| Step | Hyp | Ref
| Expression |
| 1 | | reparpht.2 |
. . 3
⊢ (𝜑 → 𝐺 ∈ (II Cn II)) |
| 2 | | reparpht.1 |
. . 3
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
| 3 | | cnco 23274 |
. . 3
⊢ ((𝐺 ∈ (II Cn II) ∧ 𝐹 ∈ (II Cn 𝐽)) → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽)) |
| 4 | 1, 2, 3 | syl2anc 584 |
. 2
⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽)) |
| 5 | | reparphtiOLD.5 |
. . 3
⊢ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)))) |
| 6 | | iitopon 24905 |
. . . . 5
⊢ II ∈
(TopOn‘(0[,]1)) |
| 7 | 6 | a1i 11 |
. . . 4
⊢ (𝜑 → II ∈
(TopOn‘(0[,]1))) |
| 8 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 9 | 8 | cnfldtop 24804 |
. . . . . . . . . 10
⊢
(TopOpen‘ℂfld) ∈ Top |
| 10 | | cnrest2r 23295 |
. . . . . . . . . 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 23677 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×t II) Cn
II)) |
| 13 | | iirevcn 24957 |
. . . . . . . . . . . 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 7439 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦)) |
| 16 | 7, 7, 12, 7, 14, 15 | cnmpt21 23679 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II
×t II) Cn II)) |
| 17 | 8 | dfii3 24909 |
. . . . . . . . . . 11
⊢ II =
((TopOpen‘ℂfld) ↾t
(0[,]1)) |
| 18 | 17 | oveq2i 7442 |
. . . . . . . . . 10
⊢ ((II
×t II) Cn II) = ((II ×t II) Cn
((TopOpen‘ℂfld) ↾t
(0[,]1))) |
| 19 | 16, 18 | eleqtrdi 2851 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II
×t II) Cn ((TopOpen‘ℂfld)
↾t (0[,]1)))) |
| 20 | 11, 19 | sseldd 3984 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II
×t II) Cn
(TopOpen‘ℂfld))) |
| 21 | 7, 7 | cnmpt1st 23676 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×t II) Cn
II)) |
| 22 | 7, 7, 21, 1 | cnmpt21f 23680 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×t II) Cn
II)) |
| 23 | 22, 18 | eleqtrdi 2851 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×t II) Cn
((TopOpen‘ℂfld) ↾t
(0[,]1)))) |
| 24 | 11, 23 | sseldd 3984 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
| 25 | 8 | mulcn 24889 |
. . . . . . . . 9
⊢ ·
∈ (((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld)) |
| 26 | 25 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → · ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld))) |
| 27 | 7, 7, 20, 24, 26 | cnmpt22f 23683 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((1 − 𝑦) · (𝐺‘𝑥))) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
| 28 | 12, 18 | eleqtrdi 2851 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×t II) Cn
((TopOpen‘ℂfld) ↾t
(0[,]1)))) |
| 29 | 11, 28 | sseldd 3984 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
| 30 | 21, 18 | eleqtrdi 2851 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×t II) Cn
((TopOpen‘ℂfld) ↾t
(0[,]1)))) |
| 31 | 11, 30 | sseldd 3984 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
| 32 | 7, 7, 29, 31, 26 | cnmpt22f 23683 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑦 · 𝑥)) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
| 33 | 8 | addcn 24887 |
. . . . . . . 8
⊢ + ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld)) |
| 34 | 33 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → + ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld))) |
| 35 | 7, 7, 27, 32, 34 | cnmpt22f 23683 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×t II) Cn
(TopOpen‘ℂfld))) |
| 36 | 8 | cnfldtopon 24803 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
| 37 | 36 | a1i 11 |
. . . . . . 7
⊢ (𝜑 →
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ)) |
| 38 | | iiuni 24907 |
. . . . . . . . . . . . . . 15
⊢ (0[,]1) =
∪ II |
| 39 | 38, 38 | cnf 23254 |
. . . . . . . . . . . . . 14
⊢ (𝐺 ∈ (II Cn II) → 𝐺:(0[,]1)⟶(0[,]1)) |
| 40 | 1, 39 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺:(0[,]1)⟶(0[,]1)) |
| 41 | 40 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → (𝐺‘𝑥) ∈ (0[,]1)) |
| 42 | 41 | adantrr 717 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (𝐺‘𝑥) ∈ (0[,]1)) |
| 43 | | simprl 771 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑥 ∈ (0[,]1)) |
| 44 | | simprr 773 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑦 ∈ (0[,]1)) |
| 45 | | 0re 11263 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℝ |
| 46 | | 1re 11261 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℝ |
| 47 | | icccvx 24981 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℝ ∧ 1 ∈ ℝ) → (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))) |
| 48 | 45, 46, 47 | mp2an 692 |
. . . . . . . . . . 11
⊢ (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1)) |
| 49 | 42, 43, 44, 48 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1)) |
| 50 | 49 | ralrimivva 3202 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1)) |
| 51 | | eqid 2737 |
. . . . . . . . . 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))) |
| 52 | 51 | fmpo 8093 |
. . . . . . . . 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)) |
| 53 | 50, 52 | sylib 218 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))):((0[,]1) ×
(0[,]1))⟶(0[,]1)) |
| 54 | 53 | frnd 6744 |
. . . . . . 7
⊢ (𝜑 → ran (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ⊆ (0[,]1)) |
| 55 | | unitssre 13539 |
. . . . . . . . 9
⊢ (0[,]1)
⊆ ℝ |
| 56 | | ax-resscn 11212 |
. . . . . . . . 9
⊢ ℝ
⊆ ℂ |
| 57 | 55, 56 | sstri 3993 |
. . . . . . . 8
⊢ (0[,]1)
⊆ ℂ |
| 58 | 57 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (0[,]1) ⊆
ℂ) |
| 59 | | cnrest2 23294 |
. . . . . . 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 | 37, 54, 58, 59 | syl3anc 1373 |
. . . . . 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 | 35, 60 | mpbid 232 |
. . . . 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 2852 |
. . . 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 23680 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)))) ∈ ((II ×t II) Cn
𝐽)) |
| 64 | 5, 63 | eqeltrid 2845 |
. 2
⊢ (𝜑 → 𝐻 ∈ ((II ×t II) Cn
𝐽)) |
| 65 | 40 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ (0[,]1)) |
| 66 | 57, 65 | sselid 3981 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ ℂ) |
| 67 | 66 | mullidd 11279 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝐺‘𝑠)) = (𝐺‘𝑠)) |
| 68 | 57 | sseli 3979 |
. . . . . . . 8
⊢ (𝑠 ∈ (0[,]1) → 𝑠 ∈
ℂ) |
| 69 | 68 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ ℂ) |
| 70 | 69 | mul02d 11459 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · 𝑠) = 0) |
| 71 | 67, 70 | oveq12d 7449 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = ((𝐺‘𝑠) + 0)) |
| 72 | 66 | addridd 11461 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐺‘𝑠) + 0) = (𝐺‘𝑠)) |
| 73 | 71, 72 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = (𝐺‘𝑠)) |
| 74 | 73 | fveq2d 6910 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) = (𝐹‘(𝐺‘𝑠))) |
| 75 | | simpr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1)) |
| 76 | | 0elunit 13509 |
. . . 4
⊢ 0 ∈
(0[,]1) |
| 77 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0) |
| 78 | 77 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = (1 − 0)) |
| 79 | | 1m0e1 12387 |
. . . . . . . . 9
⊢ (1
− 0) = 1 |
| 80 | 78, 79 | eqtrdi 2793 |
. . . . . . . 8
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = 1) |
| 81 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠) |
| 82 | 81 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐺‘𝑥) = (𝐺‘𝑠)) |
| 83 | 80, 82 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((1 − 𝑦) · (𝐺‘𝑥)) = (1 · (𝐺‘𝑠))) |
| 84 | 77, 81 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑦 · 𝑥) = (0 · 𝑠)) |
| 85 | 83, 84 | oveq12d 7449 |
. . . . . 6
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((1 · (𝐺‘𝑠)) + (0 · 𝑠))) |
| 86 | 85 | fveq2d 6910 |
. . . . 5
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠)))) |
| 87 | | fvex 6919 |
. . . . 5
⊢ (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) ∈ V |
| 88 | 86, 5, 87 | ovmpoa 7588 |
. . . 4
⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈
(0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠)))) |
| 89 | 75, 76, 88 | sylancl 586 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠)))) |
| 90 | | fvco3 7008 |
. . . 4
⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧
𝑠 ∈ (0[,]1)) →
((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠))) |
| 91 | 40, 90 | sylan 580 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠))) |
| 92 | 74, 89, 91 | 3eqtr4d 2787 |
. 2
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = ((𝐹 ∘ 𝐺)‘𝑠)) |
| 93 | | 1elunit 13510 |
. . . 4
⊢ 1 ∈
(0[,]1) |
| 94 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1) |
| 95 | 94 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = (1 − 1)) |
| 96 | | 1m1e0 12338 |
. . . . . . . . 9
⊢ (1
− 1) = 0 |
| 97 | 95, 96 | eqtrdi 2793 |
. . . . . . . 8
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = 0) |
| 98 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠) |
| 99 | 98 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐺‘𝑥) = (𝐺‘𝑠)) |
| 100 | 97, 99 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((1 − 𝑦) · (𝐺‘𝑥)) = (0 · (𝐺‘𝑠))) |
| 101 | 94, 98 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑦 · 𝑥) = (1 · 𝑠)) |
| 102 | 100, 101 | oveq12d 7449 |
. . . . . 6
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((0 · (𝐺‘𝑠)) + (1 · 𝑠))) |
| 103 | 102 | fveq2d 6910 |
. . . . 5
⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠)))) |
| 104 | | fvex 6919 |
. . . . 5
⊢ (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) ∈ V |
| 105 | 103, 5, 104 | ovmpoa 7588 |
. . . 4
⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈
(0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠)))) |
| 106 | 75, 93, 105 | sylancl 586 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠)))) |
| 107 | 66 | mul02d 11459 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝐺‘𝑠)) = 0) |
| 108 | 69 | mullidd 11279 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · 𝑠) = 𝑠) |
| 109 | 107, 108 | oveq12d 7449 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = (0 + 𝑠)) |
| 110 | 69 | addlidd 11462 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 + 𝑠) = 𝑠) |
| 111 | 109, 110 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = 𝑠) |
| 112 | 111 | fveq2d 6910 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) = (𝐹‘𝑠)) |
| 113 | 106, 112 | eqtrd 2777 |
. 2
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘𝑠)) |
| 114 | | reparpht.3 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘0) = 0) |
| 115 | 114 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘0) = 0) |
| 116 | 115 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = ((1 − 𝑠) · 0)) |
| 117 | | ax-1cn 11213 |
. . . . . . . . 9
⊢ 1 ∈
ℂ |
| 118 | | subcl 11507 |
. . . . . . . . 9
⊢ ((1
∈ ℂ ∧ 𝑠
∈ ℂ) → (1 − 𝑠) ∈ ℂ) |
| 119 | 117, 69, 118 | sylancr 587 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 − 𝑠) ∈
ℂ) |
| 120 | 119 | mul01d 11460 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 0) =
0) |
| 121 | 116, 120 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = 0) |
| 122 | 69 | mul01d 11460 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 0) = 0) |
| 123 | 121, 122 | oveq12d 7449 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = (0 + 0)) |
| 124 | | 00id 11436 |
. . . . 5
⊢ (0 + 0) =
0 |
| 125 | 123, 124 | eqtrdi 2793 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = 0) |
| 126 | 125 | fveq2d 6910 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) = (𝐹‘0)) |
| 127 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠) |
| 128 | 127 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠)) |
| 129 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 = 0) |
| 130 | 129 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘0)) |
| 131 | 128, 130 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘0))) |
| 132 | 127, 129 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 0)) |
| 133 | 131, 132 | oveq12d 7449 |
. . . . . 6
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) |
| 134 | 133 | fveq2d 6910 |
. . . . 5
⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)))) |
| 135 | | fvex 6919 |
. . . . 5
⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) ∈ V |
| 136 | 134, 5, 135 | ovmpoa 7588 |
. . . 4
⊢ ((0
∈ (0[,]1) ∧ 𝑠
∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)))) |
| 137 | 76, 75, 136 | sylancr 587 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)))) |
| 138 | | fvco3 7008 |
. . . . . 6
⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 0
∈ (0[,]1)) → ((𝐹
∘ 𝐺)‘0) =
(𝐹‘(𝐺‘0))) |
| 139 | 40, 76, 138 | sylancl 586 |
. . . . 5
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘(𝐺‘0))) |
| 140 | 114 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 → (𝐹‘(𝐺‘0)) = (𝐹‘0)) |
| 141 | 139, 140 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0)) |
| 142 | 141 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0)) |
| 143 | 126, 137,
142 | 3eqtr4d 2787 |
. 2
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = ((𝐹 ∘ 𝐺)‘0)) |
| 144 | | reparpht.4 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘1) = 1) |
| 145 | 144 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘1) = 1) |
| 146 | 145 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = ((1 − 𝑠) · 1)) |
| 147 | 119 | mulridd 11278 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 1) = (1 − 𝑠)) |
| 148 | 146, 147 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = (1 − 𝑠)) |
| 149 | 69 | mulridd 11278 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 1) = 𝑠) |
| 150 | 148, 149 | oveq12d 7449 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = ((1 − 𝑠) + 𝑠)) |
| 151 | | npcan 11517 |
. . . . . 6
⊢ ((1
∈ ℂ ∧ 𝑠
∈ ℂ) → ((1 − 𝑠) + 𝑠) = 1) |
| 152 | 117, 69, 151 | sylancr 587 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) + 𝑠) = 1) |
| 153 | 150, 152 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = 1) |
| 154 | 153 | fveq2d 6910 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) = (𝐹‘1)) |
| 155 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠) |
| 156 | 155 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠)) |
| 157 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑥 = 1) |
| 158 | 157 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘1)) |
| 159 | 156, 158 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘1))) |
| 160 | 155, 157 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 1)) |
| 161 | 159, 160 | oveq12d 7449 |
. . . . . 6
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) |
| 162 | 161 | fveq2d 6910 |
. . . . 5
⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)))) |
| 163 | | fvex 6919 |
. . . . 5
⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) ∈ V |
| 164 | 162, 5, 163 | ovmpoa 7588 |
. . . 4
⊢ ((1
∈ (0[,]1) ∧ 𝑠
∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)))) |
| 165 | 93, 75, 164 | sylancr 587 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)))) |
| 166 | | fvco3 7008 |
. . . . . 6
⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 1
∈ (0[,]1)) → ((𝐹
∘ 𝐺)‘1) =
(𝐹‘(𝐺‘1))) |
| 167 | 40, 93, 166 | sylancl 586 |
. . . . 5
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘(𝐺‘1))) |
| 168 | 144 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 → (𝐹‘(𝐺‘1)) = (𝐹‘1)) |
| 169 | 167, 168 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1)) |
| 170 | 169 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1)) |
| 171 | 154, 165,
170 | 3eqtr4d 2787 |
. 2
⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = ((𝐹 ∘ 𝐺)‘1)) |
| 172 | 4, 2, 64, 92, 113, 143, 171 | isphtpy2d 25019 |
1
⊢ (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹)) |