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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  cc 9813  cr 9814   · cmul 9820  *cxr 9952   < clt 9953  cle 9954  cmin 10145   / cdiv 10563  +crp 11708  (,)cioo 12046  [,]cicc 12049  abscabs 13822  TopOpenctopn 15905  topGenctg 15921  fldccnfld 19567  intcnt 20631  cnccncf 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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