Proof of Theorem dvbdfbdioolem1
| Step | Hyp | Ref
| Expression |
| 1 | | ioossre 13448 |
. . . 4
⊢ (𝐴(,)𝐵) ⊆ ℝ |
| 2 | | dvbdfbdioolem1.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) |
| 3 | 1, 2 | sselid 3981 |
. . 3
⊢ (𝜑 → 𝐶 ∈ ℝ) |
| 4 | | ioossre 13448 |
. . . 4
⊢ (𝐶(,)𝐵) ⊆ ℝ |
| 5 | | dvbdfbdioolem1.d |
. . . 4
⊢ (𝜑 → 𝐷 ∈ (𝐶(,)𝐵)) |
| 6 | 4, 5 | sselid 3981 |
. . 3
⊢ (𝜑 → 𝐷 ∈ ℝ) |
| 7 | 3 | rexrd 11311 |
. . . 4
⊢ (𝜑 → 𝐶 ∈
ℝ*) |
| 8 | | dvbdfbdioolem1.b |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 9 | 8 | rexrd 11311 |
. . . 4
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 10 | | ioogtlb 45508 |
. . . 4
⊢ ((𝐶 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐷
∈ (𝐶(,)𝐵)) → 𝐶 < 𝐷) |
| 11 | 7, 9, 5, 10 | syl3anc 1373 |
. . 3
⊢ (𝜑 → 𝐶 < 𝐷) |
| 12 | | dvbdfbdioolem1.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 13 | 12 | rexrd 11311 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 14 | | ioogtlb 45508 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐶
∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶) |
| 15 | 13, 9, 2, 14 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → 𝐴 < 𝐶) |
| 16 | | iooltub 45523 |
. . . . . 6
⊢ ((𝐶 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐷
∈ (𝐶(,)𝐵)) → 𝐷 < 𝐵) |
| 17 | 7, 9, 5, 16 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → 𝐷 < 𝐵) |
| 18 | | iccssioo 13456 |
. . . . 5
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵)) |
| 19 | 13, 9, 15, 17, 18 | syl22anc 839 |
. . . 4
⊢ (𝜑 → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵)) |
| 20 | | dvbdfbdioolem1.f |
. . . . 5
⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ) |
| 21 | | ax-resscn 11212 |
. . . . . . 7
⊢ ℝ
⊆ ℂ |
| 22 | 21 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℝ ⊆
ℂ) |
| 23 | 20, 22 | fssd 6753 |
. . . . . . 7
⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ) |
| 24 | 1 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ) |
| 25 | | dvbdfbdioolem1.dmdv |
. . . . . . 7
⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) |
| 26 | | dvcn 25957 |
. . . . . . 7
⊢
(((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ ℝ) ∧ dom (ℝ D
𝐹) = (𝐴(,)𝐵)) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 27 | 22, 23, 24, 25, 26 | syl31anc 1375 |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 28 | | cncfcdm 24924 |
. . . . . 6
⊢ ((ℝ
⊆ ℂ ∧ 𝐹
∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ)) |
| 29 | 22, 27, 28 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ)) |
| 30 | 20, 29 | mpbird 257 |
. . . 4
⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ)) |
| 31 | | rescncf 24923 |
. . . 4
⊢ ((𝐶[,]𝐷) ⊆ (𝐴(,)𝐵) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → (𝐹 ↾ (𝐶[,]𝐷)) ∈ ((𝐶[,]𝐷)–cn→ℝ))) |
| 32 | 19, 30, 31 | sylc 65 |
. . 3
⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) ∈ ((𝐶[,]𝐷)–cn→ℝ)) |
| 33 | 19, 24 | sstrd 3994 |
. . . . . . 7
⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ) |
| 34 | | eqid 2737 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 35 | | tgioo4 24826 |
. . . . . . . 8
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
| 36 | 34, 35 | dvres 25946 |
. . . . . . 7
⊢
(((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ) ∧ ((𝐴(,)𝐵) ⊆ ℝ ∧ (𝐶[,]𝐷) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran
(,)))‘(𝐶[,]𝐷)))) |
| 37 | 22, 23, 24, 33, 36 | syl22anc 839 |
. . . . . 6
⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran
(,)))‘(𝐶[,]𝐷)))) |
| 38 | | iccntr 24843 |
. . . . . . . 8
⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) →
((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷)) |
| 39 | 3, 6, 38 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 →
((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷)) |
| 40 | 39 | reseq2d 5997 |
. . . . . 6
⊢ (𝜑 → ((ℝ D 𝐹) ↾
((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷))) |
| 41 | 37, 40 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = ((ℝ D 𝐹) ↾ (𝐶(,)𝐷))) |
| 42 | 41 | dmeqd 5916 |
. . . 4
⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷))) |
| 43 | 12, 3, 15 | ltled 11409 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ≤ 𝐶) |
| 44 | 6, 8, 17 | ltled 11409 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ≤ 𝐵) |
| 45 | | ioossioo 13481 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) |
| 46 | 13, 9, 43, 44, 45 | syl22anc 839 |
. . . . . 6
⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) |
| 47 | 46, 25 | sseqtrrd 4021 |
. . . . 5
⊢ (𝜑 → (𝐶(,)𝐷) ⊆ dom (ℝ D 𝐹)) |
| 48 | | ssdmres 6031 |
. . . . 5
⊢ ((𝐶(,)𝐷) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝐶(,)𝐷)) |
| 49 | 47, 48 | sylib 218 |
. . . 4
⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝐶(,)𝐷)) |
| 50 | 42, 49 | eqtrd 2777 |
. . 3
⊢ (𝜑 → dom (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝐶(,)𝐷)) |
| 51 | 3, 6, 11, 32, 50 | mvth 26031 |
. 2
⊢ (𝜑 → ∃𝑥 ∈ (𝐶(,)𝐷)((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) |
| 52 | 41 | fveq1d 6908 |
. . . . . . . . 9
⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥)) |
| 53 | | fvres 6925 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐶(,)𝐷) → (((ℝ D 𝐹) ↾ (𝐶(,)𝐷))‘𝑥) = ((ℝ D 𝐹)‘𝑥)) |
| 54 | 52, 53 | sylan9eq 2797 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((ℝ D 𝐹)‘𝑥)) |
| 55 | 54 | eqcomd 2743 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥)) |
| 56 | 55 | 3adant3 1133 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥)) |
| 57 | | simp3 1139 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) |
| 58 | 6 | rexrd 11311 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈
ℝ*) |
| 59 | 3, 6, 11 | ltled 11409 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐶 ≤ 𝐷) |
| 60 | | ubicc2 13505 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ ℝ*
∧ 𝐷 ∈
ℝ* ∧ 𝐶
≤ 𝐷) → 𝐷 ∈ (𝐶[,]𝐷)) |
| 61 | 7, 58, 59, 60 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ∈ (𝐶[,]𝐷)) |
| 62 | | fvres 6925 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (𝐶[,]𝐷) → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) = (𝐹‘𝐷)) |
| 63 | 61, 62 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) = (𝐹‘𝐷)) |
| 64 | | lbicc2 13504 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ ℝ*
∧ 𝐷 ∈
ℝ* ∧ 𝐶
≤ 𝐷) → 𝐶 ∈ (𝐶[,]𝐷)) |
| 65 | 7, 58, 59, 64 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ∈ (𝐶[,]𝐷)) |
| 66 | | fvres 6925 |
. . . . . . . . . 10
⊢ (𝐶 ∈ (𝐶[,]𝐷) → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶) = (𝐹‘𝐶)) |
| 67 | 65, 66 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶) = (𝐹‘𝐶)) |
| 68 | 63, 67 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝜑 → (((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) = ((𝐹‘𝐷) − (𝐹‘𝐶))) |
| 69 | 68 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) |
| 70 | 69 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) |
| 71 | 56, 57, 70 | 3eqtrd 2781 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) |
| 72 | | simp3 1139 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) |
| 73 | 72 | eqcomd 2743 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)) = ((ℝ D 𝐹)‘𝑥)) |
| 74 | 19, 61 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐷 ∈ (𝐴(,)𝐵)) |
| 75 | 20, 74 | ffvelcdmd 7105 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐹‘𝐷) ∈ ℝ) |
| 76 | 20, 2 | ffvelcdmd 7105 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ) |
| 77 | 75, 76 | resubcld 11691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℝ) |
| 78 | 77 | recnd 11289 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℂ) |
| 79 | 78 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((𝐹‘𝐷) − (𝐹‘𝐶)) ∈ ℂ) |
| 80 | | dvfre 25989 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
| 81 | 20, 24, 80 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
| 82 | 25 | feq2d 6722 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ
D 𝐹):(𝐴(,)𝐵)⟶ℝ)) |
| 83 | 81, 82 | mpbid 232 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ) |
| 84 | 83 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ) |
| 85 | 46 | sselda 3983 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 𝑥 ∈ (𝐴(,)𝐵)) |
| 86 | 84, 85 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
| 87 | 86 | recnd 11289 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ) |
| 88 | 87 | 3adant3 1133 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ) |
| 89 | 6, 3 | resubcld 11691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℝ) |
| 90 | 89 | recnd 11289 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐷 − 𝐶) ∈ ℂ) |
| 91 | 90 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (𝐷 − 𝐶) ∈ ℂ) |
| 92 | 3, 6 | posdifd 11850 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐶 < 𝐷 ↔ 0 < (𝐷 − 𝐶))) |
| 93 | 11, 92 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < (𝐷 − 𝐶)) |
| 94 | 93 | gt0ne0d 11827 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐷 − 𝐶) ≠ 0) |
| 95 | 94 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (𝐷 − 𝐶) ≠ 0) |
| 96 | 79, 88, 91, 95 | divmul3d 12077 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶)) = ((ℝ D 𝐹)‘𝑥) ↔ ((𝐹‘𝐷) − (𝐹‘𝐶)) = (((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶)))) |
| 97 | 73, 96 | mpbid 232 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((𝐹‘𝐷) − (𝐹‘𝐶)) = (((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))) |
| 98 | 97 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) = (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶)))) |
| 99 | 90 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐷 − 𝐶) ∈ ℂ) |
| 100 | 87, 99 | absmuld 15493 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶)))) |
| 101 | 100 | 3adant3 1133 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘(((ℝ D 𝐹)‘𝑥) · (𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶)))) |
| 102 | 98, 101 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶)))) |
| 103 | 3, 6, 59 | abssubge0d 15470 |
. . . . . . . . 9
⊢ (𝜑 → (abs‘(𝐷 − 𝐶)) = (𝐷 − 𝐶)) |
| 104 | 103 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → ((abs‘((ℝ D
𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶))) |
| 105 | 104 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶))) |
| 106 | 102, 105 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) = ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶))) |
| 107 | 87 | abscld 15475 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘((ℝ D 𝐹)‘𝑥)) ∈ ℝ) |
| 108 | | dvbdfbdioolem1.k |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ ℝ) |
| 109 | 108 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 𝐾 ∈ ℝ) |
| 110 | 89 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐷 − 𝐶) ∈ ℝ) |
| 111 | | 0red 11264 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈
ℝ) |
| 112 | 111, 89, 93 | ltled 11409 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (𝐷 − 𝐶)) |
| 113 | 112 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (𝐷 − 𝐶)) |
| 114 | | dvbdfbdioolem1.dvbd |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾) |
| 115 | 114 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾) |
| 116 | | rspa 3248 |
. . . . . . . . 9
⊢
((∀𝑥 ∈
(𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾) |
| 117 | 115, 85, 116 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾) |
| 118 | 107, 109,
110, 113, 117 | lemul1ad 12207 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)) ≤ (𝐾 · (𝐷 − 𝐶))) |
| 119 | 118 | 3adant3 1133 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (𝐷 − 𝐶)) ≤ (𝐾 · (𝐷 − 𝐶))) |
| 120 | 106, 119 | eqbrtrd 5165 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶))) |
| 121 | 71, 120 | syld3an3 1411 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶))) |
| 122 | 99 | abscld 15475 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(𝐷 − 𝐶)) ∈ ℝ) |
| 123 | 8, 12 | resubcld 11691 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
| 124 | 123 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (𝐵 − 𝐴) ∈ ℝ) |
| 125 | 87 | absge0d 15483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘((ℝ D
𝐹)‘𝑥))) |
| 126 | 99 | absge0d 15483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → 0 ≤ (abs‘(𝐷 − 𝐶))) |
| 127 | 6, 12, 8, 3, 44, 43 | le2subd 11883 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐷 − 𝐶) ≤ (𝐵 − 𝐴)) |
| 128 | 103, 127 | eqbrtrd 5165 |
. . . . . . . . 9
⊢ (𝜑 → (abs‘(𝐷 − 𝐶)) ≤ (𝐵 − 𝐴)) |
| 129 | 128 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → (abs‘(𝐷 − 𝐶)) ≤ (𝐵 − 𝐴)) |
| 130 | 107, 109,
122, 124, 125, 126, 117, 129 | lemul12ad 12210 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷)) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))) |
| 131 | 130 | 3adant3 1133 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((ℝ D 𝐹)‘𝑥)) · (abs‘(𝐷 − 𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))) |
| 132 | 102, 131 | eqbrtrd 5165 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐷) − (𝐹‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))) |
| 133 | 71, 132 | syld3an3 1411 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))) |
| 134 | 121, 133 | jca 511 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,)𝐷) ∧ ((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶))) → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))) |
| 135 | 134 | rexlimdv3a 3159 |
. 2
⊢ (𝜑 → (∃𝑥 ∈ (𝐶(,)𝐷)((ℝ D (𝐹 ↾ (𝐶[,]𝐷)))‘𝑥) = ((((𝐹 ↾ (𝐶[,]𝐷))‘𝐷) − ((𝐹 ↾ (𝐶[,]𝐷))‘𝐶)) / (𝐷 − 𝐶)) → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))))) |
| 136 | 51, 135 | mpd 15 |
1
⊢ (𝜑 → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴)))) |