MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rolle Structured version   Visualization version   GIF version

Theorem rolle 24887
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 10980 . . . 4 (𝜑𝐴𝐵)
5 rolle.f . . . 4 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
61, 2, 4, 5evthicc 24356 . . 3 (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
7 reeanv 3279 . . 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 237 . 2 (𝜑 → ∃𝑢 ∈ (𝐴[,]𝐵)∃𝑣 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
9 r19.26 3092 . . . 4 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ↔ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
101ad2antrr 726 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐴 ∈ ℝ)
112ad2antrr 726 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐵 ∈ ℝ)
123ad2antrr 726 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐴 < 𝐵)
135ad2antrr 726 . . . . . . . 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 726 . . . . . . . 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 486 . . . . . . . . . . 11 (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → (𝐹𝑦) ≤ (𝐹𝑢))
1716ralimi 3083 . . . . . . . . . 10 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢))
18 fveq2 6717 . . . . . . . . . . . 12 (𝑦 = 𝑡 → (𝐹𝑦) = (𝐹𝑡))
1918breq1d 5063 . . . . . . . . . . 11 (𝑦 = 𝑡 → ((𝐹𝑦) ≤ (𝐹𝑢) ↔ (𝐹𝑡) ≤ (𝐹𝑢)))
2019cbvralvw 3358 . . . . . . . . . 10 (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹𝑡) ≤ (𝐹𝑢))
2117, 20sylib 221 . . . . . . . . 9 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹𝑡) ≤ (𝐹𝑢))
2221ad2antrl 728 . . . . . . . 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 777 . . . . . . . 8 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝑢 ∈ (𝐴[,]𝐵))
24 simprr 773 . . . . . . . 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 24886 . . . . . . 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 460 . . . . . 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 726 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐴 ∈ ℝ)
282ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐵 ∈ ℝ)
293ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐴 < 𝐵)
30 cncff 23790 . . . . . . . . . . . . . . 15 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
315, 30syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
3231ffvelrnda 6904 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝐹𝑢) ∈ ℝ)
3332renegcld 11259 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹𝑢) ∈ ℝ)
3433fmpttd 6932 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)):(𝐴[,]𝐵)⟶ℝ)
35 ax-resscn 10786 . . . . . . . . . . . 12 ℝ ⊆ ℂ
36 ssid 3923 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
37 cncfss 23796 . . . . . . . . . . . . . . 15 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
3835, 36, 37mp2an 692 . . . . . . . . . . . . . 14 ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
3938, 5sseldi 3899 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
40 eqid 2737 . . . . . . . . . . . . . 14 (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) = (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))
4140negfcncf 23820 . . . . . . . . . . . . 13 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
4239, 41syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
43 cncffvrn 23795 . . . . . . . . . . . 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))
4435, 42, 43sylancr 590 . . . . . . . . . . 11 (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)):(𝐴[,]𝐵)⟶ℝ))
4534, 44mpbird 260 . . . . . . . . . 10 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
4645ad2antrr 726 . . . . . . . . 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))
4735a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℝ ⊆ ℂ)
48 iccssre 13017 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
491, 2, 48syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
50 fss 6562 . . . . . . . . . . . . . . . . 17 ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
5131, 35, 50sylancl 589 . . . . . . . . . . . . . . . 16 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
5251ffvelrnda 6904 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝐹𝑢) ∈ ℂ)
5352negcld 11176 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹𝑢) ∈ ℂ)
54 eqid 2737 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
5554tgioo2 23700 . . . . . . . . . . . . . 14 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
56 iccntr 23718 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
571, 2, 56syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
5847, 49, 53, 55, 54, 57dvmptntr 24868 . . . . . . . . . . . . 13 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹𝑢))))
59 reelprrecn 10821 . . . . . . . . . . . . . . 15 ℝ ∈ {ℝ, ℂ}
6059a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℝ ∈ {ℝ, ℂ})
61 ioossicc 13021 . . . . . . . . . . . . . . . 16 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
6261sseli 3896 . . . . . . . . . . . . . . 15 (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ (𝐴[,]𝐵))
6362, 52sylan2 596 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴(,)𝐵)) → (𝐹𝑢) ∈ ℂ)
64 fvexd 6732 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑢) ∈ V)
6531feqmptd 6780 . . . . . . . . . . . . . . . 16 (𝜑𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢)))
6665oveq2d 7229 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢))))
67 dvf 24804 . . . . . . . . . . . . . . . . 17 (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
6814feq2d 6531 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
6967, 68mpbii 236 . . . . . . . . . . . . . . . 16 (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
7069feqmptd 6780 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
7147, 49, 52, 55, 54, 57dvmptntr 24868 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑢))))
7266, 70, 713eqtr3rd 2786 . . . . . . . . . . . . . 14 (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
7360, 63, 64, 72dvmptneg 24863 . . . . . . . . . . . . 13 (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
7458, 73eqtrd 2777 . . . . . . . . . . . 12 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
7574dmeqd 5774 . . . . . . . . . . 11 (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
76 dmmptg 6105 . . . . . . . . . . . 12 (∀𝑢 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑢) ∈ V → dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵))
77 negex 11076 . . . . . . . . . . . . 13 -((ℝ D 𝐹)‘𝑢) ∈ V
7877a1i 11 . . . . . . . . . . . 12 (𝑢 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑢) ∈ V)
7976, 78mprg 3075 . . . . . . . . . . 11 dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵)
8075, 79eqtrdi 2794 . . . . . . . . . 10 (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (𝐴(,)𝐵))
8180ad2antrr 726 . . . . . . . . 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 488 . . . . . . . . . . . . . 14 (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → (𝐹𝑣) ≤ (𝐹𝑦))
8331ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
84 simplrr 778 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑣 ∈ (𝐴[,]𝐵))
8583, 84ffvelrnd 6905 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝑣) ∈ ℝ)
8631adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
8786ffvelrnda 6904 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝑦) ∈ ℝ)
8885, 87lenegd 11411 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ -(𝐹𝑦) ≤ -(𝐹𝑣)))
89 fveq2 6717 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑦 → (𝐹𝑢) = (𝐹𝑦))
9089negeqd 11072 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑦 → -(𝐹𝑢) = -(𝐹𝑦))
91 negex 11076 . . . . . . . . . . . . . . . . . 18 -(𝐹𝑦) ∈ V
9290, 40, 91fvmpt 6818 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = -(𝐹𝑦))
9392adantl 485 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = -(𝐹𝑦))
94 fveq2 6717 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑣 → (𝐹𝑢) = (𝐹𝑣))
9594negeqd 11072 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑣 → -(𝐹𝑢) = -(𝐹𝑣))
96 negex 11076 . . . . . . . . . . . . . . . . . 18 -(𝐹𝑣) ∈ V
9795, 40, 96fvmpt 6818 . . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . 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 285 . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . 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 410 . . . . . . . . . . 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 6717 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡))
105104breq1d 5063 . . . . . . . . . . . 12 (𝑦 = 𝑡 → (((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) ↔ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣)))
106105cbvralvw 3358 . . . . . . . . . . 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 221 . . . . . . . . . 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 717 . . . . . . . . 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 778 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝑣 ∈ (𝐴[,]𝐵))
110 simprr 773 . . . . . . . . 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 24886 . . . . . . . 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 6719 . . . . . . . . . . . . 13 (𝜑 → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥))
113 fveq2 6717 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ((ℝ D 𝐹)‘𝑢) = ((ℝ D 𝐹)‘𝑥))
114113negeqd 11072 . . . . . . . . . . . . . 14 (𝑢 = 𝑥 → -((ℝ D 𝐹)‘𝑢) = -((ℝ D 𝐹)‘𝑥))
115 eqid 2737 . . . . . . . . . . . . . 14 (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))
116 negex 11076 . . . . . . . . . . . . . 14 -((ℝ D 𝐹)‘𝑥) ∈ V
117114, 115, 116fvmpt 6818 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
118112, 117sylan9eq 2798 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
119118eqeq1d 2739 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
12014eleq2d 2823 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ dom (ℝ D 𝐹) ↔ 𝑥 ∈ (𝐴(,)𝐵)))
121120biimpar 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ dom (ℝ D 𝐹))
12267ffvelrni 6903 . . . . . . . . . . . . 13 (𝑥 ∈ dom (ℝ D 𝐹) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
123121, 122syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
124123negeq0d 11181 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
125119, 124bitr4d 285 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑥) = 0))
126125rexbidva 3215 . . . . . . . . 9 (𝜑 → (∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
127126ad2antrr 726 . . . . . . . 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 235 . . . . . . 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 460 . . . . . 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 3412 . . . . . . . . . . 11 𝑢 ∈ V
131130elpr 4564 . . . . . . . . . 10 (𝑢 ∈ {𝐴, 𝐵} ↔ (𝑢 = 𝐴𝑢 = 𝐵))
132 fveq2 6717 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (𝐹𝑢) = (𝐹𝐴))
133132a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑢 = 𝐴 → (𝐹𝑢) = (𝐹𝐴)))
134 rolle.e . . . . . . . . . . . . 13 (𝜑 → (𝐹𝐴) = (𝐹𝐵))
135134eqcomd 2743 . . . . . . . . . . . 12 (𝜑 → (𝐹𝐵) = (𝐹𝐴))
136 fveqeq2 6726 . . . . . . . . . . . 12 (𝑢 = 𝐵 → ((𝐹𝑢) = (𝐹𝐴) ↔ (𝐹𝐵) = (𝐹𝐴)))
137135, 136syl5ibrcom 250 . . . . . . . . . . 11 (𝜑 → (𝑢 = 𝐵 → (𝐹𝑢) = (𝐹𝐴)))
138133, 137jaod 859 . . . . . . . . . 10 (𝜑 → ((𝑢 = 𝐴𝑢 = 𝐵) → (𝐹𝑢) = (𝐹𝐴)))
139131, 138syl5bi 245 . . . . . . . . 9 (𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴)))
140 eleq1w 2820 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (𝑢 ∈ {𝐴, 𝐵} ↔ 𝑣 ∈ {𝐴, 𝐵}))
141 fveqeq2 6726 . . . . . . . . . . . 12 (𝑢 = 𝑣 → ((𝐹𝑢) = (𝐹𝐴) ↔ (𝐹𝑣) = (𝐹𝐴)))
142140, 141imbi12d 348 . . . . . . . . . . 11 (𝑢 = 𝑣 → ((𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴)) ↔ (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴))))
143142imbi2d 344 . . . . . . . . . 10 (𝑢 = 𝑣 → ((𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴))) ↔ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴)))))
144143, 139chvarvv 2007 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴)))
145139, 144anim12d 612 . . . . . . . 8 (𝜑 → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))))
146145ad2antrr 726 . . . . . . 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))))
1471rexrd 10883 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ ℝ*)
1482rexrd 10883 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ ℝ*)
149 lbicc2 13052 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
150147, 148, 4, 149syl3anc 1373 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ (𝐴[,]𝐵))
15131, 150ffvelrnd 6905 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝐴) ∈ ℝ)
152151ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝐴) ∈ ℝ)
15387, 152letri3d 10974 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦))))
154 breq2 5057 . . . . . . . . . . . . . . 15 ((𝐹𝑢) = (𝐹𝐴) → ((𝐹𝑦) ≤ (𝐹𝑢) ↔ (𝐹𝑦) ≤ (𝐹𝐴)))
155 breq1 5056 . . . . . . . . . . . . . . 15 ((𝐹𝑣) = (𝐹𝐴) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ (𝐹𝐴) ≤ (𝐹𝑦)))
156154, 155bi2anan9 639 . . . . . . . . . . . . . 14 (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦))))
157156bibi2d 346 . . . . . . . . . . . . 13 (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → (((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) ↔ ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦)))))
158153, 157syl5ibrcom 250 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)))))
159158impancom 455 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) → (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)))))
160159imp 410 . . . . . . . . . 10 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))))
161160ralbidva 3117 . . . . . . . . 9 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))))
16231ffnd 6546 . . . . . . . . . . . . 13 (𝜑𝐹 Fn (𝐴[,]𝐵))
163 fnconstg 6607 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ ℝ → ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵))
164151, 163syl 17 . . . . . . . . . . . . 13 (𝜑 → ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵))
165 eqfnfv 6852 . . . . . . . . . . . . 13 ((𝐹 Fn (𝐴[,]𝐵) ∧ ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵)) → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦)))
166162, 164, 165syl2anc 587 . . . . . . . . . . . 12 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦)))
167 fvex 6730 . . . . . . . . . . . . . . 15 (𝐹𝐴) ∈ V
168167fvconst2 7019 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) = (𝐹𝐴))
169168eqeq2d 2748 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) ↔ (𝐹𝑦) = (𝐹𝐴)))
170169ralbiia 3087 . . . . . . . . . . . 12 (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴))
171166, 170bitrdi 290 . . . . . . . . . . 11 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴)))
172 ioon0 12961 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
173147, 148, 172syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
1743, 173mpbird 260 . . . . . . . . . . . . 13 (𝜑 → (𝐴(,)𝐵) ≠ ∅)
175 fconstmpt 5611 . . . . . . . . . . . . . . . . . . . 20 ((𝐴[,]𝐵) × {(𝐹𝐴)}) = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))
176175eqeq2i 2750 . . . . . . . . . . . . . . . . . . 19 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴)))
177176biimpi 219 . . . . . . . . . . . . . . . . . 18 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴)))
178177oveq2d 7229 . . . . . . . . . . . . . . . . 17 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))))
179151recnd 10861 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐹𝐴) ∈ ℂ)
180179adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ℝ) → (𝐹𝐴) ∈ ℂ)
181 0cnd 10826 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ℝ) → 0 ∈ ℂ)
18260, 179dvmptc 24855 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℝ D (𝑢 ∈ ℝ ↦ (𝐹𝐴))) = (𝑢 ∈ ℝ ↦ 0))
18360, 180, 181, 182, 49, 55, 54, 57dvmptres2 24859 . . . . . . . . . . . . . . . . 17 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
184178, 183sylan9eqr 2800 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
185184fveq1d 6719 . . . . . . . . . . . . . . 15 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ((ℝ D 𝐹)‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥))
186 eqidd 2738 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → 0 = 0)
187 eqid 2737 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (𝐴(,)𝐵) ↦ 0) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0)
188 c0ex 10827 . . . . . . . . . . . . . . . 16 0 ∈ V
189186, 187, 188fvmpt 6818 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥) = 0)
190185, 189sylan9eq 2798 . . . . . . . . . . . . . 14 (((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) = 0)
191190ralrimiva 3105 . . . . . . . . . . . . 13 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
192 r19.2z 4406 . . . . . . . . . . . . 13 (((𝐴(,)𝐵) ≠ ∅ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
193174, 191, 192syl2an2r 685 . . . . . . . . . . . 12 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
194193ex 416 . . . . . . . . . . 11 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
195171, 194sylbird 263 . . . . . . . . . 10 (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
196195ad2antrr 726 . . . . . . . . 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))
197161, 196sylbird 263 . . . . . . . 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))
198197impancom 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))
199146, 198syld 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))
20026, 129, 199ecased 1035 . . . . 5 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
201200ex 416 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
2029, 201syl5bir 246 . . 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))
203202rexlimdvva 3213 . 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))
2048, 203mpd 15 1 (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wcel 2110  wne 2940  wral 3061  wrex 3062  Vcvv 3408  wss 3866  c0 4237  {csn 4541  {cpr 4543   class class class wbr 5053  cmpt 5135   × cxp 5549  dom cdm 5551  ran crn 5552   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  cc 10727  cr 10728  0cc0 10729  *cxr 10866   < clt 10867  cle 10868  -cneg 11063  (,)cioo 12935  [,]cicc 12938  TopOpenctopn 16926  topGenctg 16942  fldccnfld 20363  intcnt 21914  cnccncf 23773   D cdv 24760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-addf 10808  ax-mulf 10809
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-fi 9027  df-sup 9058  df-inf 9059  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-ioo 12939  df-ico 12941  df-icc 12942  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-rest 16927  df-topn 16928  df-0g 16946  df-gsum 16947  df-topgen 16948  df-pt 16949  df-prds 16952  df-xrs 17007  df-qtop 17012  df-imas 17013  df-xps 17015  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-mulg 18489  df-cntz 18711  df-cmn 19172  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-fbas 20360  df-fg 20361  df-cnfld 20364  df-top 21791  df-topon 21808  df-topsp 21830  df-bases 21843  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995  df-lp 22033  df-perf 22034  df-cn 22124  df-cnp 22125  df-haus 22212  df-cmp 22284  df-tx 22459  df-hmeo 22652  df-fil 22743  df-fm 22835  df-flim 22836  df-flf 22837  df-xms 23218  df-ms 23219  df-tms 23220  df-cncf 23775  df-limc 24763  df-dv 24764
This theorem is referenced by:  cmvth  24888  lhop1lem  24910
  Copyright terms: Public domain W3C validator