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Theorem rolle 23952
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 10377 . . . 4 (𝜑𝐴𝐵)
5 rolle.f . . . 4 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
61, 2, 4, 5evthicc 23428 . . 3 (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
7 reeanv 3245 . . 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 224 . 2 (𝜑 → ∃𝑢 ∈ (𝐴[,]𝐵)∃𝑣 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
9 r19.26 3202 . . . 4 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ↔ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
101ad2antrr 764 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐴 ∈ ℝ)
112ad2antrr 764 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐵 ∈ ℝ)
123ad2antrr 764 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐴 < 𝐵)
135ad2antrr 764 . . . . . . . 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 764 . . . . . . . 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 474 . . . . . . . . . . 11 (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → (𝐹𝑦) ≤ (𝐹𝑢))
1716ralimi 3090 . . . . . . . . . 10 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢))
18 fveq2 6352 . . . . . . . . . . . 12 (𝑦 = 𝑡 → (𝐹𝑦) = (𝐹𝑡))
1918breq1d 4814 . . . . . . . . . . 11 (𝑦 = 𝑡 → ((𝐹𝑦) ≤ (𝐹𝑢) ↔ (𝐹𝑡) ≤ (𝐹𝑢)))
2019cbvralv 3310 . . . . . . . . . 10 (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹𝑡) ≤ (𝐹𝑢))
2117, 20sylib 208 . . . . . . . . 9 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹𝑡) ≤ (𝐹𝑢))
2221ad2antrl 766 . . . . . . . 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 819 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝑢 ∈ (𝐴[,]𝐵))
24 simprr 813 . . . . . . . 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 23951 . . . . . . 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 644 . . . . . 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 764 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐴 ∈ ℝ)
282ad2antrr 764 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐵 ∈ ℝ)
293ad2antrr 764 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐴 < 𝐵)
30 cncff 22897 . . . . . . . . . . . . . . 15 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
315, 30syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
3231ffvelrnda 6522 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝐹𝑢) ∈ ℝ)
3332renegcld 10649 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹𝑢) ∈ ℝ)
34 eqid 2760 . . . . . . . . . . . 12 (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) = (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))
3533, 34fmptd 6548 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)):(𝐴[,]𝐵)⟶ℝ)
36 ax-resscn 10185 . . . . . . . . . . . 12 ℝ ⊆ ℂ
37 ssid 3765 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
38 cncfss 22903 . . . . . . . . . . . . . . 15 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
3936, 37, 38mp2an 710 . . . . . . . . . . . . . 14 ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
4039, 5sseldi 3742 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
4134negfcncf 22923 . . . . . . . . . . . . 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 22902 . . . . . . . . . . . 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 698 . . . . . . . . . . 11 (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)):(𝐴[,]𝐵)⟶ℝ))
4535, 44mpbird 247 . . . . . . . . . 10 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
4645ad2antrr 764 . . . . . . . . 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 12448 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
491, 2, 48syl2anc 696 . . . . . . . . . . . . . 14 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
50 fss 6217 . . . . . . . . . . . . . . . . 17 ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
5131, 36, 50sylancl 697 . . . . . . . . . . . . . . . 16 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
5251ffvelrnda 6522 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝐹𝑢) ∈ ℂ)
5352negcld 10571 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹𝑢) ∈ ℂ)
54 eqid 2760 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
5554tgioo2 22807 . . . . . . . . . . . . . 14 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
56 iccntr 22825 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
571, 2, 56syl2anc 696 . . . . . . . . . . . . . 14 (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
5847, 49, 53, 55, 54, 57dvmptntr 23933 . . . . . . . . . . . . 13 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹𝑢))))
59 reelprrecn 10220 . . . . . . . . . . . . . . 15 ℝ ∈ {ℝ, ℂ}
6059a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℝ ∈ {ℝ, ℂ})
61 ioossicc 12452 . . . . . . . . . . . . . . . 16 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
6261sseli 3740 . . . . . . . . . . . . . . 15 (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ (𝐴[,]𝐵))
6362, 52sylan2 492 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴(,)𝐵)) → (𝐹𝑢) ∈ ℂ)
64 fvexd 6364 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑢) ∈ V)
6531feqmptd 6411 . . . . . . . . . . . . . . . 16 (𝜑𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢)))
6665oveq2d 6829 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢))))
67 dvf 23870 . . . . . . . . . . . . . . . . 17 (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
6814feq2d 6192 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
6967, 68mpbii 223 . . . . . . . . . . . . . . . 16 (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
7069feqmptd 6411 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
7147, 49, 52, 55, 54, 57dvmptntr 23933 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑢))))
7266, 70, 713eqtr3rd 2803 . . . . . . . . . . . . . 14 (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
7360, 63, 64, 72dvmptneg 23928 . . . . . . . . . . . . 13 (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
7458, 73eqtrd 2794 . . . . . . . . . . . 12 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
7574dmeqd 5481 . . . . . . . . . . 11 (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
76 dmmptg 5793 . . . . . . . . . . . 12 (∀𝑢 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑢) ∈ V → dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵))
77 negex 10471 . . . . . . . . . . . . 13 -((ℝ D 𝐹)‘𝑢) ∈ V
7877a1i 11 . . . . . . . . . . . 12 (𝑢 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑢) ∈ V)
7976, 78mprg 3064 . . . . . . . . . . 11 dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵)
8075, 79syl6eq 2810 . . . . . . . . . 10 (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (𝐴(,)𝐵))
8180ad2antrr 764 . . . . . . . . 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))
82 simpr 479 . . . . . . . . . . . . . 14 (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → (𝐹𝑣) ≤ (𝐹𝑦))
8331ad2antrr 764 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
84 simplrr 820 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑣 ∈ (𝐴[,]𝐵))
8583, 84ffvelrnd 6523 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝑣) ∈ ℝ)
8631adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
8786ffvelrnda 6522 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝑦) ∈ ℝ)
8885, 87lenegd 10798 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ -(𝐹𝑦) ≤ -(𝐹𝑣)))
89 fveq2 6352 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑦 → (𝐹𝑢) = (𝐹𝑦))
9089negeqd 10467 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑦 → -(𝐹𝑢) = -(𝐹𝑦))
91 negex 10471 . . . . . . . . . . . . . . . . . 18 -(𝐹𝑦) ∈ V
9290, 34, 91fvmpt 6444 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = -(𝐹𝑦))
9392adantl 473 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = -(𝐹𝑦))
94 fveq2 6352 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑣 → (𝐹𝑢) = (𝐹𝑣))
9594negeqd 10467 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑣 → -(𝐹𝑢) = -(𝐹𝑣))
96 negex 10471 . . . . . . . . . . . . . . . . . 18 -(𝐹𝑣) ∈ V
9795, 34, 96fvmpt 6444 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) = -(𝐹𝑣))
9884, 97syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) = -(𝐹𝑣))
9993, 98breq12d 4817 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) ↔ -(𝐹𝑦) ≤ -(𝐹𝑣)))
10088, 99bitr4d 271 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣)))
10182, 100syl5ib 234 . . . . . . . . . . . . 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)))
102101ralimdva 3100 . . . . . . . . . . . 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)))
103102imp 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))
104 fveq2 6352 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡))
105104breq1d 4814 . . . . . . . . . . . 12 (𝑦 = 𝑡 → (((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) ↔ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣)))
106105cbvralv 3310 . . . . . . . . . . 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))
107103, 106sylib 208 . . . . . . . . . 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))
108107adantrr 755 . . . . . . . . 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))
109 simplrr 820 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝑣 ∈ (𝐴[,]𝐵))
110 simprr 813 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → ¬ 𝑣 ∈ {𝐴, 𝐵})
11127, 28, 29, 46, 81, 108, 109, 110rollelem 23951 . . . . . . . 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)
11274fveq1d 6354 . . . . . . . . . . . . 13 (𝜑 → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥))
113 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ((ℝ D 𝐹)‘𝑢) = ((ℝ D 𝐹)‘𝑥))
114113negeqd 10467 . . . . . . . . . . . . . 14 (𝑢 = 𝑥 → -((ℝ D 𝐹)‘𝑢) = -((ℝ D 𝐹)‘𝑥))
115 eqid 2760 . . . . . . . . . . . . . 14 (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))
116 negex 10471 . . . . . . . . . . . . . 14 -((ℝ D 𝐹)‘𝑥) ∈ V
117114, 115, 116fvmpt 6444 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
118112, 117sylan9eq 2814 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
119118eqeq1d 2762 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
12014eleq2d 2825 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ dom (ℝ D 𝐹) ↔ 𝑥 ∈ (𝐴(,)𝐵)))
121120biimpar 503 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ dom (ℝ D 𝐹))
12267ffvelrni 6521 . . . . . . . . . . . . 13 (𝑥 ∈ dom (ℝ D 𝐹) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
123121, 122syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
124123negeq0d 10576 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
125119, 124bitr4d 271 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑥) = 0))
126125rexbidva 3187 . . . . . . . . 9 (𝜑 → (∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
127126ad2antrr 764 . . . . . . . 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))
128111, 127mpbid 222 . . . . . . 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)
129128expr 644 . . . . . 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))
130 vex 3343 . . . . . . . . . . 11 𝑢 ∈ V
131130elpr 4343 . . . . . . . . . 10 (𝑢 ∈ {𝐴, 𝐵} ↔ (𝑢 = 𝐴𝑢 = 𝐵))
132 fveq2 6352 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (𝐹𝑢) = (𝐹𝐴))
133132a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑢 = 𝐴 → (𝐹𝑢) = (𝐹𝐴)))
134 rolle.e . . . . . . . . . . . . 13 (𝜑 → (𝐹𝐴) = (𝐹𝐵))
135134eqcomd 2766 . . . . . . . . . . . 12 (𝜑 → (𝐹𝐵) = (𝐹𝐴))
136 fveq2 6352 . . . . . . . . . . . . 13 (𝑢 = 𝐵 → (𝐹𝑢) = (𝐹𝐵))
137136eqeq1d 2762 . . . . . . . . . . . 12 (𝑢 = 𝐵 → ((𝐹𝑢) = (𝐹𝐴) ↔ (𝐹𝐵) = (𝐹𝐴)))
138135, 137syl5ibrcom 237 . . . . . . . . . . 11 (𝜑 → (𝑢 = 𝐵 → (𝐹𝑢) = (𝐹𝐴)))
139133, 138jaod 394 . . . . . . . . . 10 (𝜑 → ((𝑢 = 𝐴𝑢 = 𝐵) → (𝐹𝑢) = (𝐹𝐴)))
140131, 139syl5bi 232 . . . . . . . . 9 (𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴)))
141 eleq1w 2822 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (𝑢 ∈ {𝐴, 𝐵} ↔ 𝑣 ∈ {𝐴, 𝐵}))
14294eqeq1d 2762 . . . . . . . . . . . 12 (𝑢 = 𝑣 → ((𝐹𝑢) = (𝐹𝐴) ↔ (𝐹𝑣) = (𝐹𝐴)))
143141, 142imbi12d 333 . . . . . . . . . . 11 (𝑢 = 𝑣 → ((𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴)) ↔ (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴))))
144143imbi2d 329 . . . . . . . . . 10 (𝑢 = 𝑣 → ((𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴))) ↔ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴)))))
145144, 140chvarv 2408 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴)))
146140, 145anim12d 587 . . . . . . . 8 (𝜑 → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))))
147146ad2antrr 764 . . . . . . 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))))
1481rexrd 10281 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ ℝ*)
1492rexrd 10281 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ ℝ*)
150 lbicc2 12481 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
151148, 149, 4, 150syl3anc 1477 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ (𝐴[,]𝐵))
15231, 151ffvelrnd 6523 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝐴) ∈ ℝ)
153152ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝐴) ∈ ℝ)
15487, 153letri3d 10371 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦))))
155 breq2 4808 . . . . . . . . . . . . . . 15 ((𝐹𝑢) = (𝐹𝐴) → ((𝐹𝑦) ≤ (𝐹𝑢) ↔ (𝐹𝑦) ≤ (𝐹𝐴)))
156 breq1 4807 . . . . . . . . . . . . . . 15 ((𝐹𝑣) = (𝐹𝐴) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ (𝐹𝐴) ≤ (𝐹𝑦)))
157155, 156bi2anan9 953 . . . . . . . . . . . . . 14 (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦))))
158157bibi2d 331 . . . . . . . . . . . . 13 (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → (((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) ↔ ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦)))))
159154, 158syl5ibrcom 237 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)))))
160159impancom 455 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) → (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)))))
161160imp 444 . . . . . . . . . 10 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))))
162161ralbidva 3123 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))))
163 ffn 6206 . . . . . . . . . . . . . 14 (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
16431, 163syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn (𝐴[,]𝐵))
165 fnconstg 6254 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ ℝ → ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵))
166152, 165syl 17 . . . . . . . . . . . . 13 (𝜑 → ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵))
167 eqfnfv 6474 . . . . . . . . . . . . 13 ((𝐹 Fn (𝐴[,]𝐵) ∧ ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵)) → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦)))
168164, 166, 167syl2anc 696 . . . . . . . . . . . 12 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦)))
169 fvex 6362 . . . . . . . . . . . . . . 15 (𝐹𝐴) ∈ V
170169fvconst2 6633 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) = (𝐹𝐴))
171170eqeq2d 2770 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) ↔ (𝐹𝑦) = (𝐹𝐴)))
172171ralbiia 3117 . . . . . . . . . . . 12 (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴))
173168, 172syl6bb 276 . . . . . . . . . . 11 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴)))
174 fconstmpt 5320 . . . . . . . . . . . . . . . . . . . 20 ((𝐴[,]𝐵) × {(𝐹𝐴)}) = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))
175174eqeq2i 2772 . . . . . . . . . . . . . . . . . . 19 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴)))
176175biimpi 206 . . . . . . . . . . . . . . . . . 18 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴)))
177176oveq2d 6829 . . . . . . . . . . . . . . . . 17 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))))
178152recnd 10260 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐹𝐴) ∈ ℂ)
179178adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ℝ) → (𝐹𝐴) ∈ ℂ)
180 0cnd 10225 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ℝ) → 0 ∈ ℂ)
18160, 178dvmptc 23920 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℝ D (𝑢 ∈ ℝ ↦ (𝐹𝐴))) = (𝑢 ∈ ℝ ↦ 0))
18260, 179, 180, 181, 49, 55, 54, 57dvmptres2 23924 . . . . . . . . . . . . . . . . 17 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
183177, 182sylan9eqr 2816 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
184183fveq1d 6354 . . . . . . . . . . . . . . 15 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ((ℝ D 𝐹)‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥))
185 eqidd 2761 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → 0 = 0)
186 eqid 2760 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (𝐴(,)𝐵) ↦ 0) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0)
187 c0ex 10226 . . . . . . . . . . . . . . . 16 0 ∈ V
188185, 186, 187fvmpt 6444 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥) = 0)
189184, 188sylan9eq 2814 . . . . . . . . . . . . . 14 (((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) = 0)
190189ralrimiva 3104 . . . . . . . . . . . . 13 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
191 ioon0 12394 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
192148, 149, 191syl2anc 696 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
1933, 192mpbird 247 . . . . . . . . . . . . . 14 (𝜑 → (𝐴(,)𝐵) ≠ ∅)
194 r19.2z 4204 . . . . . . . . . . . . . 14 (((𝐴(,)𝐵) ≠ ∅ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
195193, 194sylan 489 . . . . . . . . . . . . 13 ((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
196190, 195syldan 488 . . . . . . . . . . . 12 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
197196ex 449 . . . . . . . . . . 11 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
198173, 197sylbird 250 . . . . . . . . . 10 (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
199198ad2antrr 764 . . . . . . . . 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))
200162, 199sylbird 250 . . . . . . . 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))
201200impancom 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))
202147, 201syld 47 . . . . . 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))
20326, 129, 202ecased 1022 . . . . 5 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
204203ex 449 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
2059, 204syl5bir 233 . . 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))
206205rexlimdvva 3176 . 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))
2078, 206mpd 15 1 (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  wss 3715  c0 4058  {csn 4321  {cpr 4323   class class class wbr 4804  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  cc 10126  cr 10127  0cc0 10128  *cxr 10265   < clt 10266  cle 10267  -cneg 10459  (,)cioo 12368  [,]cicc 12371  TopOpenctopn 16284  topGenctg 16300  fldccnfld 19948  intcnt 21023  cnccncf 22880   D cdv 23826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-fi 8482  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ioo 12372  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-starv 16158  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-unif 16167  df-hom 16168  df-cco 16169  df-rest 16285  df-topn 16286  df-0g 16304  df-gsum 16305  df-topgen 16306  df-pt 16307  df-prds 16310  df-xrs 16364  df-qtop 16369  df-imas 16370  df-xps 16372  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-fbas 19945  df-fg 19946  df-cnfld 19949  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027  df-nei 21104  df-lp 21142  df-perf 21143  df-cn 21233  df-cnp 21234  df-haus 21321  df-cmp 21392  df-tx 21567  df-hmeo 21760  df-fil 21851  df-fm 21943  df-flim 21944  df-flf 21945  df-xms 22326  df-ms 22327  df-tms 22328  df-cncf 22882  df-limc 23829  df-dv 23830
This theorem is referenced by:  cmvth  23953  lhop1lem  23975
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