Step | Hyp | Ref
| Expression |
1 | | reparpht.2 |
. . 3
⊢ (𝜑 → 𝐺 ∈ (II Cn II)) |
2 | | reparpht.1 |
. . 3
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
3 | | cnco 23256 |
. . 3
⊢ ((𝐺 ∈ (II Cn II) ∧ 𝐹 ∈ (II Cn 𝐽)) → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽)) |
4 | 1, 2, 3 | syl2anc 582 |
. 2
⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽)) |
5 | | reparphti.5 |
. . 3
⊢ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)))) |
6 | | iitopon 24885 |
. . . . 5
⊢ II ∈
(TopOn‘(0[,]1)) |
7 | 6 | a1i 11 |
. . . 4
⊢ (𝜑 → II ∈
(TopOn‘(0[,]1))) |
8 | | eqid 2726 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
9 | 8 | cnfldtop 24786 |
. . . . . . . . . 10
⊢
(TopOpen‘ℂfld) ∈ Top |
10 | | cnrest2r 23277 |
. . . . . . . . . 10
⊢
((TopOpen‘ℂfld) ∈ Top → ((II
×t II) Cn ((TopOpen‘ℂfld)
↾t (0[,]1))) ⊆ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
11 | 9, 10 | mp1i 13 |
. . . . . . . . 9
⊢ (𝜑 → ((II ×t
II) Cn ((TopOpen‘ℂfld) ↾t (0[,]1)))
⊆ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
12 | 7, 7 | cnmpt2nd 23659 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×t II) Cn
II)) |
13 | | iirevcn 24937 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ (0[,]1) ↦ (1
− 𝑧)) ∈ (II Cn
II) |
14 | 13 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑧 ∈ (0[,]1) ↦ (1 − 𝑧)) ∈ (II Cn
II)) |
15 | | oveq2 7422 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦)) |
16 | 7, 7, 12, 7, 14, 15 | cnmpt21 23661 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II
×t II) Cn II)) |
17 | 8 | dfii3 24889 |
. . . . . . . . . . 11
⊢ II =
((TopOpen‘ℂfld) ↾t
(0[,]1)) |
18 | 17 | oveq2i 7425 |
. . . . . . . . . 10
⊢ ((II
×t II) Cn II) = ((II ×t II) Cn
((TopOpen‘ℂfld) ↾t
(0[,]1))) |
19 | 16, 18 | eleqtrdi 2836 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II
×t II) Cn ((TopOpen‘ℂfld)
↾t (0[,]1)))) |
20 | 11, 19 | sseldd 3980 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II
×t II) Cn
(TopOpen‘ℂfld))) |
21 | 7, 7 | cnmpt1st 23658 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×t II) Cn
II)) |
22 | 7, 7, 21, 1 | cnmpt21f 23662 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×t II) Cn
II)) |
23 | 22, 18 | eleqtrdi 2836 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×t II) Cn
((TopOpen‘ℂfld) ↾t
(0[,]1)))) |
24 | 11, 23 | sseldd 3980 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
25 | 8 | cnfldtopon 24785 |
. . . . . . . . 9
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
26 | 25 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 →
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ)) |
27 | 8 | mpomulcn 24871 |
. . . . . . . . 9
⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld)) |
28 | 27 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld))) |
29 | | oveq12 7423 |
. . . . . . . 8
⊢ ((𝑢 = (1 − 𝑦) ∧ 𝑣 = (𝐺‘𝑥)) → (𝑢 · 𝑣) = ((1 − 𝑦) · (𝐺‘𝑥))) |
30 | 7, 7, 20, 24, 26, 26, 28, 29 | cnmpt22 23664 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((1 − 𝑦) · (𝐺‘𝑥))) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
31 | 9, 10 | ax-mp 5 |
. . . . . . . . . 10
⊢ ((II
×t II) Cn ((TopOpen‘ℂfld)
↾t (0[,]1))) ⊆ ((II ×t II) Cn
(TopOpen‘ℂfld)) |
32 | 18, 31 | eqsstri 4014 |
. . . . . . . . 9
⊢ ((II
×t II) Cn II) ⊆ ((II ×t II) Cn
(TopOpen‘ℂfld)) |
33 | 32, 12 | sselid 3977 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
34 | 32, 21 | sselid 3977 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
35 | | oveq12 7423 |
. . . . . . . 8
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → (𝑢 · 𝑣) = (𝑦 · 𝑥)) |
36 | 7, 7, 33, 34, 26, 26, 28, 35 | cnmpt22 23664 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑦 · 𝑥)) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
37 | 8 | addcn 24867 |
. . . . . . . 8
⊢ + ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld)) |
38 | 37 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → + ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld))) |
39 | 7, 7, 30, 36, 38 | cnmpt22f 23665 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
40 | | iiuni 24887 |
. . . . . . . . . . . . . . 15
⊢ (0[,]1) =
∪ II |
41 | 40, 40 | cnf 23236 |
. . . . . . . . . . . . . 14
⊢ (𝐺 ∈ (II Cn II) → 𝐺:(0[,]1)⟶(0[,]1)) |
42 | 1, 41 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺:(0[,]1)⟶(0[,]1)) |
43 | 42 | ffvelcdmda 7088 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → (𝐺‘𝑥) ∈ (0[,]1)) |
44 | 43 | adantrr 715 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (𝐺‘𝑥) ∈ (0[,]1)) |
45 | | simprl 769 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑥 ∈ (0[,]1)) |
46 | | simprr 771 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑦 ∈ (0[,]1)) |
47 | | 0re 11255 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℝ |
48 | | 1re 11253 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℝ |
49 | | icccvx 24961 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℝ ∧ 1 ∈ ℝ) → (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))) |
50 | 47, 48, 49 | mp2an 690 |
. . . . . . . . . . 11
⊢ (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1)) |
51 | 44, 45, 46, 50 | syl3anc 1368 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1)) |
52 | 51 | ralrimivva 3191 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1)) |
53 | | eqid 2726 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1
− 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) |
54 | 53 | fmpo 8072 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
(0[,]1)∀𝑦 ∈
(0[,]1)(((1 − 𝑦)
· (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1) ↔ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))):((0[,]1) ×
(0[,]1))⟶(0[,]1)) |
55 | 52, 54 | sylib 217 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))):((0[,]1) ×
(0[,]1))⟶(0[,]1)) |
56 | 55 | frnd 6726 |
. . . . . . 7
⊢ (𝜑 → ran (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ⊆ (0[,]1)) |
57 | | unitsscn 13523 |
. . . . . . . 8
⊢ (0[,]1)
⊆ ℂ |
58 | 57 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (0[,]1) ⊆
ℂ) |
59 | | cnrest2 23276 |
. . . . . . 7
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ ran (𝑥 ∈
(0[,]1), 𝑦 ∈ (0[,]1)
↦ (((1 − 𝑦)
· (𝐺‘𝑥)) + (𝑦 · 𝑥))) ⊆ (0[,]1) ∧ (0[,]1) ⊆
ℂ) → ((𝑥 ∈
(0[,]1), 𝑦 ∈ (0[,]1)
↦ (((1 − 𝑦)
· (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld)) ↔ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×t II) Cn
((TopOpen‘ℂfld) ↾t
(0[,]1))))) |
60 | 26, 56, 58, 59 | syl3anc 1368 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld)) ↔ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×t II) Cn
((TopOpen‘ℂfld) ↾t
(0[,]1))))) |
61 | 39, 60 | mpbid 231 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×t II) Cn
((TopOpen‘ℂfld) ↾t
(0[,]1)))) |
62 | 61, 18 | eleqtrrdi 2837 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×t II) Cn
II)) |
63 | 7, 7, 62, 2 | cnmpt21f 23662 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)))) ∈ ((II ×t II) Cn
𝐽)) |
64 | 5, 63 | eqeltrid 2830 |
. 2
⊢ (𝜑 → 𝐻 ∈ ((II ×t II) Cn
𝐽)) |
65 | 42 | ffvelcdmda 7088 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ (0[,]1)) |
66 | 57, 65 | sselid 3977 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ ℂ) |
67 | 66 | mullidd 11271 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝐺‘𝑠)) = (𝐺‘𝑠)) |
68 | | elunitcn 13491 |
. . . . . . . 8
⊢ (𝑠 ∈ (0[,]1) → 𝑠 ∈
ℂ) |
69 | 68 | adantl 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ ℂ) |
70 | 69 | mul02d 11451 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · 𝑠) = 0) |
71 | 67, 70 | oveq12d 7432 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = ((𝐺‘𝑠) + 0)) |
72 | 66 | addridd 11453 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐺‘𝑠) + 0) = (𝐺‘𝑠)) |
73 | 71, 72 | eqtrd 2766 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = (𝐺‘𝑠)) |
74 | 73 | fveq2d 6895 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) = (𝐹‘(𝐺‘𝑠))) |
75 | | simpr 483 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1)) |
76 | | 0elunit 13492 |
. . . 4
⊢ 0 ∈
(0[,]1) |
77 | | simpr 483 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0) |
78 | 77 | oveq2d 7430 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = (1 − 0)) |
79 | | 1m0e1 12377 |
. . . . . . . . 9
⊢ (1
− 0) = 1 |
80 | 78, 79 | eqtrdi 2782 |
. . . . . . . 8
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = 1) |
81 | | simpl 481 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠) |
82 | 81 | fveq2d 6895 |
. . . . . . . 8
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐺‘𝑥) = (𝐺‘𝑠)) |
83 | 80, 82 | oveq12d 7432 |
. . . . . . 7
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((1 − 𝑦) · (𝐺‘𝑥)) = (1 · (𝐺‘𝑠))) |
84 | 77, 81 | oveq12d 7432 |
. . . . . . 7
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑦 · 𝑥) = (0 · 𝑠)) |
85 | 83, 84 | oveq12d 7432 |
. . . . . 6
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((1 · (𝐺‘𝑠)) + (0 · 𝑠))) |
86 | 85 | fveq2d 6895 |
. . . . 5
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠)))) |
87 | | fvex 6904 |
. . . . 5
⊢ (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) ∈ V |
88 | 86, 5, 87 | ovmpoa 7571 |
. . . 4
⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈
(0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠)))) |
89 | 75, 76, 88 | sylancl 584 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠)))) |
90 | | fvco3 6991 |
. . . 4
⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧
𝑠 ∈ (0[,]1)) →
((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠))) |
91 | 42, 90 | sylan 578 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠))) |
92 | 74, 89, 91 | 3eqtr4d 2776 |
. 2
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = ((𝐹 ∘ 𝐺)‘𝑠)) |
93 | | 1elunit 13493 |
. . . 4
⊢ 1 ∈
(0[,]1) |
94 | | simpr 483 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1) |
95 | 94 | oveq2d 7430 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = (1 − 1)) |
96 | | 1m1e0 12328 |
. . . . . . . . 9
⊢ (1
− 1) = 0 |
97 | 95, 96 | eqtrdi 2782 |
. . . . . . . 8
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = 0) |
98 | | simpl 481 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠) |
99 | 98 | fveq2d 6895 |
. . . . . . . 8
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐺‘𝑥) = (𝐺‘𝑠)) |
100 | 97, 99 | oveq12d 7432 |
. . . . . . 7
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((1 − 𝑦) · (𝐺‘𝑥)) = (0 · (𝐺‘𝑠))) |
101 | 94, 98 | oveq12d 7432 |
. . . . . . 7
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑦 · 𝑥) = (1 · 𝑠)) |
102 | 100, 101 | oveq12d 7432 |
. . . . . 6
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((0 · (𝐺‘𝑠)) + (1 · 𝑠))) |
103 | 102 | fveq2d 6895 |
. . . . 5
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠)))) |
104 | | fvex 6904 |
. . . . 5
⊢ (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) ∈ V |
105 | 103, 5, 104 | ovmpoa 7571 |
. . . 4
⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈
(0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠)))) |
106 | 75, 93, 105 | sylancl 584 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠)))) |
107 | 66 | mul02d 11451 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝐺‘𝑠)) = 0) |
108 | 69 | mullidd 11271 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · 𝑠) = 𝑠) |
109 | 107, 108 | oveq12d 7432 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = (0 + 𝑠)) |
110 | 69 | addlidd 11454 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 + 𝑠) = 𝑠) |
111 | 109, 110 | eqtrd 2766 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = 𝑠) |
112 | 111 | fveq2d 6895 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) = (𝐹‘𝑠)) |
113 | 106, 112 | eqtrd 2766 |
. 2
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘𝑠)) |
114 | | reparpht.3 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘0) = 0) |
115 | 114 | adantr 479 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘0) = 0) |
116 | 115 | oveq2d 7430 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = ((1 − 𝑠) · 0)) |
117 | | ax-1cn 11205 |
. . . . . . . . 9
⊢ 1 ∈
ℂ |
118 | | subcl 11498 |
. . . . . . . . 9
⊢ ((1
∈ ℂ ∧ 𝑠
∈ ℂ) → (1 − 𝑠) ∈ ℂ) |
119 | 117, 69, 118 | sylancr 585 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 − 𝑠) ∈
ℂ) |
120 | 119 | mul01d 11452 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 0) =
0) |
121 | 116, 120 | eqtrd 2766 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = 0) |
122 | 69 | mul01d 11452 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 0) = 0) |
123 | 121, 122 | oveq12d 7432 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = (0 + 0)) |
124 | | 00id 11428 |
. . . . 5
⊢ (0 + 0) =
0 |
125 | 123, 124 | eqtrdi 2782 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = 0) |
126 | 125 | fveq2d 6895 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) = (𝐹‘0)) |
127 | | simpr 483 |
. . . . . . . . 9
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠) |
128 | 127 | oveq2d 7430 |
. . . . . . . 8
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠)) |
129 | | simpl 481 |
. . . . . . . . 9
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 = 0) |
130 | 129 | fveq2d 6895 |
. . . . . . . 8
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘0)) |
131 | 128, 130 | oveq12d 7432 |
. . . . . . 7
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘0))) |
132 | 127, 129 | oveq12d 7432 |
. . . . . . 7
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 0)) |
133 | 131, 132 | oveq12d 7432 |
. . . . . 6
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) |
134 | 133 | fveq2d 6895 |
. . . . 5
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)))) |
135 | | fvex 6904 |
. . . . 5
⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) ∈ V |
136 | 134, 5, 135 | ovmpoa 7571 |
. . . 4
⊢ ((0
∈ (0[,]1) ∧ 𝑠
∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)))) |
137 | 76, 75, 136 | sylancr 585 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)))) |
138 | | fvco3 6991 |
. . . . . 6
⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 0
∈ (0[,]1)) → ((𝐹
∘ 𝐺)‘0) =
(𝐹‘(𝐺‘0))) |
139 | 42, 76, 138 | sylancl 584 |
. . . . 5
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘(𝐺‘0))) |
140 | 114 | fveq2d 6895 |
. . . . 5
⊢ (𝜑 → (𝐹‘(𝐺‘0)) = (𝐹‘0)) |
141 | 139, 140 | eqtrd 2766 |
. . . 4
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0)) |
142 | 141 | adantr 479 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0)) |
143 | 126, 137,
142 | 3eqtr4d 2776 |
. 2
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = ((𝐹 ∘ 𝐺)‘0)) |
144 | | reparpht.4 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘1) = 1) |
145 | 144 | adantr 479 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘1) = 1) |
146 | 145 | oveq2d 7430 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = ((1 − 𝑠) · 1)) |
147 | 119 | mulridd 11270 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 1) = (1 − 𝑠)) |
148 | 146, 147 | eqtrd 2766 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = (1 − 𝑠)) |
149 | 69 | mulridd 11270 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 1) = 𝑠) |
150 | 148, 149 | oveq12d 7432 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = ((1 − 𝑠) + 𝑠)) |
151 | | npcan 11508 |
. . . . . 6
⊢ ((1
∈ ℂ ∧ 𝑠
∈ ℂ) → ((1 − 𝑠) + 𝑠) = 1) |
152 | 117, 69, 151 | sylancr 585 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) + 𝑠) = 1) |
153 | 150, 152 | eqtrd 2766 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = 1) |
154 | 153 | fveq2d 6895 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) = (𝐹‘1)) |
155 | | simpr 483 |
. . . . . . . . 9
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠) |
156 | 155 | oveq2d 7430 |
. . . . . . . 8
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠)) |
157 | | simpl 481 |
. . . . . . . . 9
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑥 = 1) |
158 | 157 | fveq2d 6895 |
. . . . . . . 8
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘1)) |
159 | 156, 158 | oveq12d 7432 |
. . . . . . 7
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘1))) |
160 | 155, 157 | oveq12d 7432 |
. . . . . . 7
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 1)) |
161 | 159, 160 | oveq12d 7432 |
. . . . . 6
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) |
162 | 161 | fveq2d 6895 |
. . . . 5
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)))) |
163 | | fvex 6904 |
. . . . 5
⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) ∈ V |
164 | 162, 5, 163 | ovmpoa 7571 |
. . . 4
⊢ ((1
∈ (0[,]1) ∧ 𝑠
∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)))) |
165 | 93, 75, 164 | sylancr 585 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)))) |
166 | | fvco3 6991 |
. . . . . 6
⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 1
∈ (0[,]1)) → ((𝐹
∘ 𝐺)‘1) =
(𝐹‘(𝐺‘1))) |
167 | 42, 93, 166 | sylancl 584 |
. . . . 5
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘(𝐺‘1))) |
168 | 144 | fveq2d 6895 |
. . . . 5
⊢ (𝜑 → (𝐹‘(𝐺‘1)) = (𝐹‘1)) |
169 | 167, 168 | eqtrd 2766 |
. . . 4
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1)) |
170 | 169 | adantr 479 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1)) |
171 | 154, 165,
170 | 3eqtr4d 2776 |
. 2
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = ((𝐹 ∘ 𝐺)‘1)) |
172 | 4, 2, 64, 92, 113, 143, 171 | isphtpy2d 24999 |
1
⊢ (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹)) |