| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | taylfval.s | . . . . . . 7
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | 
| 2 |  | taylfval.f | . . . . . . 7
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | 
| 3 |  | taylfval.a | . . . . . . 7
⊢ (𝜑 → 𝐴 ⊆ 𝑆) | 
| 4 |  | taylfval.n | . . . . . . 7
⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | 
| 5 |  | taylfval.b | . . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) | 
| 6 |  | taylfval.t | . . . . . . 7
⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) | 
| 7 | 1, 2, 3, 4, 5, 6 | taylfval 26401 | . . . . . 6
⊢ (𝜑 → 𝑇 = ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂfld
tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))) | 
| 8 |  | simpr 484 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | 
| 9 | 8 | snssd 4808 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → {𝑥} ⊆ ℂ) | 
| 10 | 1, 2, 3, 4, 5 | taylfvallem 26400 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) →
(ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) ⊆ ℂ) | 
| 11 |  | xpss12 5699 | . . . . . . . . 9
⊢ (({𝑥} ⊆ ℂ ∧
(ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) ⊆ ℂ) → ({𝑥} × (ℂfld
tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ ×
ℂ)) | 
| 12 | 9, 10, 11 | syl2anc 584 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ ×
ℂ)) | 
| 13 | 12 | ralrimiva 3145 | . . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ ×
ℂ)) | 
| 14 |  | iunss 5044 | . . . . . . 7
⊢ (∪ 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ × ℂ)
↔ ∀𝑥 ∈
ℂ ({𝑥} ×
(ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ ×
ℂ)) | 
| 15 | 13, 14 | sylibr 234 | . . . . . 6
⊢ (𝜑 → ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ ×
ℂ)) | 
| 16 | 7, 15 | eqsstrd 4017 | . . . . 5
⊢ (𝜑 → 𝑇 ⊆ (ℂ ×
ℂ)) | 
| 17 |  | relxp 5702 | . . . . 5
⊢ Rel
(ℂ × ℂ) | 
| 18 |  | relss 5790 | . . . . 5
⊢ (𝑇 ⊆ (ℂ ×
ℂ) → (Rel (ℂ × ℂ) → Rel 𝑇)) | 
| 19 | 16, 17, 18 | mpisyl 21 | . . . 4
⊢ (𝜑 → Rel 𝑇) | 
| 20 | 1, 2, 3, 4, 5, 6 | eltayl 26402 | . . . . . . . 8
⊢ (𝜑 → (𝑥𝑇𝑦 ↔ (𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))))) | 
| 21 | 20 | biimpd 229 | . . . . . . 7
⊢ (𝜑 → (𝑥𝑇𝑦 → (𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))))) | 
| 22 | 21 | alrimiv 1926 | . . . . . 6
⊢ (𝜑 → ∀𝑦(𝑥𝑇𝑦 → (𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))))) | 
| 23 |  | cnfldbas 21369 | . . . . . . . . 9
⊢ ℂ =
(Base‘ℂfld) | 
| 24 |  | cnring 21404 | . . . . . . . . . 10
⊢
ℂfld ∈ Ring | 
| 25 |  | ringcmn 20280 | . . . . . . . . . 10
⊢
(ℂfld ∈ Ring → ℂfld ∈
CMnd) | 
| 26 | 24, 25 | mp1i 13 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂfld
∈ CMnd) | 
| 27 |  | cnfldtps 24799 | . . . . . . . . . 10
⊢
ℂfld ∈ TopSp | 
| 28 | 27 | a1i 11 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂfld
∈ TopSp) | 
| 29 |  | ovex 7465 | . . . . . . . . . . 11
⊢
(0[,]𝑁) ∈
V | 
| 30 | 29 | inex1 5316 | . . . . . . . . . 10
⊢
((0[,]𝑁) ∩
ℤ) ∈ V | 
| 31 | 30 | a1i 11 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((0[,]𝑁) ∩ ℤ) ∈
V) | 
| 32 | 1, 2, 3, 4, 5 | taylfvallem1 26399 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ ℂ) | 
| 33 | 32 | fmpttd 7134 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))):((0[,]𝑁) ∩
ℤ)⟶ℂ) | 
| 34 |  | eqid 2736 | . . . . . . . . 9
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) | 
| 35 | 34 | cnfldhaus 24806 | . . . . . . . . . 10
⊢
(TopOpen‘ℂfld) ∈ Haus | 
| 36 | 35 | a1i 11 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) →
(TopOpen‘ℂfld) ∈ Haus) | 
| 37 | 23, 26, 28, 31, 33, 34, 36 | haustsms 24145 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∃*𝑦 𝑦 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) | 
| 38 | 37 | ex 412 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ ℂ → ∃*𝑦 𝑦 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))) | 
| 39 |  | moanimv 2618 | . . . . . . 7
⊢
(∃*𝑦(𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂfld
tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ↔ (𝑥 ∈ ℂ → ∃*𝑦 𝑦 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))) | 
| 40 | 38, 39 | sylibr 234 | . . . . . 6
⊢ (𝜑 → ∃*𝑦(𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))) | 
| 41 |  | moim 2543 | . . . . . 6
⊢
(∀𝑦(𝑥𝑇𝑦 → (𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))) → (∃*𝑦(𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) → ∃*𝑦 𝑥𝑇𝑦)) | 
| 42 | 22, 40, 41 | sylc 65 | . . . . 5
⊢ (𝜑 → ∃*𝑦 𝑥𝑇𝑦) | 
| 43 | 42 | alrimiv 1926 | . . . 4
⊢ (𝜑 → ∀𝑥∃*𝑦 𝑥𝑇𝑦) | 
| 44 |  | dffun6 6573 | . . . 4
⊢ (Fun
𝑇 ↔ (Rel 𝑇 ∧ ∀𝑥∃*𝑦 𝑥𝑇𝑦)) | 
| 45 | 19, 43, 44 | sylanbrc 583 | . . 3
⊢ (𝜑 → Fun 𝑇) | 
| 46 | 45 | funfnd 6596 | . 2
⊢ (𝜑 → 𝑇 Fn dom 𝑇) | 
| 47 |  | rnss 5949 | . . . 4
⊢ (𝑇 ⊆ (ℂ ×
ℂ) → ran 𝑇
⊆ ran (ℂ × ℂ)) | 
| 48 | 16, 47 | syl 17 | . . 3
⊢ (𝜑 → ran 𝑇 ⊆ ran (ℂ ×
ℂ)) | 
| 49 |  | rnxpss 6191 | . . 3
⊢ ran
(ℂ × ℂ) ⊆ ℂ | 
| 50 | 48, 49 | sstrdi 3995 | . 2
⊢ (𝜑 → ran 𝑇 ⊆ ℂ) | 
| 51 |  | df-f 6564 | . 2
⊢ (𝑇:dom 𝑇⟶ℂ ↔ (𝑇 Fn dom 𝑇 ∧ ran 𝑇 ⊆ ℂ)) | 
| 52 | 46, 50, 51 | sylanbrc 583 | 1
⊢ (𝜑 → 𝑇:dom 𝑇⟶ℂ) |