| Step | Hyp | Ref
| Expression |
| 1 | | dvadd.bf |
. . . . . 6
⊢ (𝜑 → 𝐶(𝑆 D 𝐹)𝐾) |
| 2 | | eqid 2737 |
. . . . . . 7
⊢ (𝐽 ↾t 𝑆) = (𝐽 ↾t 𝑆) |
| 3 | | dvadd.j |
. . . . . . 7
⊢ 𝐽 =
(TopOpen‘ℂfld) |
| 4 | | eqid 2737 |
. . . . . . 7
⊢ (𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) |
| 5 | | dvaddbr.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 6 | | dvadd.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
| 7 | | dvadd.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
| 8 | 2, 3, 4, 5, 6, 7 | eldv 25933 |
. . . . . 6
⊢ (𝜑 → (𝐶(𝑆 D 𝐹)𝐾 ↔ (𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)))) |
| 9 | 1, 8 | mpbid 232 |
. . . . 5
⊢ (𝜑 → (𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶))) |
| 10 | 9 | simpld 494 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘𝑋)) |
| 11 | | dvadd.bg |
. . . . . 6
⊢ (𝜑 → 𝐶(𝑆 D 𝐺)𝐿) |
| 12 | | eqid 2737 |
. . . . . . 7
⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) |
| 13 | | dvadd.g |
. . . . . . 7
⊢ (𝜑 → 𝐺:𝑌⟶ℂ) |
| 14 | | dvadd.y |
. . . . . . 7
⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
| 15 | 2, 3, 12, 5, 13, 14 | eldv 25933 |
. . . . . 6
⊢ (𝜑 → (𝐶(𝑆 D 𝐺)𝐿 ↔ (𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)))) |
| 16 | 11, 15 | mpbid 232 |
. . . . 5
⊢ (𝜑 → (𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶))) |
| 17 | 16 | simpld 494 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘𝑌)) |
| 18 | 10, 17 | elind 4200 |
. . 3
⊢ (𝜑 → 𝐶 ∈ (((int‘(𝐽 ↾t 𝑆))‘𝑋) ∩ ((int‘(𝐽 ↾t 𝑆))‘𝑌))) |
| 19 | 3 | cnfldtopon 24803 |
. . . . . 6
⊢ 𝐽 ∈
(TopOn‘ℂ) |
| 20 | | resttopon 23169 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝑆 ⊆ ℂ)
→ (𝐽
↾t 𝑆)
∈ (TopOn‘𝑆)) |
| 21 | 19, 5, 20 | sylancr 587 |
. . . . 5
⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
| 22 | | topontop 22919 |
. . . . 5
⊢ ((𝐽 ↾t 𝑆) ∈ (TopOn‘𝑆) → (𝐽 ↾t 𝑆) ∈ Top) |
| 23 | 21, 22 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ Top) |
| 24 | | toponuni 22920 |
. . . . . 6
⊢ ((𝐽 ↾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ (𝐽 ↾t 𝑆)) |
| 25 | 21, 24 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑆 = ∪ (𝐽 ↾t 𝑆)) |
| 26 | 7, 25 | sseqtrd 4020 |
. . . 4
⊢ (𝜑 → 𝑋 ⊆ ∪ (𝐽 ↾t 𝑆)) |
| 27 | 14, 25 | sseqtrd 4020 |
. . . 4
⊢ (𝜑 → 𝑌 ⊆ ∪ (𝐽 ↾t 𝑆)) |
| 28 | | eqid 2737 |
. . . . 5
⊢ ∪ (𝐽
↾t 𝑆) =
∪ (𝐽 ↾t 𝑆) |
| 29 | 28 | ntrin 23069 |
. . . 4
⊢ (((𝐽 ↾t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ (𝐽
↾t 𝑆)
∧ 𝑌 ⊆ ∪ (𝐽
↾t 𝑆))
→ ((int‘(𝐽
↾t 𝑆))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾t 𝑆))‘𝑋) ∩ ((int‘(𝐽 ↾t 𝑆))‘𝑌))) |
| 30 | 23, 26, 27, 29 | syl3anc 1373 |
. . 3
⊢ (𝜑 → ((int‘(𝐽 ↾t 𝑆))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾t 𝑆))‘𝑋) ∩ ((int‘(𝐽 ↾t 𝑆))‘𝑌))) |
| 31 | 18, 30 | eleqtrrd 2844 |
. 2
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘(𝑋 ∩ 𝑌))) |
| 32 | | inss1 4237 |
. . . . . . 7
⊢ (𝑋 ∩ 𝑌) ⊆ 𝑋 |
| 33 | | ssdif 4144 |
. . . . . . 7
⊢ ((𝑋 ∩ 𝑌) ⊆ 𝑋 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑋 ∖ {𝐶})) |
| 34 | 32, 33 | mp1i 13 |
. . . . . 6
⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑋 ∖ {𝐶})) |
| 35 | 34 | sselda 3983 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ∈ (𝑋 ∖ {𝐶})) |
| 36 | 7, 5 | sstrd 3994 |
. . . . . 6
⊢ (𝜑 → 𝑋 ⊆ ℂ) |
| 37 | 28 | ntrss2 23065 |
. . . . . . . 8
⊢ (((𝐽 ↾t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ (𝐽
↾t 𝑆))
→ ((int‘(𝐽
↾t 𝑆))‘𝑋) ⊆ 𝑋) |
| 38 | 23, 26, 37 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((int‘(𝐽 ↾t 𝑆))‘𝑋) ⊆ 𝑋) |
| 39 | 38, 10 | sseldd 3984 |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ 𝑋) |
| 40 | 6, 36, 39 | dvlem 25931 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 ∖ {𝐶})) → (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ) |
| 41 | 35, 40 | syldan 591 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ) |
| 42 | | inss2 4238 |
. . . . . . 7
⊢ (𝑋 ∩ 𝑌) ⊆ 𝑌 |
| 43 | | ssdif 4144 |
. . . . . . 7
⊢ ((𝑋 ∩ 𝑌) ⊆ 𝑌 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑌 ∖ {𝐶})) |
| 44 | 42, 43 | mp1i 13 |
. . . . . 6
⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑌 ∖ {𝐶})) |
| 45 | 44 | sselda 3983 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ∈ (𝑌 ∖ {𝐶})) |
| 46 | 14, 5 | sstrd 3994 |
. . . . . 6
⊢ (𝜑 → 𝑌 ⊆ ℂ) |
| 47 | 28 | ntrss2 23065 |
. . . . . . . 8
⊢ (((𝐽 ↾t 𝑆) ∈ Top ∧ 𝑌 ⊆ ∪ (𝐽
↾t 𝑆))
→ ((int‘(𝐽
↾t 𝑆))‘𝑌) ⊆ 𝑌) |
| 48 | 23, 27, 47 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((int‘(𝐽 ↾t 𝑆))‘𝑌) ⊆ 𝑌) |
| 49 | 48, 17 | sseldd 3984 |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ 𝑌) |
| 50 | 13, 46, 49 | dvlem 25931 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ) |
| 51 | 45, 50 | syldan 591 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ) |
| 52 | | ssidd 4007 |
. . . 4
⊢ (𝜑 → ℂ ⊆
ℂ) |
| 53 | | txtopon 23599 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝐽 ∈
(TopOn‘ℂ)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(ℂ ×
ℂ))) |
| 54 | 19, 19, 53 | mp2an 692 |
. . . . 5
⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℂ
× ℂ)) |
| 55 | 54 | toponrestid 22927 |
. . . 4
⊢ (𝐽 ×t 𝐽) = ((𝐽 ×t 𝐽) ↾t (ℂ ×
ℂ)) |
| 56 | 9 | simprd 495 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 57 | 40 | fmpttd 7135 |
. . . . . . 7
⊢ (𝜑 → (𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))):(𝑋 ∖ {𝐶})⟶ℂ) |
| 58 | 36 | ssdifssd 4147 |
. . . . . . 7
⊢ (𝜑 → (𝑋 ∖ {𝐶}) ⊆ ℂ) |
| 59 | | eqid 2737 |
. . . . . . 7
⊢ (𝐽 ↾t ((𝑋 ∖ {𝐶}) ∪ {𝐶})) = (𝐽 ↾t ((𝑋 ∖ {𝐶}) ∪ {𝐶})) |
| 60 | 32, 7 | sstrid 3995 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ 𝑆) |
| 61 | 60, 25 | sseqtrd 4020 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ ∪
(𝐽 ↾t
𝑆)) |
| 62 | | difssd 4137 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (∪ (𝐽
↾t 𝑆)
∖ 𝑋) ⊆ ∪ (𝐽
↾t 𝑆)) |
| 63 | 61, 62 | unssd 4192 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋)) ⊆ ∪
(𝐽 ↾t
𝑆)) |
| 64 | | ssun1 4178 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋)) |
| 65 | 64 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋))) |
| 66 | 28 | ntrss 23063 |
. . . . . . . . . . . 12
⊢ (((𝐽 ↾t 𝑆) ∈ Top ∧ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋)) ⊆ ∪
(𝐽 ↾t
𝑆) ∧ (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋))) → ((int‘(𝐽 ↾t 𝑆))‘(𝑋 ∩ 𝑌)) ⊆ ((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋)))) |
| 67 | 23, 63, 65, 66 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → ((int‘(𝐽 ↾t 𝑆))‘(𝑋 ∩ 𝑌)) ⊆ ((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋)))) |
| 68 | 67, 31 | sseldd 3984 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋)))) |
| 69 | 68, 39 | elind 4200 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ (((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋))) ∩ 𝑋)) |
| 70 | 32 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ 𝑋) |
| 71 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ ((𝐽 ↾t 𝑆) ↾t 𝑋) = ((𝐽 ↾t 𝑆) ↾t 𝑋) |
| 72 | 28, 71 | restntr 23190 |
. . . . . . . . . . 11
⊢ (((𝐽 ↾t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ (𝐽
↾t 𝑆)
∧ (𝑋 ∩ 𝑌) ⊆ 𝑋) → ((int‘((𝐽 ↾t 𝑆) ↾t 𝑋))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋))) ∩ 𝑋)) |
| 73 | 23, 26, 70, 72 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → ((int‘((𝐽 ↾t 𝑆) ↾t 𝑋))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋))) ∩ 𝑋)) |
| 74 | 3 | cnfldtop 24804 |
. . . . . . . . . . . . . 14
⊢ 𝐽 ∈ Top |
| 75 | 74 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐽 ∈ Top) |
| 76 | | cnex 11236 |
. . . . . . . . . . . . . 14
⊢ ℂ
∈ V |
| 77 | | ssexg 5323 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ⊆ ℂ ∧ ℂ
∈ V) → 𝑆 ∈
V) |
| 78 | 5, 76, 77 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ∈ V) |
| 79 | | restabs 23173 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ V) → ((𝐽 ↾t 𝑆) ↾t 𝑋) = (𝐽 ↾t 𝑋)) |
| 80 | 75, 7, 78, 79 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐽 ↾t 𝑆) ↾t 𝑋) = (𝐽 ↾t 𝑋)) |
| 81 | 80 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ (𝜑 → (int‘((𝐽 ↾t 𝑆) ↾t 𝑋)) = (int‘(𝐽 ↾t 𝑋))) |
| 82 | 81 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝜑 → ((int‘((𝐽 ↾t 𝑆) ↾t 𝑋))‘(𝑋 ∩ 𝑌)) = ((int‘(𝐽 ↾t 𝑋))‘(𝑋 ∩ 𝑌))) |
| 83 | 73, 82 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (𝜑 → (((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑋))) ∩ 𝑋) = ((int‘(𝐽 ↾t 𝑋))‘(𝑋 ∩ 𝑌))) |
| 84 | 69, 83 | eleqtrd 2843 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t 𝑋))‘(𝑋 ∩ 𝑌))) |
| 85 | | undif1 4476 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∖ {𝐶}) ∪ {𝐶}) = (𝑋 ∪ {𝐶}) |
| 86 | 39 | snssd 4809 |
. . . . . . . . . . . . 13
⊢ (𝜑 → {𝐶} ⊆ 𝑋) |
| 87 | | ssequn2 4189 |
. . . . . . . . . . . . 13
⊢ ({𝐶} ⊆ 𝑋 ↔ (𝑋 ∪ {𝐶}) = 𝑋) |
| 88 | 86, 87 | sylib 218 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑋 ∪ {𝐶}) = 𝑋) |
| 89 | 85, 88 | eqtrid 2789 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑋 ∖ {𝐶}) ∪ {𝐶}) = 𝑋) |
| 90 | 89 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐽 ↾t ((𝑋 ∖ {𝐶}) ∪ {𝐶})) = (𝐽 ↾t 𝑋)) |
| 91 | 90 | fveq2d 6910 |
. . . . . . . . 9
⊢ (𝜑 → (int‘(𝐽 ↾t ((𝑋 ∖ {𝐶}) ∪ {𝐶}))) = (int‘(𝐽 ↾t 𝑋))) |
| 92 | | undif1 4476 |
. . . . . . . . . 10
⊢ (((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶}) = ((𝑋 ∩ 𝑌) ∪ {𝐶}) |
| 93 | 39, 49 | elind 4200 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶 ∈ (𝑋 ∩ 𝑌)) |
| 94 | 93 | snssd 4809 |
. . . . . . . . . . 11
⊢ (𝜑 → {𝐶} ⊆ (𝑋 ∩ 𝑌)) |
| 95 | | ssequn2 4189 |
. . . . . . . . . . 11
⊢ ({𝐶} ⊆ (𝑋 ∩ 𝑌) ↔ ((𝑋 ∩ 𝑌) ∪ {𝐶}) = (𝑋 ∩ 𝑌)) |
| 96 | 94, 95 | sylib 218 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∪ {𝐶}) = (𝑋 ∩ 𝑌)) |
| 97 | 92, 96 | eqtrid 2789 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶}) = (𝑋 ∩ 𝑌)) |
| 98 | 91, 97 | fveq12d 6913 |
. . . . . . . 8
⊢ (𝜑 → ((int‘(𝐽 ↾t ((𝑋 ∖ {𝐶}) ∪ {𝐶})))‘(((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶})) = ((int‘(𝐽 ↾t 𝑋))‘(𝑋 ∩ 𝑌))) |
| 99 | 84, 98 | eleqtrrd 2844 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t ((𝑋 ∖ {𝐶}) ∪ {𝐶})))‘(((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶}))) |
| 100 | 57, 34, 58, 3, 59, 99 | limcres 25921 |
. . . . . 6
⊢ (𝜑 → (((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) limℂ 𝐶) = ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 101 | 34 | resmptd 6058 |
. . . . . . 7
⊢ (𝜑 → ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) = (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)))) |
| 102 | 101 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → (((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) limℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 103 | 100, 102 | eqtr3d 2779 |
. . . . 5
⊢ (𝜑 → ((𝑧 ∈ (𝑋 ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 104 | 56, 103 | eleqtrd 2843 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 105 | 16 | simprd 495 |
. . . . 5
⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 106 | 50 | fmpttd 7135 |
. . . . . . 7
⊢ (𝜑 → (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))):(𝑌 ∖ {𝐶})⟶ℂ) |
| 107 | 46 | ssdifssd 4147 |
. . . . . . 7
⊢ (𝜑 → (𝑌 ∖ {𝐶}) ⊆ ℂ) |
| 108 | | eqid 2737 |
. . . . . . 7
⊢ (𝐽 ↾t ((𝑌 ∖ {𝐶}) ∪ {𝐶})) = (𝐽 ↾t ((𝑌 ∖ {𝐶}) ∪ {𝐶})) |
| 109 | | difssd 4137 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (∪ (𝐽
↾t 𝑆)
∖ 𝑌) ⊆ ∪ (𝐽
↾t 𝑆)) |
| 110 | 61, 109 | unssd 4192 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌)) ⊆ ∪
(𝐽 ↾t
𝑆)) |
| 111 | | ssun1 4178 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌)) |
| 112 | 111 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌))) |
| 113 | 28 | ntrss 23063 |
. . . . . . . . . . . 12
⊢ (((𝐽 ↾t 𝑆) ∈ Top ∧ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌)) ⊆ ∪
(𝐽 ↾t
𝑆) ∧ (𝑋 ∩ 𝑌) ⊆ ((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌))) → ((int‘(𝐽 ↾t 𝑆))‘(𝑋 ∩ 𝑌)) ⊆ ((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌)))) |
| 114 | 23, 110, 112, 113 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → ((int‘(𝐽 ↾t 𝑆))‘(𝑋 ∩ 𝑌)) ⊆ ((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌)))) |
| 115 | 114, 31 | sseldd 3984 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌)))) |
| 116 | 115, 49 | elind 4200 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ (((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌))) ∩ 𝑌)) |
| 117 | 42 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑋 ∩ 𝑌) ⊆ 𝑌) |
| 118 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ ((𝐽 ↾t 𝑆) ↾t 𝑌) = ((𝐽 ↾t 𝑆) ↾t 𝑌) |
| 119 | 28, 118 | restntr 23190 |
. . . . . . . . . . 11
⊢ (((𝐽 ↾t 𝑆) ∈ Top ∧ 𝑌 ⊆ ∪ (𝐽
↾t 𝑆)
∧ (𝑋 ∩ 𝑌) ⊆ 𝑌) → ((int‘((𝐽 ↾t 𝑆) ↾t 𝑌))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌))) ∩ 𝑌)) |
| 120 | 23, 27, 117, 119 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → ((int‘((𝐽 ↾t 𝑆) ↾t 𝑌))‘(𝑋 ∩ 𝑌)) = (((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌))) ∩ 𝑌)) |
| 121 | | restabs 23173 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑆 ∧ 𝑆 ∈ V) → ((𝐽 ↾t 𝑆) ↾t 𝑌) = (𝐽 ↾t 𝑌)) |
| 122 | 75, 14, 78, 121 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐽 ↾t 𝑆) ↾t 𝑌) = (𝐽 ↾t 𝑌)) |
| 123 | 122 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ (𝜑 → (int‘((𝐽 ↾t 𝑆) ↾t 𝑌)) = (int‘(𝐽 ↾t 𝑌))) |
| 124 | 123 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝜑 → ((int‘((𝐽 ↾t 𝑆) ↾t 𝑌))‘(𝑋 ∩ 𝑌)) = ((int‘(𝐽 ↾t 𝑌))‘(𝑋 ∩ 𝑌))) |
| 125 | 120, 124 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (𝜑 → (((int‘(𝐽 ↾t 𝑆))‘((𝑋 ∩ 𝑌) ∪ (∪ (𝐽 ↾t 𝑆) ∖ 𝑌))) ∩ 𝑌) = ((int‘(𝐽 ↾t 𝑌))‘(𝑋 ∩ 𝑌))) |
| 126 | 116, 125 | eleqtrd 2843 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t 𝑌))‘(𝑋 ∩ 𝑌))) |
| 127 | | undif1 4476 |
. . . . . . . . . . . 12
⊢ ((𝑌 ∖ {𝐶}) ∪ {𝐶}) = (𝑌 ∪ {𝐶}) |
| 128 | 49 | snssd 4809 |
. . . . . . . . . . . . 13
⊢ (𝜑 → {𝐶} ⊆ 𝑌) |
| 129 | | ssequn2 4189 |
. . . . . . . . . . . . 13
⊢ ({𝐶} ⊆ 𝑌 ↔ (𝑌 ∪ {𝐶}) = 𝑌) |
| 130 | 128, 129 | sylib 218 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑌 ∪ {𝐶}) = 𝑌) |
| 131 | 127, 130 | eqtrid 2789 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑌 ∖ {𝐶}) ∪ {𝐶}) = 𝑌) |
| 132 | 131 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐽 ↾t ((𝑌 ∖ {𝐶}) ∪ {𝐶})) = (𝐽 ↾t 𝑌)) |
| 133 | 132 | fveq2d 6910 |
. . . . . . . . 9
⊢ (𝜑 → (int‘(𝐽 ↾t ((𝑌 ∖ {𝐶}) ∪ {𝐶}))) = (int‘(𝐽 ↾t 𝑌))) |
| 134 | 133, 97 | fveq12d 6913 |
. . . . . . . 8
⊢ (𝜑 → ((int‘(𝐽 ↾t ((𝑌 ∖ {𝐶}) ∪ {𝐶})))‘(((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶})) = ((int‘(𝐽 ↾t 𝑌))‘(𝑋 ∩ 𝑌))) |
| 135 | 126, 134 | eleqtrrd 2844 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t ((𝑌 ∖ {𝐶}) ∪ {𝐶})))‘(((𝑋 ∩ 𝑌) ∖ {𝐶}) ∪ {𝐶}))) |
| 136 | 106, 44, 107, 3, 108, 135 | limcres 25921 |
. . . . . 6
⊢ (𝜑 → (((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) limℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 137 | 44 | resmptd 6058 |
. . . . . . 7
⊢ (𝜑 → ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) = (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) |
| 138 | 137 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → (((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) ↾ ((𝑋 ∩ 𝑌) ∖ {𝐶})) limℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 139 | 136, 138 | eqtr3d 2779 |
. . . . 5
⊢ (𝜑 → ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 140 | 105, 139 | eleqtrd 2843 |
. . . 4
⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 141 | 3 | addcn 24887 |
. . . . 5
⊢ + ∈
((𝐽 ×t
𝐽) Cn 𝐽) |
| 142 | 5, 6, 7 | dvcl 25934 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐶(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ) |
| 143 | 1, 142 | mpdan 687 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ ℂ) |
| 144 | 5, 13, 14 | dvcl 25934 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐶(𝑆 D 𝐺)𝐿) → 𝐿 ∈ ℂ) |
| 145 | 11, 144 | mpdan 687 |
. . . . . 6
⊢ (𝜑 → 𝐿 ∈ ℂ) |
| 146 | 143, 145 | opelxpd 5724 |
. . . . 5
⊢ (𝜑 → 〈𝐾, 𝐿〉 ∈ (ℂ ×
ℂ)) |
| 147 | 54 | toponunii 22922 |
. . . . . 6
⊢ (ℂ
× ℂ) = ∪ (𝐽 ×t 𝐽) |
| 148 | 147 | cncnpi 23286 |
. . . . 5
⊢ (( +
∈ ((𝐽
×t 𝐽) Cn
𝐽) ∧ 〈𝐾, 𝐿〉 ∈ (ℂ × ℂ))
→ + ∈ (((𝐽
×t 𝐽) CnP
𝐽)‘〈𝐾, 𝐿〉)) |
| 149 | 141, 146,
148 | sylancr 587 |
. . . 4
⊢ (𝜑 → + ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘〈𝐾, 𝐿〉)) |
| 150 | 41, 51, 52, 52, 3, 55, 104, 140, 149 | limccnp2 25927 |
. . 3
⊢ (𝜑 → (𝐾 + 𝐿) ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) limℂ 𝐶)) |
| 151 | | eldifi 4131 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) → 𝑧 ∈ (𝑋 ∩ 𝑌)) |
| 152 | 151 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ∈ (𝑋 ∩ 𝑌)) |
| 153 | 6 | ffnd 6737 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹 Fn 𝑋) |
| 154 | 153 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝐹 Fn 𝑋) |
| 155 | 13 | ffnd 6737 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 Fn 𝑌) |
| 156 | 155 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝐺 Fn 𝑌) |
| 157 | | ssexg 5323 |
. . . . . . . . . . . . 13
⊢ ((𝑋 ⊆ ℂ ∧ ℂ
∈ V) → 𝑋 ∈
V) |
| 158 | 36, 76, 157 | sylancl 586 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑋 ∈ V) |
| 159 | 158 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑋 ∈ V) |
| 160 | | ssexg 5323 |
. . . . . . . . . . . . 13
⊢ ((𝑌 ⊆ ℂ ∧ ℂ
∈ V) → 𝑌 ∈
V) |
| 161 | 46, 76, 160 | sylancl 586 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑌 ∈ V) |
| 162 | 161 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑌 ∈ V) |
| 163 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑋 ∩ 𝑌) = (𝑋 ∩ 𝑌) |
| 164 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) = (𝐹‘𝑧)) |
| 165 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝑧 ∈ 𝑌) → (𝐺‘𝑧) = (𝐺‘𝑧)) |
| 166 | 154, 156,
159, 162, 163, 164, 165 | ofval 7708 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝑧 ∈ (𝑋 ∩ 𝑌)) → ((𝐹 ∘f + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 167 | 152, 166 | mpdan 687 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((𝐹 ∘f + 𝐺)‘𝑧) = ((𝐹‘𝑧) + (𝐺‘𝑧))) |
| 168 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝐶 ∈ 𝑋) → (𝐹‘𝐶) = (𝐹‘𝐶)) |
| 169 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝐶 ∈ 𝑌) → (𝐺‘𝐶) = (𝐺‘𝐶)) |
| 170 | 154, 156,
159, 162, 163, 168, 169 | ofval 7708 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) ∧ 𝐶 ∈ (𝑋 ∩ 𝑌)) → ((𝐹 ∘f + 𝐺)‘𝐶) = ((𝐹‘𝐶) + (𝐺‘𝐶))) |
| 171 | 93, 170 | mpidan 689 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((𝐹 ∘f + 𝐺)‘𝐶) = ((𝐹‘𝐶) + (𝐺‘𝐶))) |
| 172 | 167, 171 | oveq12d 7449 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) = (((𝐹‘𝑧) + (𝐺‘𝑧)) − ((𝐹‘𝐶) + (𝐺‘𝐶)))) |
| 173 | | difss 4136 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ (𝑋 ∩ 𝑌) |
| 174 | 173, 32 | sstri 3993 |
. . . . . . . . . . 11
⊢ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ 𝑋 |
| 175 | 174 | sseli 3979 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) → 𝑧 ∈ 𝑋) |
| 176 | | ffvelcdm 7101 |
. . . . . . . . . 10
⊢ ((𝐹:𝑋⟶ℂ ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℂ) |
| 177 | 6, 175, 176 | syl2an 596 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝐹‘𝑧) ∈ ℂ) |
| 178 | 173, 42 | sstri 3993 |
. . . . . . . . . . 11
⊢ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ 𝑌 |
| 179 | 178 | sseli 3979 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) → 𝑧 ∈ 𝑌) |
| 180 | | ffvelcdm 7101 |
. . . . . . . . . 10
⊢ ((𝐺:𝑌⟶ℂ ∧ 𝑧 ∈ 𝑌) → (𝐺‘𝑧) ∈ ℂ) |
| 181 | 13, 179, 180 | syl2an 596 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝐺‘𝑧) ∈ ℂ) |
| 182 | 6, 39 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ) |
| 183 | 182 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝐹‘𝐶) ∈ ℂ) |
| 184 | 13, 49 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺‘𝐶) ∈ ℂ) |
| 185 | 184 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝐺‘𝐶) ∈ ℂ) |
| 186 | 177, 181,
183, 185 | addsub4d 11667 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐹‘𝑧) + (𝐺‘𝑧)) − ((𝐹‘𝐶) + (𝐺‘𝐶))) = (((𝐹‘𝑧) − (𝐹‘𝐶)) + ((𝐺‘𝑧) − (𝐺‘𝐶)))) |
| 187 | 172, 186 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) = (((𝐹‘𝑧) − (𝐹‘𝐶)) + ((𝐺‘𝑧) − (𝐺‘𝐶)))) |
| 188 | 187 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) / (𝑧 − 𝐶)) = ((((𝐹‘𝑧) − (𝐹‘𝐶)) + ((𝐺‘𝑧) − (𝐺‘𝐶))) / (𝑧 − 𝐶))) |
| 189 | 177, 183 | subcld 11620 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((𝐹‘𝑧) − (𝐹‘𝐶)) ∈ ℂ) |
| 190 | 181, 185 | subcld 11620 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ) |
| 191 | 174, 36 | sstrid 3995 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑋 ∩ 𝑌) ∖ {𝐶}) ⊆ ℂ) |
| 192 | 191 | sselda 3983 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ∈ ℂ) |
| 193 | 36, 39 | sseldd 3984 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 194 | 193 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝐶 ∈ ℂ) |
| 195 | 192, 194 | subcld 11620 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝑧 − 𝐶) ∈ ℂ) |
| 196 | | eldifsni 4790 |
. . . . . . . . 9
⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) → 𝑧 ≠ 𝐶) |
| 197 | 196 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → 𝑧 ≠ 𝐶) |
| 198 | 192, 194,
197 | subne0d 11629 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → (𝑧 − 𝐶) ≠ 0) |
| 199 | 189, 190,
195, 198 | divdird 12081 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((((𝐹‘𝑧) − (𝐹‘𝐶)) + ((𝐺‘𝑧) − (𝐺‘𝐶))) / (𝑧 − 𝐶)) = ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) |
| 200 | 188, 199 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶})) → ((((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) / (𝑧 − 𝐶)) = ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) |
| 201 | 200 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))))) |
| 202 | 201 | oveq1d 7446 |
. . 3
⊢ (𝜑 → ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶) = ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹‘𝑧) − (𝐹‘𝐶)) / (𝑧 − 𝐶)) + (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) limℂ 𝐶)) |
| 203 | 150, 202 | eleqtrrd 2844 |
. 2
⊢ (𝜑 → (𝐾 + 𝐿) ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
| 204 | | eqid 2737 |
. . 3
⊢ (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) |
| 205 | | addcl 11237 |
. . . . 5
⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) |
| 206 | 205 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ) |
| 207 | 206, 6, 13, 158, 161, 163 | off 7715 |
. . 3
⊢ (𝜑 → (𝐹 ∘f + 𝐺):(𝑋 ∩ 𝑌)⟶ℂ) |
| 208 | 2, 3, 204, 5, 207, 60 | eldv 25933 |
. 2
⊢ (𝜑 → (𝐶(𝑆 D (𝐹 ∘f + 𝐺))(𝐾 + 𝐿) ↔ (𝐶 ∈ ((int‘(𝐽 ↾t 𝑆))‘(𝑋 ∩ 𝑌)) ∧ (𝐾 + 𝐿) ∈ ((𝑧 ∈ ((𝑋 ∩ 𝑌) ∖ {𝐶}) ↦ ((((𝐹 ∘f + 𝐺)‘𝑧) − ((𝐹 ∘f + 𝐺)‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)))) |
| 209 | 31, 203, 208 | mpbir2and 713 |
1
⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘f + 𝐺))(𝐾 + 𝐿)) |