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Theorem c1liplem1 23563
 Description: Lemma for c1lip1 23564. (Contributed by Stefan O'Rear, 15-Nov-2014.)
Hypotheses
Ref Expression
c1liplem1.a (𝜑𝐴 ∈ ℝ)
c1liplem1.b (𝜑𝐵 ∈ ℝ)
c1liplem1.le (𝜑𝐴𝐵)
c1liplem1.f (𝜑𝐹 ∈ (ℂ ↑pm ℝ))
c1liplem1.dv (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
c1liplem1.cn (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
c1liplem1.k 𝐾 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )
Assertion
Ref Expression
c1liplem1 (𝜑 → (𝐾 ∈ ℝ ∧ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝐾 · (abs‘(𝑦𝑥))))))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐾(𝑥,𝑦)

Proof of Theorem c1liplem1
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 c1liplem1.k . . 3 𝐾 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )
2 imassrn 5396 . . . . . 6 (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ran abs
3 absf 13925 . . . . . . 7 abs:ℂ⟶ℝ
4 frn 5966 . . . . . . 7 (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
53, 4ax-mp 5 . . . . . 6 ran abs ⊆ ℝ
62, 5sstri 3577 . . . . 5 (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ
76a1i 11 . . . 4 (𝜑 → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ)
8 dvf 23477 . . . . . . . 8 (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
9 ffun 5961 . . . . . . . 8 ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ → Fun (ℝ D 𝐹))
108, 9ax-mp 5 . . . . . . 7 Fun (ℝ D 𝐹)
1110a1i 11 . . . . . 6 (𝜑 → Fun (ℝ D 𝐹))
12 c1liplem1.dv . . . . . . . 8 (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
13 cncff 22504 . . . . . . . 8 (((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
14 fdm 5964 . . . . . . . 8 (((ℝ D 𝐹) ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ → dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
1512, 13, 143syl 18 . . . . . . 7 (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
16 ssdmres 5340 . . . . . . 7 ((𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝐴[,]𝐵))
1715, 16sylibr 223 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
18 c1liplem1.a . . . . . . . 8 (𝜑𝐴 ∈ ℝ)
1918rexrd 9968 . . . . . . 7 (𝜑𝐴 ∈ ℝ*)
20 c1liplem1.b . . . . . . . 8 (𝜑𝐵 ∈ ℝ)
2120rexrd 9968 . . . . . . 7 (𝜑𝐵 ∈ ℝ*)
22 c1liplem1.le . . . . . . 7 (𝜑𝐴𝐵)
23 lbicc2 12159 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
2419, 21, 22, 23syl3anc 1318 . . . . . 6 (𝜑𝐴 ∈ (𝐴[,]𝐵))
25 funfvima2 6397 . . . . . . 7 ((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) → (𝐴 ∈ (𝐴[,]𝐵) → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
2625imp 444 . . . . . 6 (((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) ∧ 𝐴 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
2711, 17, 24, 26syl21anc 1317 . . . . 5 (𝜑 → ((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
28 ffun 5961 . . . . . . 7 (abs:ℂ⟶ℝ → Fun abs)
293, 28ax-mp 5 . . . . . 6 Fun abs
30 imassrn 5396 . . . . . . . 8 ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ ran (ℝ D 𝐹)
31 frn 5966 . . . . . . . . 9 ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ → ran (ℝ D 𝐹) ⊆ ℂ)
328, 31ax-mp 5 . . . . . . . 8 ran (ℝ D 𝐹) ⊆ ℂ
3330, 32sstri 3577 . . . . . . 7 ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ ℂ
343fdmi 5965 . . . . . . 7 dom abs = ℂ
3533, 34sseqtr4i 3601 . . . . . 6 ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ dom abs
36 funfvima2 6397 . . . . . 6 ((Fun abs ∧ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ⊆ dom abs) → (((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))))
3729, 35, 36mp2an 704 . . . . 5 (((ℝ D 𝐹)‘𝐴) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
38 ne0i 3880 . . . . 5 ((abs‘((ℝ D 𝐹)‘𝐴)) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
3927, 37, 383syl 18 . . . 4 (𝜑 → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
40 ax-resscn 9872 . . . . . . . 8 ℝ ⊆ ℂ
41 ssid 3587 . . . . . . . 8 ℂ ⊆ ℂ
42 cncfss 22510 . . . . . . . 8 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
4340, 41, 42mp2an 704 . . . . . . 7 ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
4443, 12sseldi 3566 . . . . . 6 (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
45 cniccbdd 23037 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎)
4618, 20, 44, 45syl3anc 1318 . . . . 5 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎)
47 fvelima 6158 . . . . . . . . . 10 ((Fun abs ∧ 𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))) → ∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏)
4829, 47mpan 702 . . . . . . . . 9 (𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏)
49 fvelima 6158 . . . . . . . . . . . . . 14 ((Fun (ℝ D 𝐹) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦)
5010, 49mpan 702 . . . . . . . . . . . . 13 (𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → ∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦)
51 fvres 6117 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ (𝐴[,]𝐵) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏) = ((ℝ D 𝐹)‘𝑏))
5251adantl 481 . . . . . . . . . . . . . . . . . 18 ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎𝑏 ∈ (𝐴[,]𝐵)) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏) = ((ℝ D 𝐹)‘𝑏))
5352fveq2d 6107 . . . . . . . . . . . . . . . . 17 ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎𝑏 ∈ (𝐴[,]𝐵)) → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) = (abs‘((ℝ D 𝐹)‘𝑏)))
54 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑏 → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥) = (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏))
5554fveq2d 6107 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑏 → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) = (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)))
5655breq1d 4593 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → ((abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 ↔ (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) ≤ 𝑎))
5756rspccva 3281 . . . . . . . . . . . . . . . . 17 ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎𝑏 ∈ (𝐴[,]𝐵)) → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑏)) ≤ 𝑎)
5853, 57eqbrtrrd 4607 . . . . . . . . . . . . . . . 16 ((∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎𝑏 ∈ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎)
5958adantll 746 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎)
60 fveq2 6103 . . . . . . . . . . . . . . . 16 (((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘((ℝ D 𝐹)‘𝑏)) = (abs‘𝑦))
6160breq1d 4593 . . . . . . . . . . . . . . 15 (((ℝ D 𝐹)‘𝑏) = 𝑦 → ((abs‘((ℝ D 𝐹)‘𝑏)) ≤ 𝑎 ↔ (abs‘𝑦) ≤ 𝑎))
6259, 61syl5ibcom 234 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑏 ∈ (𝐴[,]𝐵)) → (((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘𝑦) ≤ 𝑎))
6362rexlimdva 3013 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (∃𝑏 ∈ (𝐴[,]𝐵)((ℝ D 𝐹)‘𝑏) = 𝑦 → (abs‘𝑦) ≤ 𝑎))
6450, 63syl5 33 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘𝑦) ≤ 𝑎))
6564imp 444 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs‘𝑦) ≤ 𝑎)
66 breq1 4586 . . . . . . . . . . 11 ((abs‘𝑦) = 𝑏 → ((abs‘𝑦) ≤ 𝑎𝑏𝑎))
6765, 66syl5ibcom 234 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) ∧ 𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → ((abs‘𝑦) = 𝑏𝑏𝑎))
6867rexlimdva 3013 . . . . . . . . 9 (((𝜑𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (∃𝑦 ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))(abs‘𝑦) = 𝑏𝑏𝑎))
6948, 68syl5 33 . . . . . . . 8 (((𝜑𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → (𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → 𝑏𝑎))
7069ralrimiv 2948 . . . . . . 7 (((𝜑𝑎 ∈ ℝ) ∧ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎) → ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏𝑎)
7170ex 449 . . . . . 6 ((𝜑𝑎 ∈ ℝ) → (∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 → ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏𝑎))
7271reximdva 3000 . . . . 5 (𝜑 → (∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑥)) ≤ 𝑎 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏𝑎))
7346, 72mpd 15 . . . 4 (𝜑 → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏𝑎)
74 suprcl 10862 . . . 4 (((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ ∧ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏𝑎) → sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ)
757, 39, 73, 74syl3anc 1318 . . 3 (𝜑 → sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ∈ ℝ)
761, 75syl5eqel 2692 . 2 (𝜑𝐾 ∈ ℝ)
77 simplrr 797 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝐴[,]𝐵))
78 fvres 6117 . . . . . . . . . . 11 (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹𝑦))
7977, 78syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) = (𝐹𝑦))
80 c1liplem1.cn . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
81 cncff 22504 . . . . . . . . . . . . . 14 ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
8280, 81syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
8382ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℝ)
8483, 77ffvelrnd 6268 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℝ)
8584recnd 9947 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑦) ∈ ℂ)
8679, 85eqeltrrd 2689 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹𝑦) ∈ ℂ)
87 simplrl 796 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝐴[,]𝐵))
88 fvres 6117 . . . . . . . . . . 11 (𝑥 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹𝑥))
8987, 88syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) = (𝐹𝑥))
9083, 87ffvelrnd 6268 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℝ)
9190recnd 9947 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝑥) ∈ ℂ)
9289, 91eqeltrrd 2689 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹𝑥) ∈ ℂ)
9386, 92subcld 10271 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹𝑦) − (𝐹𝑥)) ∈ ℂ)
94 iccssre 12126 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
9518, 20, 94syl2anc 691 . . . . . . . . . . . 12 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
9695ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ ℝ)
9796, 77sseldd 3569 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
9896, 87sseldd 3569 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
9997, 98resubcld 10337 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦𝑥) ∈ ℝ)
10099recnd 9947 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦𝑥) ∈ ℂ)
101 simpr 476 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
102 difrp 11744 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℝ+))
10398, 97, 102syl2anc 691 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℝ+))
104101, 103mpbid 221 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦𝑥) ∈ ℝ+)
105104rpne0d 11753 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦𝑥) ≠ 0)
10693, 100, 105absdivd 14042 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥))) = ((abs‘((𝐹𝑦) − (𝐹𝑥))) / (abs‘(𝑦𝑥))))
1076a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ)
10839ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅)
10973ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏𝑎)
11029a1i 11 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → Fun abs)
11193, 100, 105divcld 10680 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)) ∈ ℂ)
112111, 34syl6eleqr 2699 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)) ∈ dom abs)
11398rexrd 9968 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ*)
11497rexrd 9968 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ*)
11598, 97, 101ltled 10064 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥𝑦)
116 ubicc2 12160 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑥𝑦) → 𝑦 ∈ (𝑥[,]𝑦))
117113, 114, 115, 116syl3anc 1318 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝑥[,]𝑦))
118 fvres 6117 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑥[,]𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) = (𝐹𝑦))
119117, 118syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) = (𝐹𝑦))
120 lbicc2 12159 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑥𝑦) → 𝑥 ∈ (𝑥[,]𝑦))
121113, 114, 115, 120syl3anc 1318 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝑥[,]𝑦))
122 fvres 6117 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑥[,]𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥) = (𝐹𝑥))
123121, 122syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥) = (𝐹𝑥))
124119, 123oveq12d 6567 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) = ((𝐹𝑦) − (𝐹𝑥)))
125124oveq1d 6564 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) = (((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)))
126 iccss2 12115 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥[,]𝑦) ⊆ (𝐴[,]𝐵))
127126ad2antlr 759 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥[,]𝑦) ⊆ (𝐴[,]𝐵))
128127resabs1d 5348 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) = (𝐹 ↾ (𝑥[,]𝑦)))
12980ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
130 rescncf 22508 . . . . . . . . . . . . . . 15 ((𝑥[,]𝑦) ⊆ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ)))
131127, 129, 130sylc 63 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ))
132128, 131eqeltrrd 2689 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹 ↾ (𝑥[,]𝑦)) ∈ ((𝑥[,]𝑦)–cn→ℝ))
13340a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ℝ ⊆ ℂ)
134 c1liplem1.f . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹 ∈ (ℂ ↑pm ℝ))
135134ad2antrr 758 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹 ∈ (ℂ ↑pm ℝ))
136 cnex 9896 . . . . . . . . . . . . . . . . . . . 20 ℂ ∈ V
137 reex 9906 . . . . . . . . . . . . . . . . . . . 20 ℝ ∈ V
138136, 137elpm2 7775 . . . . . . . . . . . . . . . . . . 19 (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
139138simplbi 475 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (ℂ ↑pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
140135, 139syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹:dom 𝐹⟶ℂ)
141138simprbi 479 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (ℂ ↑pm ℝ) → dom 𝐹 ⊆ ℝ)
142135, 141syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom 𝐹 ⊆ ℝ)
143 iccssre 12126 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥[,]𝑦) ⊆ ℝ)
14498, 97, 143syl2anc 691 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥[,]𝑦) ⊆ ℝ)
145 eqid 2610 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
146145tgioo2 22414 . . . . . . . . . . . . . . . . . 18 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
147145, 146dvres 23481 . . . . . . . . . . . . . . . . 17 (((ℝ ⊆ ℂ ∧ 𝐹:dom 𝐹⟶ℂ) ∧ (dom 𝐹 ⊆ ℝ ∧ (𝑥[,]𝑦) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦))))
148133, 140, 142, 144, 147syl22anc 1319 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦))))
149 iccntr 22432 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
15098, 97, 149syl2anc 691 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦)) = (𝑥(,)𝑦))
151150reseq2d 5317 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
152148, 151eqtrd 2644 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
153152dmeqd 5248 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)))
154 ioossicc 12130 . . . . . . . . . . . . . . . . 17 (𝑥(,)𝑦) ⊆ (𝑥[,]𝑦)
155154, 127syl5ss 3579 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥(,)𝑦) ⊆ (𝐴[,]𝐵))
15617ad2antrr 758 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
157155, 156sstrd 3578 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥(,)𝑦) ⊆ dom (ℝ D 𝐹))
158 ssdmres 5340 . . . . . . . . . . . . . . 15 ((𝑥(,)𝑦) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)) = (𝑥(,)𝑦))
159157, 158sylib 207 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom ((ℝ D 𝐹) ↾ (𝑥(,)𝑦)) = (𝑥(,)𝑦))
160153, 159eqtrd 2644 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → dom (ℝ D (𝐹 ↾ (𝑥[,]𝑦))) = (𝑥(,)𝑦))
16198, 97, 101, 132, 160mvth 23559 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ∃𝑎 ∈ (𝑥(,)𝑦)((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)))
162152fveq1d 6105 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎))
163162adantrr 749 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎))
164 fvres 6117 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ (𝑥(,)𝑦) → (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
165164ad2antll 761 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦𝑎 ∈ (𝑥(,)𝑦))) → (((ℝ D 𝐹) ↾ (𝑥(,)𝑦))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
166163, 165eqtrd 2644 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((ℝ D 𝐹)‘𝑎))
16710a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦𝑎 ∈ (𝑥(,)𝑦))) → Fun (ℝ D 𝐹))
16817ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦𝑎 ∈ (𝑥(,)𝑦))) → (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹))
169155sseld 3567 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑎 ∈ (𝑥(,)𝑦) → 𝑎 ∈ (𝐴[,]𝐵)))
170169impr 647 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦𝑎 ∈ (𝑥(,)𝑦))) → 𝑎 ∈ (𝐴[,]𝐵))
171 funfvima2 6397 . . . . . . . . . . . . . . . . . 18 ((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) → (𝑎 ∈ (𝐴[,]𝐵) → ((ℝ D 𝐹)‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
172171imp 444 . . . . . . . . . . . . . . . . 17 (((Fun (ℝ D 𝐹) ∧ (𝐴[,]𝐵) ⊆ dom (ℝ D 𝐹)) ∧ 𝑎 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
173167, 168, 170, 172syl21anc 1317 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D 𝐹)‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
174166, 173eqeltrd 2688 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦𝑎 ∈ (𝑥(,)𝑦))) → ((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
175 eleq1 2676 . . . . . . . . . . . . . . 15 (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) → (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) ↔ ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
176174, 175syl5ibcom 234 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ (𝑥 < 𝑦𝑎 ∈ (𝑥(,)𝑦))) → (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
177176expr 641 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑎 ∈ (𝑥(,)𝑦) → (((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))))
178177rexlimdv 3012 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (∃𝑎 ∈ (𝑥(,)𝑦)((ℝ D (𝐹 ↾ (𝑥[,]𝑦)))‘𝑎) = ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
179161, 178mpd 15 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((((𝐹 ↾ (𝑥[,]𝑦))‘𝑦) − ((𝐹 ↾ (𝑥[,]𝑦))‘𝑥)) / (𝑦𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
180125, 179eqeltrrd 2689 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))
181 funfvima 6396 . . . . . . . . . . 11 ((Fun abs ∧ (((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)) ∈ dom abs) → ((((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵)) → (abs‘(((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))))
182181imp 444 . . . . . . . . . 10 (((Fun abs ∧ (((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)) ∈ dom abs) ∧ (((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)) ∈ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) → (abs‘(((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
183110, 112, 180, 182syl21anc 1317 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))))
184 suprub 10863 . . . . . . . . 9 ((((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ⊆ ℝ ∧ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))) ≠ ∅ ∧ ∃𝑎 ∈ ℝ ∀𝑏 ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))𝑏𝑎) ∧ (abs‘(((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥))) ∈ (abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵)))) → (abs‘(((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥))) ≤ sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ))
185107, 108, 109, 183, 184syl31anc 1321 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥))) ≤ sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ))
186185, 1syl6breqr 4625 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥))) ≤ 𝐾)
187106, 186eqbrtrrd 4607 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((abs‘((𝐹𝑦) − (𝐹𝑥))) / (abs‘(𝑦𝑥))) ≤ 𝐾)
18893abscld 14023 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ∈ ℝ)
18976ad2antrr 758 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℝ)
190100, 105absrpcld 14035 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(𝑦𝑥)) ∈ ℝ+)
191188, 189, 190ledivmuld 11801 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((abs‘((𝐹𝑦) − (𝐹𝑥))) / (abs‘(𝑦𝑥))) ≤ 𝐾 ↔ (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ ((abs‘(𝑦𝑥)) · 𝐾)))
192187, 191mpbid 221 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ ((abs‘(𝑦𝑥)) · 𝐾))
193190rpcnd 11750 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘(𝑦𝑥)) ∈ ℂ)
194189recnd 9947 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℂ)
195193, 194mulcomd 9940 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((abs‘(𝑦𝑥)) · 𝐾) = (𝐾 · (abs‘(𝑦𝑥))))
196192, 195breqtrd 4609 . . . 4 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝐾 · (abs‘(𝑦𝑥))))
197196ex 449 . . 3 ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝐾 · (abs‘(𝑦𝑥)))))
198197ralrimivva 2954 . 2 (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝐾 · (abs‘(𝑦𝑥)))))
19976, 198jca 553 1 (𝜑 → (𝐾 ∈ ℝ ∧ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝐾 · (abs‘(𝑦𝑥))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑pm cpm 7745  supcsup 8229  ℂcc 9813  ℝcr 9814   · cmul 9820  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℝ+crp 11708  (,)cioo 12046  [,]cicc 12049  abscabs 13822  TopOpenctopn 15905  topGenctg 15921  ℂfldccnfld 19567  intcnt 20631  –cn→ccncf 22487   D cdv 23433 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437 This theorem is referenced by:  c1lip1  23564
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