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Theorem rolle 23557
 Description: Rolle's theorem. If 𝐹 is a real continuous function on [𝐴, 𝐵] which is differentiable on (𝐴, 𝐵), and 𝐹(𝐴) = 𝐹(𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that (ℝ D 𝐹)‘𝑥 = 0. (Contributed by Mario Carneiro, 1-Sep-2014.)
Hypotheses
Ref Expression
rolle.a (𝜑𝐴 ∈ ℝ)
rolle.b (𝜑𝐵 ∈ ℝ)
rolle.lt (𝜑𝐴 < 𝐵)
rolle.f (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
rolle.d (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
rolle.e (𝜑 → (𝐹𝐴) = (𝐹𝐵))
Assertion
Ref Expression
rolle (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐵   𝑥,𝐹

Proof of Theorem rolle
Dummy variables 𝑢 𝑡 𝑣 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rolle.a . . . 4 (𝜑𝐴 ∈ ℝ)
2 rolle.b . . . 4 (𝜑𝐵 ∈ ℝ)
3 rolle.lt . . . . 5 (𝜑𝐴 < 𝐵)
41, 2, 3ltled 10064 . . . 4 (𝜑𝐴𝐵)
5 rolle.f . . . 4 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
61, 2, 4, 5evthicc 23035 . . 3 (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
7 reeanv 3086 . . 3 (∃𝑢 ∈ (𝐴[,]𝐵)∃𝑣 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)) ↔ (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
86, 7sylibr 223 . 2 (𝜑 → ∃𝑢 ∈ (𝐴[,]𝐵)∃𝑣 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
9 r19.26 3046 . . . 4 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ↔ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
101ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐴 ∈ ℝ)
112ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐵 ∈ ℝ)
123ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐴 < 𝐵)
135ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
14 rolle.d . . . . . . . . 9 (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
1514ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
16 simpl 472 . . . . . . . . . . 11 (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → (𝐹𝑦) ≤ (𝐹𝑢))
1716ralimi 2936 . . . . . . . . . 10 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢))
18 fveq2 6103 . . . . . . . . . . . 12 (𝑦 = 𝑡 → (𝐹𝑦) = (𝐹𝑡))
1918breq1d 4593 . . . . . . . . . . 11 (𝑦 = 𝑡 → ((𝐹𝑦) ≤ (𝐹𝑢) ↔ (𝐹𝑡) ≤ (𝐹𝑢)))
2019cbvralv 3147 . . . . . . . . . 10 (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹𝑡) ≤ (𝐹𝑢))
2117, 20sylib 207 . . . . . . . . 9 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹𝑡) ≤ (𝐹𝑢))
2221ad2antrl 760 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹𝑡) ≤ (𝐹𝑢))
23 simplrl 796 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝑢 ∈ (𝐴[,]𝐵))
24 simprr 792 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → ¬ 𝑢 ∈ {𝐴, 𝐵})
2510, 11, 12, 13, 15, 22, 23, 24rollelem 23556 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
2625expr 641 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → (¬ 𝑢 ∈ {𝐴, 𝐵} → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
271ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐴 ∈ ℝ)
282ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐵 ∈ ℝ)
293ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐴 < 𝐵)
30 cncff 22504 . . . . . . . . . . . . . . 15 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
315, 30syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
3231ffvelrnda 6267 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝐹𝑢) ∈ ℝ)
3332renegcld 10336 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹𝑢) ∈ ℝ)
34 eqid 2610 . . . . . . . . . . . 12 (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) = (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))
3533, 34fmptd 6292 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)):(𝐴[,]𝐵)⟶ℝ)
36 ax-resscn 9872 . . . . . . . . . . . 12 ℝ ⊆ ℂ
37 ssid 3587 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
38 cncfss 22510 . . . . . . . . . . . . . . 15 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
3936, 37, 38mp2an 704 . . . . . . . . . . . . . 14 ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
4039, 5sseldi 3566 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
4134negfcncf 22530 . . . . . . . . . . . . 13 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
4240, 41syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
43 cncffvrn 22509 . . . . . . . . . . . 12 ((ℝ ⊆ ℂ ∧ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)):(𝐴[,]𝐵)⟶ℝ))
4436, 42, 43sylancr 694 . . . . . . . . . . 11 (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)):(𝐴[,]𝐵)⟶ℝ))
4535, 44mpbird 246 . . . . . . . . . 10 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
4645ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
4736a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℝ ⊆ ℂ)
48 iccssre 12126 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
491, 2, 48syl2anc 691 . . . . . . . . . . . . . 14 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
50 fss 5969 . . . . . . . . . . . . . . . . 17 ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
5131, 36, 50sylancl 693 . . . . . . . . . . . . . . . 16 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
5251ffvelrnda 6267 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝐹𝑢) ∈ ℂ)
5352negcld 10258 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹𝑢) ∈ ℂ)
54 eqid 2610 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
5554tgioo2 22414 . . . . . . . . . . . . . 14 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
56 iccntr 22432 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
571, 2, 56syl2anc 691 . . . . . . . . . . . . . 14 (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
5847, 49, 53, 55, 54, 57dvmptntr 23540 . . . . . . . . . . . . 13 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹𝑢))))
59 reelprrecn 9907 . . . . . . . . . . . . . . 15 ℝ ∈ {ℝ, ℂ}
6059a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℝ ∈ {ℝ, ℂ})
61 ioossicc 12130 . . . . . . . . . . . . . . . 16 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
6261sseli 3564 . . . . . . . . . . . . . . 15 (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ (𝐴[,]𝐵))
6362, 52sylan2 490 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴(,)𝐵)) → (𝐹𝑢) ∈ ℂ)
64 fvex 6113 . . . . . . . . . . . . . . 15 ((ℝ D 𝐹)‘𝑢) ∈ V
6564a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑢) ∈ V)
6631feqmptd 6159 . . . . . . . . . . . . . . . 16 (𝜑𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢)))
6766oveq2d 6565 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢))))
68 dvf 23477 . . . . . . . . . . . . . . . . 17 (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
6914feq2d 5944 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
7068, 69mpbii 222 . . . . . . . . . . . . . . . 16 (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
7170feqmptd 6159 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
7247, 49, 52, 55, 54, 57dvmptntr 23540 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑢))))
7367, 71, 723eqtr3rd 2653 . . . . . . . . . . . . . 14 (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
7460, 63, 65, 73dvmptneg 23535 . . . . . . . . . . . . 13 (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
7558, 74eqtrd 2644 . . . . . . . . . . . 12 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
7675dmeqd 5248 . . . . . . . . . . 11 (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
77 dmmptg 5549 . . . . . . . . . . . 12 (∀𝑢 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑢) ∈ V → dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵))
78 negex 10158 . . . . . . . . . . . . 13 -((ℝ D 𝐹)‘𝑢) ∈ V
7978a1i 11 . . . . . . . . . . . 12 (𝑢 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑢) ∈ V)
8077, 79mprg 2910 . . . . . . . . . . 11 dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵)
8176, 80syl6eq 2660 . . . . . . . . . 10 (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (𝐴(,)𝐵))
8281ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (𝐴(,)𝐵))
83 simpr 476 . . . . . . . . . . . . . 14 (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → (𝐹𝑣) ≤ (𝐹𝑦))
8431ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
85 simplrr 797 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑣 ∈ (𝐴[,]𝐵))
8684, 85ffvelrnd 6268 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝑣) ∈ ℝ)
8731adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
8887ffvelrnda 6267 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝑦) ∈ ℝ)
8986, 88lenegd 10485 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ -(𝐹𝑦) ≤ -(𝐹𝑣)))
90 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑦 → (𝐹𝑢) = (𝐹𝑦))
9190negeqd 10154 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑦 → -(𝐹𝑢) = -(𝐹𝑦))
92 negex 10158 . . . . . . . . . . . . . . . . . 18 -(𝐹𝑦) ∈ V
9391, 34, 92fvmpt 6191 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = -(𝐹𝑦))
9493adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = -(𝐹𝑦))
95 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑣 → (𝐹𝑢) = (𝐹𝑣))
9695negeqd 10154 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑣 → -(𝐹𝑢) = -(𝐹𝑣))
97 negex 10158 . . . . . . . . . . . . . . . . . 18 -(𝐹𝑣) ∈ V
9896, 34, 97fvmpt 6191 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) = -(𝐹𝑣))
9985, 98syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) = -(𝐹𝑣))
10094, 99breq12d 4596 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) ↔ -(𝐹𝑦) ≤ -(𝐹𝑣)))
10189, 100bitr4d 270 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣)))
10283, 101syl5ib 233 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣)))
103102ralimdva 2945 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣)))
104103imp 444 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → ∀𝑦 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣))
105 fveq2 6103 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡))
106105breq1d 4593 . . . . . . . . . . . 12 (𝑦 = 𝑡 → (((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) ↔ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣)))
107106cbvralv 3147 . . . . . . . . . . 11 (∀𝑦 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣))
108104, 107sylib 207 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → ∀𝑡 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣))
109108adantrr 749 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → ∀𝑡 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣))
110 simplrr 797 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝑣 ∈ (𝐴[,]𝐵))
111 simprr 792 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → ¬ 𝑣 ∈ {𝐴, 𝐵})
11227, 28, 29, 46, 82, 109, 110, 111rollelem 23556 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0)
11375fveq1d 6105 . . . . . . . . . . . . 13 (𝜑 → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥))
114 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ((ℝ D 𝐹)‘𝑢) = ((ℝ D 𝐹)‘𝑥))
115114negeqd 10154 . . . . . . . . . . . . . 14 (𝑢 = 𝑥 → -((ℝ D 𝐹)‘𝑢) = -((ℝ D 𝐹)‘𝑥))
116 eqid 2610 . . . . . . . . . . . . . 14 (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))
117 negex 10158 . . . . . . . . . . . . . 14 -((ℝ D 𝐹)‘𝑥) ∈ V
118115, 116, 117fvmpt 6191 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
119113, 118sylan9eq 2664 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
120119eqeq1d 2612 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
12114eleq2d 2673 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ dom (ℝ D 𝐹) ↔ 𝑥 ∈ (𝐴(,)𝐵)))
122121biimpar 501 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ dom (ℝ D 𝐹))
12368ffvelrni 6266 . . . . . . . . . . . . 13 (𝑥 ∈ dom (ℝ D 𝐹) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
124122, 123syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
125124negeq0d 10263 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
126120, 125bitr4d 270 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑥) = 0))
127126rexbidva 3031 . . . . . . . . 9 (𝜑 → (∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
128127ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → (∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
129112, 128mpbid 221 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
130129expr 641 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → (¬ 𝑣 ∈ {𝐴, 𝐵} → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
131 vex 3176 . . . . . . . . . . 11 𝑢 ∈ V
132131elpr 4146 . . . . . . . . . 10 (𝑢 ∈ {𝐴, 𝐵} ↔ (𝑢 = 𝐴𝑢 = 𝐵))
133 fveq2 6103 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (𝐹𝑢) = (𝐹𝐴))
134133a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑢 = 𝐴 → (𝐹𝑢) = (𝐹𝐴)))
135 rolle.e . . . . . . . . . . . . 13 (𝜑 → (𝐹𝐴) = (𝐹𝐵))
136135eqcomd 2616 . . . . . . . . . . . 12 (𝜑 → (𝐹𝐵) = (𝐹𝐴))
137 fveq2 6103 . . . . . . . . . . . . 13 (𝑢 = 𝐵 → (𝐹𝑢) = (𝐹𝐵))
138137eqeq1d 2612 . . . . . . . . . . . 12 (𝑢 = 𝐵 → ((𝐹𝑢) = (𝐹𝐴) ↔ (𝐹𝐵) = (𝐹𝐴)))
139136, 138syl5ibrcom 236 . . . . . . . . . . 11 (𝜑 → (𝑢 = 𝐵 → (𝐹𝑢) = (𝐹𝐴)))
140134, 139jaod 394 . . . . . . . . . 10 (𝜑 → ((𝑢 = 𝐴𝑢 = 𝐵) → (𝐹𝑢) = (𝐹𝐴)))
141132, 140syl5bi 231 . . . . . . . . 9 (𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴)))
142 eleq1 2676 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (𝑢 ∈ {𝐴, 𝐵} ↔ 𝑣 ∈ {𝐴, 𝐵}))
14395eqeq1d 2612 . . . . . . . . . . . 12 (𝑢 = 𝑣 → ((𝐹𝑢) = (𝐹𝐴) ↔ (𝐹𝑣) = (𝐹𝐴)))
144142, 143imbi12d 333 . . . . . . . . . . 11 (𝑢 = 𝑣 → ((𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴)) ↔ (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴))))
145144imbi2d 329 . . . . . . . . . 10 (𝑢 = 𝑣 → ((𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴))) ↔ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴)))))
146145, 141chvarv 2251 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴)))
147141, 146anim12d 584 . . . . . . . 8 (𝜑 → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))))
148147ad2antrr 758 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))))
1491rexrd 9968 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ ℝ*)
1502rexrd 9968 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ ℝ*)
151 lbicc2 12159 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
152149, 150, 4, 151syl3anc 1318 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ (𝐴[,]𝐵))
15331, 152ffvelrnd 6268 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝐴) ∈ ℝ)
154153ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝐴) ∈ ℝ)
15588, 154letri3d 10058 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦))))
156 breq2 4587 . . . . . . . . . . . . . . 15 ((𝐹𝑢) = (𝐹𝐴) → ((𝐹𝑦) ≤ (𝐹𝑢) ↔ (𝐹𝑦) ≤ (𝐹𝐴)))
157 breq1 4586 . . . . . . . . . . . . . . 15 ((𝐹𝑣) = (𝐹𝐴) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ (𝐹𝐴) ≤ (𝐹𝑦)))
158156, 157bi2anan9 913 . . . . . . . . . . . . . 14 (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦))))
159158bibi2d 331 . . . . . . . . . . . . 13 (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → (((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) ↔ ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦)))))
160155, 159syl5ibrcom 236 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)))))
161160impancom 455 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) → (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)))))
162161imp 444 . . . . . . . . . 10 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))))
163162ralbidva 2968 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))))
164 ffn 5958 . . . . . . . . . . . . . 14 (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
16531, 164syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn (𝐴[,]𝐵))
166 fnconstg 6006 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ ℝ → ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵))
167153, 166syl 17 . . . . . . . . . . . . 13 (𝜑 → ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵))
168 eqfnfv 6219 . . . . . . . . . . . . 13 ((𝐹 Fn (𝐴[,]𝐵) ∧ ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵)) → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦)))
169165, 167, 168syl2anc 691 . . . . . . . . . . . 12 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦)))
170 fvex 6113 . . . . . . . . . . . . . . 15 (𝐹𝐴) ∈ V
171170fvconst2 6374 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) = (𝐹𝐴))
172171eqeq2d 2620 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) ↔ (𝐹𝑦) = (𝐹𝐴)))
173172ralbiia 2962 . . . . . . . . . . . 12 (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴))
174169, 173syl6bb 275 . . . . . . . . . . 11 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴)))
175 fconstmpt 5085 . . . . . . . . . . . . . . . . . . . 20 ((𝐴[,]𝐵) × {(𝐹𝐴)}) = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))
176175eqeq2i 2622 . . . . . . . . . . . . . . . . . . 19 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴)))
177176biimpi 205 . . . . . . . . . . . . . . . . . 18 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴)))
178177oveq2d 6565 . . . . . . . . . . . . . . . . 17 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))))
179153recnd 9947 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐹𝐴) ∈ ℂ)
180179adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ℝ) → (𝐹𝐴) ∈ ℂ)
181 0cnd 9912 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ℝ) → 0 ∈ ℂ)
18260, 179dvmptc 23527 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℝ D (𝑢 ∈ ℝ ↦ (𝐹𝐴))) = (𝑢 ∈ ℝ ↦ 0))
18360, 180, 181, 182, 49, 55, 54, 57dvmptres2 23531 . . . . . . . . . . . . . . . . 17 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
184178, 183sylan9eqr 2666 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
185184fveq1d 6105 . . . . . . . . . . . . . . 15 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ((ℝ D 𝐹)‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥))
186 eqidd 2611 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → 0 = 0)
187 eqid 2610 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (𝐴(,)𝐵) ↦ 0) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0)
188 c0ex 9913 . . . . . . . . . . . . . . . 16 0 ∈ V
189186, 187, 188fvmpt 6191 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥) = 0)
190185, 189sylan9eq 2664 . . . . . . . . . . . . . 14 (((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) = 0)
191190ralrimiva 2949 . . . . . . . . . . . . 13 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
192 ioon0 12072 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
193149, 150, 192syl2anc 691 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
1943, 193mpbird 246 . . . . . . . . . . . . . 14 (𝜑 → (𝐴(,)𝐵) ≠ ∅)
195 r19.2z 4012 . . . . . . . . . . . . . 14 (((𝐴(,)𝐵) ≠ ∅ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
196194, 195sylan 487 . . . . . . . . . . . . 13 ((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
197191, 196syldan 486 . . . . . . . . . . . 12 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
198197ex 449 . . . . . . . . . . 11 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
199174, 198sylbird 249 . . . . . . . . . 10 (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
200199ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
201163, 200sylbird 249 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) → (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
202201impancom 455 . . . . . . 7 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
203148, 202syld 46 . . . . . 6 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
20426, 130, 203ecased 982 . . . . 5 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
205204ex 449 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
2069, 205syl5bir 232 . . 3 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → ((∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
207206rexlimdvva 3020 . 2 (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∃𝑣 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
2088, 207mpd 15 1 (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  -cneg 10146  (,)cioo 12046  [,]cicc 12049  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:  cmvth  23558  lhop1lem  23580
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