| Step | Hyp | Ref
| Expression |
| 1 | | cvxpconn.1 |
. . 3
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 2 | | cvxpconn.2 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) |
| 3 | | cvxpconn.3 |
. . 3
⊢ 𝐽 =
(TopOpen‘ℂfld) |
| 4 | | cvxpconn.4 |
. . 3
⊢ 𝐾 = (𝐽 ↾t 𝑆) |
| 5 | 1, 2, 3, 4 | cvxpconn 35247 |
. 2
⊢ (𝜑 → 𝐾 ∈ PConn) |
| 6 | | simprl 771 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn 𝐾)) |
| 7 | | pconntop 35230 |
. . . . . . . . . 10
⊢ (𝐾 ∈ PConn → 𝐾 ∈ Top) |
| 8 | 5, 7 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Top) |
| 9 | 8 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐾 ∈ Top) |
| 10 | | toptopon2 22924 |
. . . . . . . 8
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
| 11 | 9, 10 | sylib 218 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
| 12 | | iiuni 24907 |
. . . . . . . . . 10
⊢ (0[,]1) =
∪ II |
| 13 | | eqid 2737 |
. . . . . . . . . 10
⊢ ∪ 𝐾 =
∪ 𝐾 |
| 14 | 12, 13 | cnf 23254 |
. . . . . . . . 9
⊢ (𝑓 ∈ (II Cn 𝐾) → 𝑓:(0[,]1)⟶∪
𝐾) |
| 15 | 6, 14 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶∪
𝐾) |
| 16 | | 0elunit 13509 |
. . . . . . . 8
⊢ 0 ∈
(0[,]1) |
| 17 | | ffvelcdm 7101 |
. . . . . . . 8
⊢ ((𝑓:(0[,]1)⟶∪ 𝐾
∧ 0 ∈ (0[,]1)) → (𝑓‘0) ∈ ∪
𝐾) |
| 18 | 15, 16, 17 | sylancl 586 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ ∪
𝐾) |
| 19 | | eqid 2737 |
. . . . . . . 8
⊢ ((0[,]1)
× {(𝑓‘0)}) =
((0[,]1) × {(𝑓‘0)}) |
| 20 | 19 | pcoptcl 25054 |
. . . . . . 7
⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾)
∧ (𝑓‘0) ∈
∪ 𝐾) → (((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐾) ∧ (((0[,]1) ×
{(𝑓‘0)})‘0) =
(𝑓‘0) ∧ (((0[,]1)
× {(𝑓‘0)})‘1) = (𝑓‘0))) |
| 21 | 11, 18, 20 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐾) ∧ (((0[,]1) ×
{(𝑓‘0)})‘0) =
(𝑓‘0) ∧ (((0[,]1)
× {(𝑓‘0)})‘1) = (𝑓‘0))) |
| 22 | 21 | simp1d 1143 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → ((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐾)) |
| 23 | | iitopon 24905 |
. . . . . . . . . . 11
⊢ II ∈
(TopOn‘(0[,]1)) |
| 24 | 23 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → II ∈
(TopOn‘(0[,]1))) |
| 25 | 3 | dfii3 24909 |
. . . . . . . . . . . 12
⊢ II =
(𝐽 ↾t
(0[,]1)) |
| 26 | 3 | cnfldtopon 24803 |
. . . . . . . . . . . . 13
⊢ 𝐽 ∈
(TopOn‘ℂ) |
| 27 | 26 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐽 ∈
(TopOn‘ℂ)) |
| 28 | | unitsscn 13540 |
. . . . . . . . . . . . 13
⊢ (0[,]1)
⊆ ℂ |
| 29 | 28 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (0[,]1) ⊆
ℂ) |
| 30 | 27, 27 | cnmpt2nd 23677 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ ℂ, 𝑡 ∈ ℂ ↦ 𝑡) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
| 31 | 25, 27, 29, 25, 27, 29, 30 | cnmpt2res 23685 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ 𝑡) ∈ ((II ×t II) Cn
𝐽)) |
| 32 | 1 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ⊆ ℂ) |
| 33 | | resttopon 23169 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝑆 ⊆ ℂ)
→ (𝐽
↾t 𝑆)
∈ (TopOn‘𝑆)) |
| 34 | 26, 1, 33 | sylancr 587 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
| 35 | 4, 34 | eqeltrid 2845 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆)) |
| 36 | | toponuni 22920 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐾) |
| 37 | 35, 36 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑆 = ∪ 𝐾) |
| 38 | 37 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 = ∪ 𝐾) |
| 39 | 18, 38 | eleqtrrd 2844 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ 𝑆) |
| 40 | 32, 39 | sseldd 3984 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ ℂ) |
| 41 | 24, 24, 27, 40 | cnmpt2c 23678 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ (𝑓‘0)) ∈ ((II ×t
II) Cn 𝐽)) |
| 42 | 3 | mpomulcn 24891 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) |
| 43 | 42 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
| 44 | | oveq12 7440 |
. . . . . . . . . . 11
⊢ ((𝑢 = 𝑡 ∧ 𝑣 = (𝑓‘0)) → (𝑢 · 𝑣) = (𝑡 · (𝑓‘0))) |
| 45 | 24, 24, 31, 41, 27, 27, 43, 44 | cnmpt22 23682 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ (𝑡 · (𝑓‘0))) ∈ ((II ×t
II) Cn 𝐽)) |
| 46 | | ax-1cn 11213 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℂ |
| 47 | 46 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 1 ∈
ℂ) |
| 48 | 24, 24, 27, 47 | cnmpt2c 23678 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ 1) ∈ ((II
×t II) Cn 𝐽)) |
| 49 | 3 | subcn 24888 |
. . . . . . . . . . . . 13
⊢ −
∈ ((𝐽
×t 𝐽) Cn
𝐽) |
| 50 | 49 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
| 51 | 24, 24, 48, 31, 50 | cnmpt22f 23683 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ (1 − 𝑡)) ∈ ((II
×t II) Cn 𝐽)) |
| 52 | 24, 24 | cnmpt1st 23676 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ 𝑧) ∈ ((II ×t II) Cn
II)) |
| 53 | 3 | cnfldtop 24804 |
. . . . . . . . . . . . . 14
⊢ 𝐽 ∈ Top |
| 54 | | cnrest2r 23295 |
. . . . . . . . . . . . . 14
⊢ (𝐽 ∈ Top → (II Cn (𝐽 ↾t 𝑆)) ⊆ (II Cn 𝐽)) |
| 55 | 53, 54 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ (II Cn
(𝐽 ↾t
𝑆)) ⊆ (II Cn 𝐽) |
| 56 | 4 | oveq2i 7442 |
. . . . . . . . . . . . . 14
⊢ (II Cn
𝐾) = (II Cn (𝐽 ↾t 𝑆)) |
| 57 | 6, 56 | eleqtrdi 2851 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝐽 ↾t 𝑆))) |
| 58 | 55, 57 | sselid 3981 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn 𝐽)) |
| 59 | 24, 24, 52, 58 | cnmpt21f 23680 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ (𝑓‘𝑧)) ∈ ((II ×t II) Cn
𝐽)) |
| 60 | | oveq12 7440 |
. . . . . . . . . . 11
⊢ ((𝑢 = (1 − 𝑡) ∧ 𝑣 = (𝑓‘𝑧)) → (𝑢 · 𝑣) = ((1 − 𝑡) · (𝑓‘𝑧))) |
| 61 | 24, 24, 51, 59, 27, 27, 43, 60 | cnmpt22 23682 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((1 − 𝑡) · (𝑓‘𝑧))) ∈ ((II ×t II) Cn
𝐽)) |
| 62 | 3 | addcn 24887 |
. . . . . . . . . . 11
⊢ + ∈
((𝐽 ×t
𝐽) Cn 𝐽) |
| 63 | 62 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
| 64 | 24, 24, 45, 61, 63 | cnmpt22f 23683 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×t II) Cn
𝐽)) |
| 65 | | oveq2 7439 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑓‘0) → (𝑡 · 𝑥) = (𝑡 · (𝑓‘0))) |
| 66 | 65 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑓‘0) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦))) |
| 67 | 66 | eleq1d 2826 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑓‘0) → (((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆 ↔ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)) |
| 68 | | oveq2 7439 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = (𝑓‘𝑧) → ((1 − 𝑡) · 𝑦) = ((1 − 𝑡) · (𝑓‘𝑧))) |
| 69 | 68 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑓‘𝑧) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) |
| 70 | 69 | eleq1d 2826 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑓‘𝑧) → (((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆 ↔ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) ∈ 𝑆)) |
| 71 | 2 | 3exp2 1355 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → (𝑡 ∈ (0[,]1) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)))) |
| 72 | 71 | imp42 426 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑡 ∈ (0[,]1)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) |
| 73 | 72 | an32s 652 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑡 ∈ (0[,]1)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) |
| 74 | 73 | ralrimivva 3202 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) |
| 75 | 74 | ad2ant2rl 749 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) |
| 76 | 39 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → (𝑓‘0) ∈ 𝑆) |
| 77 | 15 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → 𝑓:(0[,]1)⟶∪
𝐾) |
| 78 | | simprl 771 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → 𝑧 ∈ (0[,]1)) |
| 79 | 77, 78 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → (𝑓‘𝑧) ∈ ∪ 𝐾) |
| 80 | 38 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → 𝑆 = ∪ 𝐾) |
| 81 | 79, 80 | eleqtrrd 2844 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → (𝑓‘𝑧) ∈ 𝑆) |
| 82 | 67, 70, 75, 76, 81 | rspc2dv 3637 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) ∈ 𝑆) |
| 83 | 82 | ralrimivva 3202 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → ∀𝑧 ∈ (0[,]1)∀𝑡 ∈ (0[,]1)((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) ∈ 𝑆) |
| 84 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) = (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) |
| 85 | 84 | fmpo 8093 |
. . . . . . . . . . . 12
⊢
(∀𝑧 ∈
(0[,]1)∀𝑡 ∈
(0[,]1)((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) ∈ 𝑆 ↔ (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))):((0[,]1) × (0[,]1))⟶𝑆) |
| 86 | 83, 85 | sylib 218 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))):((0[,]1) × (0[,]1))⟶𝑆) |
| 87 | 86 | frnd 6744 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → ran (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ⊆ 𝑆) |
| 88 | | cnrest2 23294 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ ran (𝑧 ∈
(0[,]1), 𝑡 ∈ (0[,]1)
↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ⊆ 𝑆 ∧ 𝑆 ⊆ ℂ) → ((𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×t II) Cn
𝐽) ↔ (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×t II) Cn
(𝐽 ↾t
𝑆)))) |
| 89 | 26, 87, 32, 88 | mp3an2i 1468 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → ((𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×t II) Cn
𝐽) ↔ (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×t II) Cn
(𝐽 ↾t
𝑆)))) |
| 90 | 64, 89 | mpbid 232 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×t II) Cn
(𝐽 ↾t
𝑆))) |
| 91 | 4 | oveq2i 7442 |
. . . . . . . 8
⊢ ((II
×t II) Cn 𝐾) = ((II ×t II) Cn (𝐽 ↾t 𝑆)) |
| 92 | 90, 91 | eleqtrrdi 2852 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×t II) Cn
𝐾)) |
| 93 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1)) |
| 94 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → 𝑡 = 0) |
| 95 | 94 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (𝑡 · (𝑓‘0)) = (0 · (𝑓‘0))) |
| 96 | 94 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (1 − 𝑡) = (1 − 0)) |
| 97 | | 1m0e1 12387 |
. . . . . . . . . . . . 13
⊢ (1
− 0) = 1 |
| 98 | 96, 97 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (1 − 𝑡) = 1) |
| 99 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → 𝑧 = 𝑠) |
| 100 | 99 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (𝑓‘𝑧) = (𝑓‘𝑠)) |
| 101 | 98, 100 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → ((1 − 𝑡) · (𝑓‘𝑧)) = (1 · (𝑓‘𝑠))) |
| 102 | 95, 101 | oveq12d 7449 |
. . . . . . . . . 10
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠)))) |
| 103 | | ovex 7464 |
. . . . . . . . . 10
⊢ ((0
· (𝑓‘0)) + (1
· (𝑓‘𝑠))) ∈ V |
| 104 | 102, 84, 103 | ovmpoa 7588 |
. . . . . . . . 9
⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈
(0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))0) = ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠)))) |
| 105 | 93, 16, 104 | sylancl 586 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))0) = ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠)))) |
| 106 | 40 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘0) ∈ ℂ) |
| 107 | 106 | mul02d 11459 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝑓‘0)) = 0) |
| 108 | 26 | toponunii 22922 |
. . . . . . . . . . . . 13
⊢ ℂ =
∪ 𝐽 |
| 109 | 12, 108 | cnf 23254 |
. . . . . . . . . . . 12
⊢ (𝑓 ∈ (II Cn 𝐽) → 𝑓:(0[,]1)⟶ℂ) |
| 110 | 58, 109 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶ℂ) |
| 111 | 110 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘𝑠) ∈ ℂ) |
| 112 | 111 | mullidd 11279 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝑓‘𝑠)) = (𝑓‘𝑠)) |
| 113 | 107, 112 | oveq12d 7449 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))) = (0 + (𝑓‘𝑠))) |
| 114 | 111 | addlidd 11462 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 + (𝑓‘𝑠)) = (𝑓‘𝑠)) |
| 115 | 105, 113,
114 | 3eqtrd 2781 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))0) = (𝑓‘𝑠)) |
| 116 | | 1elunit 13510 |
. . . . . . . . 9
⊢ 1 ∈
(0[,]1) |
| 117 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → 𝑡 = 1) |
| 118 | 117 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (𝑡 · (𝑓‘0)) = (1 · (𝑓‘0))) |
| 119 | 117 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (1 − 𝑡) = (1 − 1)) |
| 120 | | 1m1e0 12338 |
. . . . . . . . . . . . 13
⊢ (1
− 1) = 0 |
| 121 | 119, 120 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (1 − 𝑡) = 0) |
| 122 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → 𝑧 = 𝑠) |
| 123 | 122 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (𝑓‘𝑧) = (𝑓‘𝑠)) |
| 124 | 121, 123 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → ((1 − 𝑡) · (𝑓‘𝑧)) = (0 · (𝑓‘𝑠))) |
| 125 | 118, 124 | oveq12d 7449 |
. . . . . . . . . 10
⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠)))) |
| 126 | | ovex 7464 |
. . . . . . . . . 10
⊢ ((1
· (𝑓‘0)) + (0
· (𝑓‘𝑠))) ∈ V |
| 127 | 125, 84, 126 | ovmpoa 7588 |
. . . . . . . . 9
⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈
(0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))1) = ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠)))) |
| 128 | 93, 116, 127 | sylancl 586 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))1) = ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠)))) |
| 129 | 106 | mullidd 11279 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝑓‘0)) = (𝑓‘0)) |
| 130 | 111 | mul02d 11459 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝑓‘𝑠)) = 0) |
| 131 | 129, 130 | oveq12d 7449 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))) = ((𝑓‘0) + 0)) |
| 132 | 106 | addridd 11461 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑓‘0) + 0) = (𝑓‘0)) |
| 133 | | fvex 6919 |
. . . . . . . . . . 11
⊢ (𝑓‘0) ∈
V |
| 134 | 133 | fvconst2 7224 |
. . . . . . . . . 10
⊢ (𝑠 ∈ (0[,]1) → (((0[,]1)
× {(𝑓‘0)})‘𝑠) = (𝑓‘0)) |
| 135 | 134 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) ×
{(𝑓‘0)})‘𝑠) = (𝑓‘0)) |
| 136 | 132, 135 | eqtr4d 2780 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑓‘0) + 0) = (((0[,]1) × {(𝑓‘0)})‘𝑠)) |
| 137 | 128, 131,
136 | 3eqtrd 2781 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))1) = (((0[,]1) × {(𝑓‘0)})‘𝑠)) |
| 138 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → 𝑡 = 𝑠) |
| 139 | 138 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (𝑡 · (𝑓‘0)) = (𝑠 · (𝑓‘0))) |
| 140 | 138 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (1 − 𝑡) = (1 − 𝑠)) |
| 141 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → 𝑧 = 0) |
| 142 | 141 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (𝑓‘𝑧) = (𝑓‘0)) |
| 143 | 140, 142 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → ((1 − 𝑡) · (𝑓‘𝑧)) = ((1 − 𝑠) · (𝑓‘0))) |
| 144 | 139, 143 | oveq12d 7449 |
. . . . . . . . . 10
⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0)))) |
| 145 | | ovex 7464 |
. . . . . . . . . 10
⊢ ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))) ∈ V |
| 146 | 144, 84, 145 | ovmpoa 7588 |
. . . . . . . . 9
⊢ ((0
∈ (0[,]1) ∧ 𝑠
∈ (0[,]1)) → (0(𝑧
∈ (0[,]1), 𝑡 ∈
(0[,]1) ↦ ((𝑡
· (𝑓‘0)) + ((1
− 𝑡) · (𝑓‘𝑧))))𝑠) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0)))) |
| 147 | 16, 93, 146 | sylancr 587 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0)))) |
| 148 | 28, 93 | sselid 3981 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ ℂ) |
| 149 | | pncan3 11516 |
. . . . . . . . . . 11
⊢ ((𝑠 ∈ ℂ ∧ 1 ∈
ℂ) → (𝑠 + (1
− 𝑠)) =
1) |
| 150 | 148, 46, 149 | sylancl 586 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠 + (1 − 𝑠)) = 1) |
| 151 | 150 | oveq1d 7446 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 + (1 − 𝑠)) · (𝑓‘0)) = (1 · (𝑓‘0))) |
| 152 | | subcl 11507 |
. . . . . . . . . . 11
⊢ ((1
∈ ℂ ∧ 𝑠
∈ ℂ) → (1 − 𝑠) ∈ ℂ) |
| 153 | 46, 148, 152 | sylancr 587 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 − 𝑠) ∈
ℂ) |
| 154 | 148, 153,
106 | adddird 11286 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 + (1 − 𝑠)) · (𝑓‘0)) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0)))) |
| 155 | 151, 154,
129 | 3eqtr3d 2785 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))) = (𝑓‘0)) |
| 156 | 147, 155 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = (𝑓‘0)) |
| 157 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → 𝑡 = 𝑠) |
| 158 | 157 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (𝑡 · (𝑓‘0)) = (𝑠 · (𝑓‘0))) |
| 159 | 157 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (1 − 𝑡) = (1 − 𝑠)) |
| 160 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → 𝑧 = 1) |
| 161 | 160 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (𝑓‘𝑧) = (𝑓‘1)) |
| 162 | 159, 161 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → ((1 − 𝑡) · (𝑓‘𝑧)) = ((1 − 𝑠) · (𝑓‘1))) |
| 163 | 158, 162 | oveq12d 7449 |
. . . . . . . . . 10
⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1)))) |
| 164 | | ovex 7464 |
. . . . . . . . . 10
⊢ ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))) ∈ V |
| 165 | 163, 84, 164 | ovmpoa 7588 |
. . . . . . . . 9
⊢ ((1
∈ (0[,]1) ∧ 𝑠
∈ (0[,]1)) → (1(𝑧
∈ (0[,]1), 𝑡 ∈
(0[,]1) ↦ ((𝑡
· (𝑓‘0)) + ((1
− 𝑡) · (𝑓‘𝑧))))𝑠) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1)))) |
| 166 | 116, 93, 165 | sylancr 587 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1)))) |
| 167 | | simplrr 778 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘0) = (𝑓‘1)) |
| 168 | 167 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝑓‘0)) = ((1 − 𝑠) · (𝑓‘1))) |
| 169 | 168 | oveq2d 7447 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1)))) |
| 170 | 155, 169,
167 | 3eqtr3d 2785 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))) = (𝑓‘1)) |
| 171 | 166, 170 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = (𝑓‘1)) |
| 172 | 6, 22, 92, 115, 137, 156, 171 | isphtpy2d 25019 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ (𝑓(PHtpy‘𝐾)((0[,]1) × {(𝑓‘0)}))) |
| 173 | 172 | ne0d 4342 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓(PHtpy‘𝐾)((0[,]1) × {(𝑓‘0)})) ≠ ∅) |
| 174 | | isphtpc 25026 |
. . . . 5
⊢ (𝑓(
≃ph‘𝐾)((0[,]1) × {(𝑓‘0)}) ↔ (𝑓 ∈ (II Cn 𝐾) ∧ ((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐾) ∧ (𝑓(PHtpy‘𝐾)((0[,]1) × {(𝑓‘0)})) ≠ ∅)) |
| 175 | 6, 22, 173, 174 | syl3anbrc 1344 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓( ≃ph‘𝐾)((0[,]1) × {(𝑓‘0)})) |
| 176 | 175 | expr 456 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ (II Cn 𝐾)) → ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐾)((0[,]1) × {(𝑓‘0)}))) |
| 177 | 176 | ralrimiva 3146 |
. 2
⊢ (𝜑 → ∀𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐾)((0[,]1) × {(𝑓‘0)}))) |
| 178 | | issconn 35231 |
. 2
⊢ (𝐾 ∈ SConn ↔ (𝐾 ∈ PConn ∧
∀𝑓 ∈ (II Cn
𝐾)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐾)((0[,]1) × {(𝑓‘0)})))) |
| 179 | 5, 177, 178 | sylanbrc 583 |
1
⊢ (𝜑 → 𝐾 ∈ SConn) |