| 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 11690 |
. . 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 2737 |
. . . . 5
⊢ (𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦)) = (𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦)) |
| 9 | 8 | negfcncf 24950 |
. . . 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 3983 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐷) |
| 12 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) |
| 13 | 12 | negeqd 11502 |
. . . . . 6
⊢ (𝑦 = 𝑥 → -(𝐹‘𝑦) = -(𝐹‘𝑥)) |
| 14 | | negex 11506 |
. . . . . 6
⊢ -(𝐹‘𝑥) ∈ V |
| 15 | 13, 8, 14 | fvmpt 7016 |
. . . . 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 11690 |
. . . 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 11311 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 21 | 2 | rexrd 11311 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 22 | 1, 2, 5 | ltled 11409 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| 23 | | lbicc2 13504 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
| 24 | 20, 21, 22, 23 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
| 25 | 6, 24 | sseldd 3984 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝐷) |
| 26 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) |
| 27 | 26 | negeqd 11502 |
. . . . . . 7
⊢ (𝑦 = 𝐴 → -(𝐹‘𝑦) = -(𝐹‘𝐴)) |
| 28 | | negex 11506 |
. . . . . . 7
⊢ -(𝐹‘𝐴) ∈ V |
| 29 | 27, 8, 28 | fvmpt 7016 |
. . . . . 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 495 |
. . . . . 6
⊢ (𝜑 → 𝑈 < (𝐹‘𝐴)) |
| 33 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) |
| 34 | 33 | eleq1d 2826 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ)) |
| 35 | 17 | ralrimiva 3146 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ) |
| 36 | 34, 35, 24 | rspcdva 3623 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
| 37 | 3, 36 | ltnegd 11841 |
. . . . . 6
⊢ (𝜑 → (𝑈 < (𝐹‘𝐴) ↔ -(𝐹‘𝐴) < -𝑈)) |
| 38 | 32, 37 | mpbid 232 |
. . . . 5
⊢ (𝜑 → -(𝐹‘𝐴) < -𝑈) |
| 39 | 30, 38 | eqbrtrd 5165 |
. . . 4
⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐴) < -𝑈) |
| 40 | 31 | simpld 494 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝐵) < 𝑈) |
| 41 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) |
| 42 | 41 | eleq1d 2826 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ)) |
| 43 | | ubicc2 13505 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
| 44 | 20, 21, 22, 43 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
| 45 | 42, 35, 44 | rspcdva 3623 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ) |
| 46 | 45, 3 | ltnegd 11841 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ↔ -𝑈 < -(𝐹‘𝐵))) |
| 47 | 40, 46 | mpbid 232 |
. . . . 5
⊢ (𝜑 → -𝑈 < -(𝐹‘𝐵)) |
| 48 | 6, 44 | sseldd 3984 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ 𝐷) |
| 49 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) |
| 50 | 49 | negeqd 11502 |
. . . . . . 7
⊢ (𝑦 = 𝐵 → -(𝐹‘𝑦) = -(𝐹‘𝐵)) |
| 51 | | negex 11506 |
. . . . . . 7
⊢ -(𝐹‘𝐵) ∈ V |
| 52 | 50, 8, 51 | fvmpt 7016 |
. . . . . 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 5171 |
. . . 4
⊢ (𝜑 → -𝑈 < ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐵)) |
| 55 | 39, 54 | jca 511 |
. . 3
⊢ (𝜑 → (((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐴) < -𝑈 ∧ -𝑈 < ((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝐵))) |
| 56 | 1, 2, 4, 5, 6, 10,
19, 55 | ivth 25489 |
. 2
⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -𝑈) |
| 57 | | ioossicc 13473 |
. . . . . . . 8
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
| 58 | 57, 6 | sstrid 3995 |
. . . . . . 7
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) |
| 59 | 58 | sselda 3983 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ 𝐷) |
| 60 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑦 = 𝑐 → (𝐹‘𝑦) = (𝐹‘𝑐)) |
| 61 | 60 | negeqd 11502 |
. . . . . . 7
⊢ (𝑦 = 𝑐 → -(𝐹‘𝑦) = -(𝐹‘𝑐)) |
| 62 | | negex 11506 |
. . . . . . 7
⊢ -(𝐹‘𝑐) ∈ V |
| 63 | 61, 8, 62 | fvmpt 7016 |
. . . . . 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 2739 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -𝑈 ↔ -(𝐹‘𝑐) = -𝑈)) |
| 66 | | cncff 24919 |
. . . . . . . 8
⊢ (𝐹 ∈ (𝐷–cn→ℂ) → 𝐹:𝐷⟶ℂ) |
| 67 | 7, 66 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
| 68 | 67 | ffvelcdmda 7104 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐷) → (𝐹‘𝑐) ∈ ℂ) |
| 69 | 59, 68 | syldan 591 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ) |
| 70 | 3 | recnd 11289 |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ ℂ) |
| 71 | 70 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑈 ∈ ℂ) |
| 72 | 69, 71 | neg11ad 11616 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (-(𝐹‘𝑐) = -𝑈 ↔ (𝐹‘𝑐) = 𝑈)) |
| 73 | 65, 72 | bitrd 279 |
. . 3
⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -𝑈 ↔ (𝐹‘𝑐) = 𝑈)) |
| 74 | 73 | rexbidva 3177 |
. 2
⊢ (𝜑 → (∃𝑐 ∈ (𝐴(,)𝐵)((𝑦 ∈ 𝐷 ↦ -(𝐹‘𝑦))‘𝑐) = -𝑈 ↔ ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈)) |
| 75 | 56, 74 | mpbid 232 |
1
⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈) |