 | 70, 18, 13, 71 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐾 ↾t 𝑆) ↾t 𝐴) = (𝐾 ↾t 𝐴)) |
| 73 | 69, 72 | eqtrid 2789 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑇 ↾t 𝐴) = (𝐾 ↾t 𝐴)) |
| 74 | 73 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (int‘(𝑇 ↾t 𝐴)) = (int‘(𝐾 ↾t 𝐴))) |
| 75 | 74 | fveq1d 6908 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((int‘(𝑇 ↾t 𝐴))‘(𝐴 ∩ 𝐵)) = ((int‘(𝐾 ↾t 𝐴))‘(𝐴 ∩ 𝐵))) |
| 76 | 68, 75 | eqtr3d 2779 |
. . . . . . . . . . 11
⊢ (𝜑 → (((int‘𝑇)‘((𝐴 ∩ 𝐵) ∪ (∪ 𝑇 ∖ 𝐴))) ∩ 𝐴) = ((int‘(𝐾 ↾t 𝐴))‘(𝐴 ∩ 𝐵))) |
| 77 | 63, 76 | sseqtrd 4020 |
. . . . . . . . . 10
⊢ (𝜑 → ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘(𝐾 ↾t 𝐴))‘(𝐴 ∩ 𝐵))) |
| 78 | 77 | sselda 3983 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → 𝑥 ∈ ((int‘(𝐾 ↾t 𝐴))‘(𝐴 ∩ 𝐵))) |
| 79 | | undif1 4476 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥}) |
| 80 | 29 | sselda 3983 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → 𝑥 ∈ (𝐴 ∩ 𝐵)) |
| 81 | 80 | snssd 4809 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → {𝑥} ⊆ (𝐴 ∩ 𝐵)) |
| 82 | 81, 17 | sstrdi 3996 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → {𝑥} ⊆ 𝐴) |
| 83 | | ssequn2 4189 |
. . . . . . . . . . . . . 14
⊢ ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴) |
| 84 | 82, 83 | sylib 218 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → (𝐴 ∪ {𝑥}) = 𝐴) |
| 85 | 79, 84 | eqtrid 2789 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = 𝐴) |
| 86 | 85 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → (𝐾 ↾t ((𝐴 ∖ {𝑥}) ∪ {𝑥})) = (𝐾 ↾t 𝐴)) |
| 87 | 86 | fveq2d 6910 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → (int‘(𝐾 ↾t ((𝐴 ∖ {𝑥}) ∪ {𝑥}))) = (int‘(𝐾 ↾t 𝐴))) |
| 88 | | undif1 4476 |
. . . . . . . . . . 11
⊢ (((𝐴 ∩ 𝐵) ∖ {𝑥}) ∪ {𝑥}) = ((𝐴 ∩ 𝐵) ∪ {𝑥}) |
| 89 | | ssequn2 4189 |
. . . . . . . . . . . 12
⊢ ({𝑥} ⊆ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∪ {𝑥}) = (𝐴 ∩ 𝐵)) |
| 90 | 81, 89 | sylib 218 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → ((𝐴 ∩ 𝐵) ∪ {𝑥}) = (𝐴 ∩ 𝐵)) |
| 91 | 88, 90 | eqtrid 2789 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → (((𝐴 ∩ 𝐵) ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∩ 𝐵)) |
| 92 | 87, 91 | fveq12d 6913 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → ((int‘(𝐾 ↾t ((𝐴 ∖ {𝑥}) ∪ {𝑥})))‘(((𝐴 ∩ 𝐵) ∖ {𝑥}) ∪ {𝑥})) = ((int‘(𝐾 ↾t 𝐴))‘(𝐴 ∩ 𝐵))) |
| 93 | 78, 92 | eleqtrrd 2844 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → 𝑥 ∈ ((int‘(𝐾 ↾t ((𝐴 ∖ {𝑥}) ∪ {𝑥})))‘(((𝐴 ∩ 𝐵) ∖ {𝑥}) ∪ {𝑥}))) |
| 94 | 52, 53, 55, 8, 56, 93 | limcres 25921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → ((𝐺 ↾ ((𝐴 ∩ 𝐵) ∖ {𝑥})) limℂ 𝑥) = (𝐺 limℂ 𝑥)) |
| 95 | 44, 94 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → ((𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥) = (𝐺 limℂ 𝑥)) |
| 96 | 95 | eleq2d 2827 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵))) → (𝑦 ∈ ((𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥) ↔ 𝑦 ∈ (𝐺 limℂ 𝑥))) |
| 97 | 96 | pm5.32da 579 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ∧ 𝑦 ∈ ((𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ↔ (𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ∧ 𝑦 ∈ (𝐺 limℂ 𝑥)))) |
| 98 | | dvres.b |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ⊆ 𝑆) |
| 99 | 98, 25 | sseqtrd 4020 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ⊆ ∪ 𝑇) |
| 100 | 27 | ntrin 23069 |
. . . . . . . 8
⊢ ((𝑇 ∈ Top ∧ 𝐴 ⊆ ∪ 𝑇
∧ 𝐵 ⊆ ∪ 𝑇)
→ ((int‘𝑇)‘(𝐴 ∩ 𝐵)) = (((int‘𝑇)‘𝐴) ∩ ((int‘𝑇)‘𝐵))) |
| 101 | 16, 64, 99, 100 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → ((int‘𝑇)‘(𝐴 ∩ 𝐵)) = (((int‘𝑇)‘𝐴) ∩ ((int‘𝑇)‘𝐵))) |
| 102 | 101 | eleq2d 2827 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ↔ 𝑥 ∈ (((int‘𝑇)‘𝐴) ∩ ((int‘𝑇)‘𝐵)))) |
| 103 | | elin 3967 |
. . . . . 6
⊢ (𝑥 ∈ (((int‘𝑇)‘𝐴) ∩ ((int‘𝑇)‘𝐵)) ↔ (𝑥 ∈ ((int‘𝑇)‘𝐴) ∧ 𝑥 ∈ ((int‘𝑇)‘𝐵))) |
| 104 | 102, 103 | bitrdi 287 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ ((int‘𝑇)‘𝐴) ∧ 𝑥 ∈ ((int‘𝑇)‘𝐵)))) |
| 105 | 104 | anbi1d 631 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ∧ 𝑦 ∈ (𝐺 limℂ 𝑥)) ↔ ((𝑥 ∈ ((int‘𝑇)‘𝐴) ∧ 𝑥 ∈ ((int‘𝑇)‘𝐵)) ∧ 𝑦 ∈ (𝐺 limℂ 𝑥)))) |
| 106 | 97, 105 | bitrd 279 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ∧ 𝑦 ∈ ((𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ↔ ((𝑥 ∈ ((int‘𝑇)‘𝐴) ∧ 𝑥 ∈ ((int‘𝑇)‘𝐵)) ∧ 𝑦 ∈ (𝐺 limℂ 𝑥)))) |
| 107 | | an32 646 |
. . 3
⊢ (((𝑥 ∈ ((int‘𝑇)‘𝐴) ∧ 𝑥 ∈ ((int‘𝑇)‘𝐵)) ∧ 𝑦 ∈ (𝐺 limℂ 𝑥)) ↔ ((𝑥 ∈ ((int‘𝑇)‘𝐴) ∧ 𝑦 ∈ (𝐺 limℂ 𝑥)) ∧ 𝑥 ∈ ((int‘𝑇)‘𝐵))) |
| 108 | 106, 107 | bitrdi 287 |
. 2
⊢ (𝜑 → ((𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ∧ 𝑦 ∈ ((𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ↔ ((𝑥 ∈ ((int‘𝑇)‘𝐴) ∧ 𝑦 ∈ (𝐺 limℂ 𝑥)) ∧ 𝑥 ∈ ((int‘𝑇)‘𝐵)))) |
| 109 | | eqid 2737 |
. . 3
⊢ (𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) |
| 110 | | fresin 6777 |
. . . 4
⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ) |
| 111 | 45, 110 | syl 17 |
. . 3
⊢ (𝜑 → (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ) |
| 112 | 7, 8, 109, 10, 111, 19 | eldv 25933 |
. 2
⊢ (𝜑 → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘(𝐴 ∩ 𝐵)) ∧ 𝑦 ∈ ((𝑧 ∈ ((𝐴 ∩ 𝐵) ∖ {𝑥}) ↦ ((((𝐹 ↾ 𝐵)‘𝑧) − ((𝐹 ↾ 𝐵)‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)))) |
| 113 | 7, 8, 37, 10, 45, 18 | eldv 25933 |
. . 3
⊢ (𝜑 → (𝑥(𝑆 D 𝐹)𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘𝐴) ∧ 𝑦 ∈ (𝐺 limℂ 𝑥)))) |
| 114 | 113 | anbi1cd 635 |
. 2
⊢ (𝜑 → ((𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦) ↔ ((𝑥 ∈ ((int‘𝑇)‘𝐴) ∧ 𝑦 ∈ (𝐺 limℂ 𝑥)) ∧ 𝑥 ∈ ((int‘𝑇)‘𝐵)))) |
| 115 | 108, 112,
114 | 3bitr4d 311 |
1
⊢ (𝜑 → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦))) |