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Theorem rolle 25152

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	⊢ (𝜑 → 𝐴 ∈ ℝ)
rolle.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
rolle.lt	⊢ (𝜑 → 𝐴 < 𝐵)
rolle.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
rolle.d	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
rolle.e	⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))

**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.

Step	Hyp	Ref	Expression
1		rolle.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		rolle.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		rolle.lt	. . . . 5 ⊢ (𝜑 → 𝐴 < 𝐵)
4	1, 2, 3	ltled 11121	. . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
5		rolle.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
6	1, 2, 4, 5	evthicc 24621	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)))
7		reeanv 3293	. . 3 ⊢ (∃𝑢 ∈ (𝐴[,]𝐵)∃𝑣 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)) ↔ (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)))
8	6, 7	sylibr 233	. 2 ⊢ (𝜑 → ∃𝑢 ∈ (𝐴[,]𝐵)∃𝑣 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)))
9		r19.26 3095	. . . 4 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ↔ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)))
10	1	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐴 ∈ ℝ)
11	2	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐵 ∈ ℝ)
12	3	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐴 < 𝐵)
13	5	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
14		rolle.d	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
15	14	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
16		simpl 483	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → (𝐹‘𝑦) ≤ (𝐹‘𝑢))
17	16	ralimi 3087	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢))
18		fveq2 6776	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (𝐹‘𝑦) = (𝐹‘𝑡))
19	18	breq1d 5086	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ (𝐹‘𝑡) ≤ (𝐹‘𝑢)))
20	19	cbvralvw 3382	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹‘𝑡) ≤ (𝐹‘𝑢))
21	17, 20	sylib 217	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹‘𝑡) ≤ (𝐹‘𝑢))
22	21	ad2antrl 725	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹‘𝑡) ≤ (𝐹‘𝑢))
23		simplrl 774	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → 𝑢 ∈ (𝐴[,]𝐵))
24		simprr 770	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → ¬ 𝑢 ∈ {𝐴, 𝐵})
25	10, 11, 12, 13, 15, 22, 23, 24	rollelem 25151	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑢 ∈ {𝐴, 𝐵})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
26	25	expr 457	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))) → (¬ 𝑢 ∈ {𝐴, 𝐵} → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
27	1	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐴 ∈ ℝ)
28	2	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐵 ∈ ℝ)
29	3	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝐴 < 𝐵)
30		cncff 24054	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
31	5, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
32	31	ffvelrnda 6963	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℝ)
33	32	renegcld 11400	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℝ)
34	33	fmpttd 6991	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ)
35		ax-resscn 10926	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
36		ssid 3944	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
37		cncfss 24060	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
38	35, 36, 37	mp2an 689	. . . . . . . . . . . . . 14 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
39	38, 5	sselid 3920	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
40		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) = (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))
41	40	negfcncf 24084	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
42	39, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
43		cncffvrn 24059	. . . . . . . . . . . 12 ⊢ ((ℝ ⊆ ℂ ∧ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ))
44	35, 42, 43	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ))
45	34, 44	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
46	45	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
47	35	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ⊆ ℂ)
48		iccssre 13159	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
49	1, 2, 48	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
50		fss 6619	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
51	31, 35, 50	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
52	51	ffvelrnda 6963	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℂ)
53	52	negcld 11317	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℂ)
54		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
55	54	tgioo2 23964	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
56		iccntr 23982	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
57	1, 2, 56	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
58	47, 49, 53, 55, 54, 57	dvmptntr 25133	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))))
59		reelprrecn 10961	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ {ℝ, ℂ}
60	59	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
61		ioossicc 13163	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
62	61	sseli 3918	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ (𝐴[,]𝐵))
63	62, 52	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑢) ∈ ℂ)
64		fvexd 6791	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑢) ∈ V)
65	31	feqmptd 6839	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢)))
66	65	oveq2d 7293	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))))
67		dvf 25069	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
68	14	feq2d 6588	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
69	67, 68	mpbii 232	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
70	69	feqmptd 6839	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
71	47, 49, 52, 55, 54, 57	dvmptntr 25133	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))))
72	66, 70, 71	3eqtr3rd 2787	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
73	60, 63, 64, 72	dvmptneg 25128	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
74	58, 73	eqtrd 2778	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
75	74	dmeqd 5816	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
76		dmmptg 6147	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑢) ∈ V → dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵))
77		negex 11217	. . . . . . . . . . . . 13 ⊢ -((ℝ D 𝐹)‘𝑢) ∈ V
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑢) ∈ V)
79	76, 78	mprg 3078	. . . . . . . . . . 11 ⊢ dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵)
80	75, 79	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝐴(,)𝐵))
81	80	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝐴(,)𝐵))
82		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → (𝐹‘𝑣) ≤ (𝐹‘𝑦))
83	31	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
84		simplrr 775	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑣 ∈ (𝐴[,]𝐵))
85	83, 84	ffvelrnd 6964	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑣) ∈ ℝ)
86	31	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
87	86	ffvelrnda 6963	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑦) ∈ ℝ)
88	85, 87	lenegd 11552	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ -(𝐹‘𝑦) ≤ -(𝐹‘𝑣)))
89		fveq2 6776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑦 → (𝐹‘𝑢) = (𝐹‘𝑦))
90	89	negeqd 11213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑦 → -(𝐹‘𝑢) = -(𝐹‘𝑦))
91		negex 11217	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑦) ∈ V
92	90, 40, 91	fvmpt 6877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = -(𝐹‘𝑦))
93	92	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = -(𝐹‘𝑦))
94		fveq2 6776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑣 → (𝐹‘𝑢) = (𝐹‘𝑣))
95	94	negeqd 11213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑣 → -(𝐹‘𝑢) = -(𝐹‘𝑣))
96		negex 11217	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑣) ∈ V
97	95, 40, 96	fvmpt 6877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣) = -(𝐹‘𝑣))
98	84, 97	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣) = -(𝐹‘𝑣))
99	93, 98	breq12d 5089	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣) ↔ -(𝐹‘𝑦) ≤ -(𝐹‘𝑣)))
100	88, 99	bitr4d 281	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣)))
101	82, 100	syl5ib 243	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣)))
102	101	ralimdva 3108	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣)))
103	102	imp 407	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))) → ∀𝑦 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣))
104		fveq2 6776	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑡))
105	104	breq1d 5086	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣) ↔ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣)))
106	105	cbvralvw 3382	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣))
107	103, 106	sylib 217	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))) → ∀𝑡 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣))
108	107	adantrr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → ∀𝑡 ∈ (𝐴[,]𝐵)((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑡) ≤ ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣))
109		simplrr 775	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → 𝑣 ∈ (𝐴[,]𝐵))
110		simprr 770	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → ¬ 𝑣 ∈ {𝐴, 𝐵})
111	27, 28, 29, 46, 81, 108, 109, 110	rollelem 25151	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0)
112	74	fveq1d 6778	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥))
113		fveq2 6776	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → ((ℝ D 𝐹)‘𝑢) = ((ℝ D 𝐹)‘𝑥))
114	113	negeqd 11213	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑥 → -((ℝ D 𝐹)‘𝑢) = -((ℝ D 𝐹)‘𝑥))
115		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))
116		negex 11217	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑥) ∈ V
117	114, 115, 116	fvmpt 6877	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
118	112, 117	sylan9eq 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
119	118	eqeq1d 2740	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
120	14	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ dom (ℝ D 𝐹) ↔ 𝑥 ∈ (𝐴(,)𝐵)))
121	120	biimpar 478	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ dom (ℝ D 𝐹))
122	67	ffvelrni 6962	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom (ℝ D 𝐹) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
123	121, 122	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
124	123	negeq0d 11322	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
125	119, 124	bitr4d 281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑥) = 0))
126	125	rexbidva 3224	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
127	126	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → (∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
128	111, 127	mpbid 231	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ∧ ¬ 𝑣 ∈ {𝐴, 𝐵})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
129	128	expr 457	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))) → (¬ 𝑣 ∈ {𝐴, 𝐵} → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
130		vex 3435	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
131	130	elpr 4586	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝐴, 𝐵} ↔ (𝑢 = 𝐴 ∨ 𝑢 = 𝐵))
132		fveq2 6776	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐴 → (𝐹‘𝑢) = (𝐹‘𝐴))
133	132	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 = 𝐴 → (𝐹‘𝑢) = (𝐹‘𝐴)))
134		rolle.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))
135	134	eqcomd 2744	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝐵) = (𝐹‘𝐴))
136		fveqeq2 6785	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → ((𝐹‘𝑢) = (𝐹‘𝐴) ↔ (𝐹‘𝐵) = (𝐹‘𝐴)))
137	135, 136	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 = 𝐵 → (𝐹‘𝑢) = (𝐹‘𝐴)))
138	133, 137	jaod 856	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 = 𝐴 ∨ 𝑢 = 𝐵) → (𝐹‘𝑢) = (𝐹‘𝐴)))
139	131, 138	syl5bi 241	. . . . . . . . 9 ⊢ (𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴)))
140		eleq1w 2821	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 ∈ {𝐴, 𝐵} ↔ 𝑣 ∈ {𝐴, 𝐵}))
141		fveqeq2 6785	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → ((𝐹‘𝑢) = (𝐹‘𝐴) ↔ (𝐹‘𝑣) = (𝐹‘𝐴)))
142	140, 141	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → ((𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴)) ↔ (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴))))
143	142	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → ((𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴))) ↔ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴)))))
144	143, 139	chvarvv 2002	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴)))
145	139, 144	anim12d 609	. . . . . . . 8 ⊢ (𝜑 → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))))
146	145	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))) → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))))
147	1	rexrd 11023	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
148	2	rexrd 11023	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
149		lbicc2 13194	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
150	147, 148, 4, 149	syl3anc 1370	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
151	31, 150	ffvelrnd 6964	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
152	151	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) ∈ ℝ)
153	87, 152	letri3d 11115	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑦) = (𝐹‘𝐴) ↔ ((𝐹‘𝑦) ≤ (𝐹‘𝐴) ∧ (𝐹‘𝐴) ≤ (𝐹‘𝑦))))
154		breq2 5080	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑢) = (𝐹‘𝐴) → ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ (𝐹‘𝑦) ≤ (𝐹‘𝐴)))
155		breq1 5079	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑣) = (𝐹‘𝐴) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝐴) ≤ (𝐹‘𝑦)))
156	154, 155	bi2anan9 636	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴)) → (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) ↔ ((𝐹‘𝑦) ≤ (𝐹‘𝐴) ∧ (𝐹‘𝐴) ≤ (𝐹‘𝑦))))
157	156	bibi2d 343	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴)) → (((𝐹‘𝑦) = (𝐹‘𝐴) ↔ ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))) ↔ ((𝐹‘𝑦) = (𝐹‘𝐴) ↔ ((𝐹‘𝑦) ≤ (𝐹‘𝐴) ∧ (𝐹‘𝐴) ≤ (𝐹‘𝑦)))))
158	153, 157	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴)) → ((𝐹‘𝑦) = (𝐹‘𝐴) ↔ ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)))))
159	158	impancom 452	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))) → (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹‘𝑦) = (𝐹‘𝐴) ↔ ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)))))
160	159	imp 407	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑦) = (𝐹‘𝐴) ↔ ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))))
161	160	ralbidva 3117	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))) → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))))
162	31	ffnd 6603	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
163		fnconstg 6664	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝐴) ∈ ℝ → ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) Fn (𝐴[,]𝐵))
164	151, 163	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) Fn (𝐴[,]𝐵))
165		eqfnfv 6911	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn (𝐴[,]𝐵) ∧ ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) Fn (𝐴[,]𝐵)) → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦)))
166	162, 164, 165	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦)))
167		fvex 6789	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝐴) ∈ V
168	167	fvconst2 7081	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) = (𝐹‘𝐴))
169	168	eqeq2d 2749	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝐴)))
170	169	ralbiia 3091	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴))
171	166, 170	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴)))
172		ioon0 13103	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
173	147, 148, 172	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
174	3, 173	mpbird 256	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅)
175		fconstmpt 5651	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))
176	175	eqeq2i 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴)))
177	176	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴)))
178	177	oveq2d 7293	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))))
179	151	recnd 11001	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
180	179	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → (𝐹‘𝐴) ∈ ℂ)
181		0cnd 10966	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → 0 ∈ ℂ)
182	60, 179	dvmptc 25120	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℝ D (𝑢 ∈ ℝ ↦ (𝐹‘𝐴))) = (𝑢 ∈ ℝ ↦ 0))
183	60, 180, 181, 182, 49, 55, 54, 57	dvmptres2 25124	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
184	178, 183	sylan9eqr 2800	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
185	184	fveq1d 6778	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ((ℝ D 𝐹)‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥))
186		eqidd 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → 0 = 0)
187		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ 0) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0)
188		c0ex 10967	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
189	186, 187, 188	fvmpt 6877	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥) = 0)
190	185, 189	sylan9eq 2798	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) = 0)
191	190	ralrimiva 3103	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
192		r19.2z 4427	. . . . . . . . . . . . 13 ⊢ (((𝐴(,)𝐵) ≠ ∅ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
193	174, 191, 192	syl2an2r 682	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
194	193	ex 413	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
195	171, 194	sylbird 259	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
196	195	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))) → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
197	161, 196	sylbird 259	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))) → (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
198	197	impancom 452	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))) → (((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
199	146, 198	syld 47	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))) → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
200	26, 129, 199	ecased 1032	. . . . 5 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦))) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
201	200	ex 413	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
202	9, 201	syl5bir 242	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → ((∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
203	202	rexlimdvva 3222	. 2 ⊢ (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∃𝑣 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
204	8, 203	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 Vcvv 3431 ⊆ wss 3888 ∅c0 4258 {csn 4563 {cpr 4565 class class class wbr 5076 ↦ cmpt 5159 × cxp 5589 dom cdm 5591 ran crn 5592 Fn wfn 6430 ⟶wf 6431 ‘cfv 6435 (class class class)co 7277 ℂcc 10867 ℝcr 10868 0cc0 10869 ℝ^*cxr 11006 < clt 11007 ≤ cle 11008 -cneg 11204 (,)cioo 13077 [,]cicc 13080 TopOpenctopn 17130 topGenctg 17146 ℂ_fldccnfld 20595 intcnt 22166 –cn→ccncf 24037 D cdv 25025

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 ax-addf 10948 ax-mulf 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7976 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-2o 8296 df-er 8496 df-map 8615 df-pm 8616 df-ixp 8684 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-fsupp 9127 df-fi 9168 df-sup 9199 df-inf 9200 df-oi 9267 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-uz 12581 df-q 12687 df-rp 12729 df-xneg 12846 df-xadd 12847 df-xmul 12848 df-ioo 13081 df-ico 13083 df-icc 13084 df-fz 13238 df-fzo 13381 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-lp 22285 df-perf 22286 df-cn 22376 df-cnp 22377 df-haus 22464 df-cmp 22536 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cncf 24039 df-limc 25028 df-dv 25029

This theorem is referenced by: cmvth 25153 lhop1lem 25175

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