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Theorem rolle 26043
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 11407 . . . 4 (𝜑𝐴𝐵)
5 rolle.f . . . 4 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
61, 2, 4, 5evthicc 25508 . . 3 (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
7 reeanv 3227 . . 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 234 . 2 (𝜑 → ∃𝑢 ∈ (𝐴[,]𝐵)∃𝑣 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑣) ≤ (𝐹𝑦)))
9 r19.26 3109 . . . 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 482 . . . . . . . . . . 11 (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → (𝐹𝑦) ≤ (𝐹𝑢))
1716ralimi 3081 . . . . . . . . . 10 (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢))
18 fveq2 6907 . . . . . . . . . . . 12 (𝑦 = 𝑡 → (𝐹𝑦) = (𝐹𝑡))
1918breq1d 5158 . . . . . . . . . . 11 (𝑦 = 𝑡 → ((𝐹𝑦) ≤ (𝐹𝑢) ↔ (𝐹𝑡) ≤ (𝐹𝑢)))
2019cbvralvw 3235 . . . . . . . . . 10 (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑢) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹𝑡) ≤ (𝐹𝑢))
2117, 20sylib 218 . . . . . . . . 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 26042 . . . . . . 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 456 . . . . . 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 24933 . . . . . . . . . . . . . . 15 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
315, 30syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
3231ffvelcdmda 7104 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝐹𝑢) ∈ ℝ)
3332renegcld 11688 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹𝑢) ∈ ℝ)
3433fmpttd 7135 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)):(𝐴[,]𝐵)⟶ℝ)
35 ax-resscn 11210 . . . . . . . . . . . 12 ℝ ⊆ ℂ
36 ssid 4018 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
37 cncfss 24939 . . . . . . . . . . . . . . 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, 5sselid 3993 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
40 eqid 2735 . . . . . . . . . . . . . 14 (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) = (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))
4140negfcncf 24964 . . . . . . . . . . . . 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 cncfcdm 24938 . . . . . . . . . . . 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 587 . . . . . . . . . . 11 (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)):(𝐴[,]𝐵)⟶ℝ))
4534, 44mpbird 257 . . . . . . . . . 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 13466 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
491, 2, 48syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
50 fss 6753 . . . . . . . . . . . . . . . . 17 ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
5131, 35, 50sylancl 586 . . . . . . . . . . . . . . . 16 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
5251ffvelcdmda 7104 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝐹𝑢) ∈ ℂ)
5352negcld 11605 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹𝑢) ∈ ℂ)
54 eqid 2735 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
5554tgioo2 24839 . . . . . . . . . . . . . 14 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
56 iccntr 24857 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
571, 2, 56syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
5847, 49, 53, 55, 54, 57dvmptntr 26024 . . . . . . . . . . . . 13 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹𝑢))))
59 reelprrecn 11245 . . . . . . . . . . . . . . 15 ℝ ∈ {ℝ, ℂ}
6059a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ℝ ∈ {ℝ, ℂ})
61 ioossicc 13470 . . . . . . . . . . . . . . . 16 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
6261sseli 3991 . . . . . . . . . . . . . . 15 (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ (𝐴[,]𝐵))
6362, 52sylan2 593 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴(,)𝐵)) → (𝐹𝑢) ∈ ℂ)
64 fvexd 6922 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑢) ∈ V)
6531feqmptd 6977 . . . . . . . . . . . . . . . 16 (𝜑𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢)))
6665oveq2d 7447 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢))))
67 dvf 25957 . . . . . . . . . . . . . . . . 17 (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
6814feq2d 6723 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
6967, 68mpbii 233 . . . . . . . . . . . . . . . 16 (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
7069feqmptd 6977 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
7147, 49, 52, 55, 54, 57dvmptntr 26024 . . . . . . . . . . . . . . 15 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑢))))
7266, 70, 713eqtr3rd 2784 . . . . . . . . . . . . . 14 (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
7360, 63, 64, 72dvmptneg 26019 . . . . . . . . . . . . 13 (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
7458, 73eqtrd 2775 . . . . . . . . . . . 12 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
7574dmeqd 5919 . . . . . . . . . . 11 (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))) = dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
76 dmmptg 6264 . . . . . . . . . . . 12 (∀𝑢 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑢) ∈ V → dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵))
77 negex 11504 . . . . . . . . . . . . 13 -((ℝ D 𝐹)‘𝑢) ∈ V
7877a1i 11 . . . . . . . . . . . 12 (𝑢 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑢) ∈ V)
7976, 78mprg 3065 . . . . . . . . . . 11 dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵)
8075, 79eqtrdi 2791 . . . . . . . . . 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 484 . . . . . . . . . . . . . 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, 84ffvelcdmd 7105 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝑣) ∈ ℝ)
8631adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
8786ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝑦) ∈ ℝ)
8885, 87lenegd 11840 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ -(𝐹𝑦) ≤ -(𝐹𝑣)))
89 fveq2 6907 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑦 → (𝐹𝑢) = (𝐹𝑦))
9089negeqd 11500 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑦 → -(𝐹𝑢) = -(𝐹𝑦))
91 negex 11504 . . . . . . . . . . . . . . . . . 18 -(𝐹𝑦) ∈ V
9290, 40, 91fvmpt 7016 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = -(𝐹𝑦))
9392adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = -(𝐹𝑦))
94 fveq2 6907 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑣 → (𝐹𝑢) = (𝐹𝑣))
9594negeqd 11500 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑣 → -(𝐹𝑢) = -(𝐹𝑣))
96 negex 11504 . . . . . . . . . . . . . . . . . 18 -(𝐹𝑣) ∈ V
9795, 40, 96fvmpt 7016 . . . . . . . . . . . . . . . . 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 5161 . . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣)))
10182, 100imbitrid 244 . . . . . . . . . . . . 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 3165 . . . . . . . . . . . 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 406 . . . . . . . . . . 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 6907 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) = ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡))
105104breq1d 5158 . . . . . . . . . . . 12 (𝑦 = 𝑡 → (((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣) ↔ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢))‘𝑣)))
106105cbvralvw 3235 . . . . . . . . . . 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 218 . . . . . . . . . 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 26042 . . . . . . . 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 6909 . . . . . . . . . . . . 13 (𝜑 → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥))
113 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → ((ℝ D 𝐹)‘𝑢) = ((ℝ D 𝐹)‘𝑥))
114113negeqd 11500 . . . . . . . . . . . . . 14 (𝑢 = 𝑥 → -((ℝ D 𝐹)‘𝑢) = -((ℝ D 𝐹)‘𝑥))
115 eqid 2735 . . . . . . . . . . . . . 14 (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))
116 negex 11504 . . . . . . . . . . . . . 14 -((ℝ D 𝐹)‘𝑥) ∈ V
117114, 115, 116fvmpt 7016 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
118112, 117sylan9eq 2795 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
119118eqeq1d 2737 . . . . . . . . . . 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 477 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ dom (ℝ D 𝐹))
12267ffvelcdmi 7103 . . . . . . . . . . . . 13 (𝑥 ∈ dom (ℝ D 𝐹) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
123121, 122syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
124123negeq0d 11610 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
125119, 124bitr4d 282 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹𝑢)))‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑥) = 0))
126125rexbidva 3175 . . . . . . . . 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 232 . . . . . . 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 456 . . . . . 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 3482 . . . . . . . . . . 11 𝑢 ∈ V
131130elpr 4655 . . . . . . . . . 10 (𝑢 ∈ {𝐴, 𝐵} ↔ (𝑢 = 𝐴𝑢 = 𝐵))
132 fveq2 6907 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (𝐹𝑢) = (𝐹𝐴))
133132a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑢 = 𝐴 → (𝐹𝑢) = (𝐹𝐴)))
134 rolle.e . . . . . . . . . . . . 13 (𝜑 → (𝐹𝐴) = (𝐹𝐵))
135134eqcomd 2741 . . . . . . . . . . . 12 (𝜑 → (𝐹𝐵) = (𝐹𝐴))
136 fveqeq2 6916 . . . . . . . . . . . 12 (𝑢 = 𝐵 → ((𝐹𝑢) = (𝐹𝐴) ↔ (𝐹𝐵) = (𝐹𝐴)))
137135, 136syl5ibrcom 247 . . . . . . . . . . 11 (𝜑 → (𝑢 = 𝐵 → (𝐹𝑢) = (𝐹𝐴)))
138133, 137jaod 859 . . . . . . . . . 10 (𝜑 → ((𝑢 = 𝐴𝑢 = 𝐵) → (𝐹𝑢) = (𝐹𝐴)))
139131, 138biimtrid 242 . . . . . . . . 9 (𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴)))
140 eleq1w 2822 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (𝑢 ∈ {𝐴, 𝐵} ↔ 𝑣 ∈ {𝐴, 𝐵}))
141 fveqeq2 6916 . . . . . . . . . . . 12 (𝑢 = 𝑣 → ((𝐹𝑢) = (𝐹𝐴) ↔ (𝐹𝑣) = (𝐹𝐴)))
142140, 141imbi12d 344 . . . . . . . . . . 11 (𝑢 = 𝑣 → ((𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴)) ↔ (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴))))
143142imbi2d 340 . . . . . . . . . 10 (𝑢 = 𝑣 → ((𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹𝑢) = (𝐹𝐴))) ↔ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴)))))
144143, 139chvarvv 1996 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹𝑣) = (𝐹𝐴)))
145139, 144anim12d 609 . . . . . . . 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 11309 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ ℝ*)
1482rexrd 11309 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ ℝ*)
149 lbicc2 13501 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
150147, 148, 4, 149syl3anc 1370 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ (𝐴[,]𝐵))
15131, 150ffvelcdmd 7105 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝐴) ∈ ℝ)
152151ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹𝐴) ∈ ℝ)
15387, 152letri3d 11401 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦))))
154 breq2 5152 . . . . . . . . . . . . . . 15 ((𝐹𝑢) = (𝐹𝐴) → ((𝐹𝑦) ≤ (𝐹𝑢) ↔ (𝐹𝑦) ≤ (𝐹𝐴)))
155 breq1 5151 . . . . . . . . . . . . . . 15 ((𝐹𝑣) = (𝐹𝐴) → ((𝐹𝑣) ≤ (𝐹𝑦) ↔ (𝐹𝐴) ≤ (𝐹𝑦)))
156154, 155bi2anan9 638 . . . . . . . . . . . . . 14 (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → (((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦))))
157156bibi2d 342 . . . . . . . . . . . . 13 (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → (((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))) ↔ ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝐴) ∧ (𝐹𝐴) ≤ (𝐹𝑦)))))
158153, 157syl5ibrcom 247 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)))))
159158impancom 451 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) → (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)))))
160159imp 406 . . . . . . . . . 10 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹𝑢) = (𝐹𝐴) ∧ (𝐹𝑣) = (𝐹𝐴))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹𝑦) = (𝐹𝐴) ↔ ((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦))))
161160ralbidva 3174 . . . . . . . . 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 6738 . . . . . . . . . . . . 13 (𝜑𝐹 Fn (𝐴[,]𝐵))
163 fnconstg 6797 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ ℝ → ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵))
164151, 163syl 17 . . . . . . . . . . . . 13 (𝜑 → ((𝐴[,]𝐵) × {(𝐹𝐴)}) Fn (𝐴[,]𝐵))
165 eqfnfv 7051 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 12 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦)))
167 fvex 6920 . . . . . . . . . . . . . . 15 (𝐹𝐴) ∈ V
168167fvconst2 7224 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) = (𝐹𝐴))
169168eqeq2d 2746 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) ↔ (𝐹𝑦) = (𝐹𝐴)))
170169ralbiia 3089 . . . . . . . . . . . 12 (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (((𝐴[,]𝐵) × {(𝐹𝐴)})‘𝑦) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴))
171166, 170bitrdi 287 . . . . . . . . . . 11 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) = (𝐹𝐴)))
172 ioon0 13410 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
173147, 148, 172syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
1743, 173mpbird 257 . . . . . . . . . . . . 13 (𝜑 → (𝐴(,)𝐵) ≠ ∅)
175 fconstmpt 5751 . . . . . . . . . . . . . . . . . . . 20 ((𝐴[,]𝐵) × {(𝐹𝐴)}) = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))
176175eqeq2i 2748 . . . . . . . . . . . . . . . . . . 19 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) ↔ 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴)))
177176biimpi 216 . . . . . . . . . . . . . . . . . 18 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴)))
178177oveq2d 7447 . . . . . . . . . . . . . . . . 17 (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))))
179151recnd 11287 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐹𝐴) ∈ ℂ)
180179adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ℝ) → (𝐹𝐴) ∈ ℂ)
181 0cnd 11252 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢 ∈ ℝ) → 0 ∈ ℂ)
18260, 179dvmptc 26011 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℝ D (𝑢 ∈ ℝ ↦ (𝐹𝐴))) = (𝑢 ∈ ℝ ↦ 0))
18360, 180, 181, 182, 49, 55, 54, 57dvmptres2 26015 . . . . . . . . . . . . . . . . 17 (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹𝐴))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
184178, 183sylan9eqr 2797 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
185184fveq1d 6909 . . . . . . . . . . . . . . 15 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ((ℝ D 𝐹)‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥))
186 eqidd 2736 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥 → 0 = 0)
187 eqid 2735 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (𝐴(,)𝐵) ↦ 0) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0)
188 c0ex 11253 . . . . . . . . . . . . . . . 16 0 ∈ V
189186, 187, 188fvmpt 7016 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥) = 0)
190185, 189sylan9eq 2795 . . . . . . . . . . . . . 14 (((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) = 0)
191190ralrimiva 3144 . . . . . . . . . . . . 13 ((𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)})) → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
192 r19.2z 4501 . . . . . . . . . . . . 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 412 . . . . . . . . . . 11 (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
195171, 194sylbird 260 . . . . . . . . . 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 260 . . . . . . . 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 451 . . . . . . 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 412 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹𝑦) ≤ (𝐹𝑢) ∧ (𝐹𝑣) ≤ (𝐹𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
2029, 201biimtrrid 243 . . 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 3211 . 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 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339  {csn 4631  {cpr 4633   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cc 11151  cr 11152  0cc0 11153  *cxr 11292   < clt 11293  cle 11294  -cneg 11491  (,)cioo 13384  [,]cicc 13387  TopOpenctopn 17468  topGenctg 17484  fldccnfld 21382  intcnt 23041  cnccncf 24916   D cdv 25913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917
This theorem is referenced by:  cmvth  26044  cmvthOLD  26045  lhop1lem  26067
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