Step | Hyp | Ref
| Expression |
1 | | ivth.1 |
. . 3
⊢ (𝜑 → 𝐴 ∈ ℝ) |
2 | | ivth.2 |
. . 3
⊢ (𝜑 → 𝐵 ∈ ℝ) |
3 | | ivth.3 |
. . . 4
⊢ (𝜑 → 𝑈 ∈ ℝ) |
4 | 3 | renegcld 11402 |
. . 3
⊢ (𝜑 → -𝑈 ∈ ℝ) |
5 | | ivth.4 |
. . 3
⊢ (𝜑 → 𝐴 < 𝐵) |
6 | | ivth.5 |
. . 3
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) |
7 | | ivth.7 |
. . . 4
⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) |
8 | | eqid 2740 |
. . . . 5
⊢ (𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦)) = (𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦)) |
9 | 8 | negfcncf 24084 |
. . . 4
⊢ (𝐹 ∈ (𝐷–cn→ℂ) → (𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦)) ∈ (𝐷–cn→ℂ)) |
10 | 7, 9 | syl 17 |
. . 3
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦)) ∈ (𝐷–cn→ℂ)) |
11 | 6 | sselda 3926 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐷) |
12 | | fveq2 6771 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) |
13 | 12 | negeqd 11215 |
. . . . . 6
⊢ (𝑦 = 𝑥 → -(𝐹‘𝑦) = -(𝐹‘𝑥)) |
14 | | negex 11219 |
. . . . . 6
⊢ -(𝐹‘𝑥) ∈ V |
15 | 13, 8, 14 | fvmpt 6872 |
. . . . 5
⊢ (𝑥 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑥) = -(𝐹‘𝑥)) |
16 | 11, 15 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑥) = -(𝐹‘𝑥)) |
17 | | ivth.8 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) |
18 | 17 | renegcld 11402 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑥) ∈ ℝ) |
19 | 16, 18 | eqeltrd 2841 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑥) ∈ ℝ) |
20 | 1 | rexrd 11026 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
21 | 2 | rexrd 11026 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
22 | 1, 2, 5 | ltled 11123 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
23 | | lbicc2 13195 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
24 | 20, 21, 22, 23 | syl3anc 1370 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
25 | 6, 24 | sseldd 3927 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝐷) |
26 | | fveq2 6771 |
. . . . . . . 8
⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) |
27 | 26 | negeqd 11215 |
. . . . . . 7
⊢ (𝑦 = 𝐴 → -(𝐹‘𝑦) = -(𝐹‘𝐴)) |
28 | | negex 11219 |
. . . . . . 7
⊢ -(𝐹‘𝐴) ∈ V |
29 | 27, 8, 28 | fvmpt 6872 |
. . . . . 6
⊢ (𝐴 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐴) = -(𝐹‘𝐴)) |
30 | 25, 29 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐴) = -(𝐹‘𝐴)) |
31 | | ivth2.9 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) |
32 | 31 | simprd 496 |
. . . . . 6
⊢ (𝜑 → 𝑈 < (𝐹‘𝐴)) |
33 | | fveq2 6771 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) |
34 | 33 | eleq1d 2825 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ)) |
35 | 17 | ralrimiva 3110 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ) |
36 | 34, 35, 24 | rspcdva 3563 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
37 | 3, 36 | ltnegd 11553 |
. . . . . 6
⊢ (𝜑 → (𝑈 < (𝐹‘𝐴) ↔ -(𝐹‘𝐴) < -𝑈)) |
38 | 32, 37 | mpbid 231 |
. . . . 5
⊢ (𝜑 → -(𝐹‘𝐴) < -𝑈) |
39 | 30, 38 | eqbrtrd 5101 |
. . . 4
⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐴) < -𝑈) |
40 | 31 | simpld 495 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝐵) < 𝑈) |
41 | | fveq2 6771 |
. . . . . . . . 9
⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) |
42 | 41 | eleq1d 2825 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ)) |
43 | | ubicc2 13196 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
44 | 20, 21, 22, 43 | syl3anc 1370 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
45 | 42, 35, 44 | rspcdva 3563 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ) |
46 | 45, 3 | ltnegd 11553 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ↔ -𝑈 < -(𝐹‘𝐵))) |
47 | 40, 46 | mpbid 231 |
. . . . 5
⊢ (𝜑 → -𝑈 < -(𝐹‘𝐵)) |
48 | 6, 44 | sseldd 3927 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ 𝐷) |
49 | | fveq2 6771 |
. . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) |
50 | 49 | negeqd 11215 |
. . . . . . 7
⊢ (𝑦 = 𝐵 → -(𝐹‘𝑦) = -(𝐹‘𝐵)) |
51 | | negex 11219 |
. . . . . . 7
⊢ -(𝐹‘𝐵) ∈ V |
52 | 50, 8, 51 | fvmpt 6872 |
. . . . . 6
⊢ (𝐵 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐵) = -(𝐹‘𝐵)) |
53 | 48, 52 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐵) = -(𝐹‘𝐵)) |
54 | 47, 53 | breqtrrd 5107 |
. . . 4
⊢ (𝜑 → -𝑈 < ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐵)) |
55 | 39, 54 | jca 512 |
. . 3
⊢ (𝜑 → (((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐴) < -𝑈 ∧ -𝑈 < ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐵))) |
56 | 1, 2, 4, 5, 6, 10,
19, 55 | ivth 24616 |
. 2
⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -𝑈) |
57 | | ioossicc 13164 |
. . . . . . . 8
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
58 | 57, 6 | sstrid 3937 |
. . . . . . 7
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) |
59 | 58 | sselda 3926 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ 𝐷) |
60 | | fveq2 6771 |
. . . . . . . 8
⊢ (𝑦 = 𝑐 → (𝐹‘𝑦) = (𝐹‘𝑐)) |
61 | 60 | negeqd 11215 |
. . . . . . 7
⊢ (𝑦 = 𝑐 → -(𝐹‘𝑦) = -(𝐹‘𝑐)) |
62 | | negex 11219 |
. . . . . . 7
⊢ -(𝐹‘𝑐) ∈ V |
63 | 61, 8, 62 | fvmpt 6872 |
. . . . . 6
⊢ (𝑐 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -(𝐹‘𝑐)) |
64 | 59, 63 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -(𝐹‘𝑐)) |
65 | 64 | eqeq1d 2742 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -𝑈 ↔ -(𝐹‘𝑐) = -𝑈)) |
66 | | cncff 24054 |
. . . . . . . 8
⊢ (𝐹 ∈ (𝐷–cn→ℂ) → 𝐹:𝐷⟶ℂ) |
67 | 7, 66 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
68 | 67 | ffvelrnda 6958 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐷) → (𝐹‘𝑐) ∈ ℂ) |
69 | 59, 68 | syldan 591 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ) |
70 | 3 | recnd 11004 |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ ℂ) |
71 | 70 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑈 ∈ ℂ) |
72 | 69, 71 | neg11ad 11328 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (-(𝐹‘𝑐) = -𝑈 ↔ (𝐹‘𝑐) = 𝑈)) |
73 | 65, 72 | bitrd 278 |
. . 3
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -𝑈 ↔ (𝐹‘𝑐) = 𝑈)) |
74 | 73 | rexbidva 3227 |
. 2
⊢ (𝜑 → (∃𝑐 ∈ (𝐴(,)𝐵)((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -𝑈 ↔ ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)) |
75 | 56, 74 | mpbid 231 |
1
⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈) |