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Theorem rolle 26149

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 11353	. . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
5		rolle.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
6	1, 2, 4, 5	evthicc 25618	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)))
7		reeanv 3237	. . 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 237	. 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 3125	. . . 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 738	. . . . . . . 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 738	. . . . . . . 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 738	. . . . . . . 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 738	. . . . . . . 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 738	. . . . . . . 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 487	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → (𝐹‘𝑦) ≤ (𝐹‘𝑢))
17	16	ralimi 3102	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢))
18		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (𝐹‘𝑦) = (𝐹‘𝑡))
19	18	breq1d 5119	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ (𝐹‘𝑡) ≤ (𝐹‘𝑢)))
20	19	cbvralvw 3243	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹‘𝑡) ≤ (𝐹‘𝑢))
21	17, 20	sylib 221	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹‘𝑡) ≤ (𝐹‘𝑢))
22	21	ad2antrl 740	. . . . . . . 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 788	. . . . . . . 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 784	. . . . . . . 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 26148	. . . . . . 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 461	. . . . . 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 738	. . . . . . . . 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 738	. . . . . . . . 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 738	. . . . . . . . 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 25052	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
31	5, 30	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
32	31	ffvelcdmda 7079	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℝ)
33	32	renegcld 11636	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℝ)
34	33	fmpttd 7110	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ)
35		ax-resscn 11152	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
36		ssid 3959	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
37		cncfss 25058	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
38	35, 36, 37	mp2an 704	. . . . . . . . . . . . . 14 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
39	38, 5	sselid 3935	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
40		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) = (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))
41	40	negfcncf 25082	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
42	39, 41	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
43		cncfcdm 25057	. . . . . . . . . . . 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 598	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ))
45	34, 44	mpbird 260	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
46	45	ad2antrr 738	. . . . . . . . 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 13451	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
49	1, 2, 48	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
50		fss 6722	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
51	31, 35, 50	sylancl 597	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
52	51	ffvelcdmda 7079	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℂ)
53	52	negcld 11551	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℂ)
54		tgioo4 24962	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
55		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
56		iccntr 24979	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
57	1, 2, 56	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
58	47, 49, 53, 54, 55, 57	dvmptntr 26130	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))))
59		reelprrecn 11187	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ {ℝ, ℂ}
60	59	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
61		ioossicc 13455	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
62	61	sseli 3933	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ (𝐴[,]𝐵))
63	62, 52	sylan2 604	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑢) ∈ ℂ)
64		fvexd 6896	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑢) ∈ V)
65	31	feqmptd 6949	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢)))
66	65	oveq2d 7426	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))))
67		dvf 26066	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
68	14	feq2d 6689	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
69	67, 68	mpbii 236	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
70	69	feqmptd 6949	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
71	47, 49, 52, 54, 55, 57	dvmptntr 26130	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))))
72	66, 70, 71	3eqtr3rd 2807	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
73	60, 63, 64, 72	dvmptneg 26125	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
74	58, 73	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
75	74	dmeqd 5895	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
76		dmmptg 6243	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑢) ∈ V → dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵))
77		negex 11450	. . . . . . . . . . . . 13 ⊢ -((ℝ D 𝐹)‘𝑢) ∈ V
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑢) ∈ V)
79	76, 78	mprg 3085	. . . . . . . . . . 11 ⊢ dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵)
80	75, 79	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝐴(,)𝐵))
81	80	ad2antrr 738	. . . . . . . . 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 489	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → (𝐹‘𝑣) ≤ (𝐹‘𝑦))
83	31	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
84		simplrr 789	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑣 ∈ (𝐴[,]𝐵))
85	83, 84	ffvelcdmd 7080	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑣) ∈ ℝ)
86	31	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
87	86	ffvelcdmda 7079	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑦) ∈ ℝ)
88	85, 87	lenegd 11788	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ -(𝐹‘𝑦) ≤ -(𝐹‘𝑣)))
89		fveq2 6881	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑦 → (𝐹‘𝑢) = (𝐹‘𝑦))
90	89	negeqd 11446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑦 → -(𝐹‘𝑢) = -(𝐹‘𝑦))
91		negex 11450	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑦) ∈ V
92	90, 40, 91	fvmpt 6989	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = -(𝐹‘𝑦))
93	92	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = -(𝐹‘𝑦))
94		fveq2 6881	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑣 → (𝐹‘𝑢) = (𝐹‘𝑣))
95	94	negeqd 11446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑣 → -(𝐹‘𝑢) = -(𝐹‘𝑣))
96		negex 11450	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑣) ∈ V
97	95, 40, 96	fvmpt 6989	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣) = -(𝐹‘𝑣))
98	84, 97	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑣) = -(𝐹‘𝑣))
99	93, 98	breq12d 5122	. . . . . . . . . . . . . . 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 285	. . . . . . . . . . . . . 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	imbitrid 247	. . . . . . . . . . . . 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 3177	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 6881	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑡))
105	104	breq1d 5119	. . . . . . . . . . . 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 3243	. . . . . . . . . . 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 221	. . . . . . . . . 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 729	. . . . . . . . 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 789	. . . . . . . . 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 784	. . . . . . . . 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 26148	. . . . . . . 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 6883	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥))
113		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → ((ℝ D 𝐹)‘𝑢) = ((ℝ D 𝐹)‘𝑥))
114	113	negeqd 11446	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑥 → -((ℝ D 𝐹)‘𝑢) = -((ℝ D 𝐹)‘𝑥))
115		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))
116		negex 11450	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑥) ∈ V
117	114, 115, 116	fvmpt 6989	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
118	112, 117	sylan9eq 2818	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
119	118	eqeq1d 2765	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
120	14	eleq2d 2849	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ dom (ℝ D 𝐹) ↔ 𝑥 ∈ (𝐴(,)𝐵)))
121	120	biimpar 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ dom (ℝ D 𝐹))
122	67	ffvelcdmi 7078	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom (ℝ D 𝐹) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
123	121, 122	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
124	123	negeq0d 11556	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
125	119, 124	bitr4d 285	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑥) = 0))
126	125	rexbidva 3187	. . . . . . . . 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 738	. . . . . . . 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 235	. . . . . . 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 461	. . . . . 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 3459	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
131	130	elpr 4614	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝐴, 𝐵} ↔ (𝑢 = 𝐴 ∨ 𝑢 = 𝐵))
132		fveq2 6881	. . . . . . . . . . . 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 2769	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝐵) = (𝐹‘𝐴))
136		fveqeq2 6890	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → ((𝐹‘𝑢) = (𝐹‘𝐴) ↔ (𝐹‘𝐵) = (𝐹‘𝐴)))
137	135, 136	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 = 𝐵 → (𝐹‘𝑢) = (𝐹‘𝐴)))
138	133, 137	jaod 872	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 = 𝐴 ∨ 𝑢 = 𝐵) → (𝐹‘𝑢) = (𝐹‘𝐴)))
139	131, 138	biimtrid 245	. . . . . . . . 9 ⊢ (𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴)))
140		eleq1w 2846	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 ∈ {𝐴, 𝐵} ↔ 𝑣 ∈ {𝐴, 𝐵}))
141		fveqeq2 6890	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → ((𝐹‘𝑢) = (𝐹‘𝐴) ↔ (𝐹‘𝑣) = (𝐹‘𝐴)))
142	140, 141	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → ((𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴)) ↔ (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴))))
143	142	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → ((𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴))) ↔ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴)))))
144	143, 139	chvarvv 2019	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴)))
145	139, 144	anim12d 620	. . . . . . . 8 ⊢ (𝜑 → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))))
146	145	ad2antrr 738	. . . . . . 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 11254	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
148	2	rexrd 11254	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
149		lbicc2 13486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
150	147, 148, 4, 149	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
151	31, 150	ffvelcdmd 7080	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
152	151	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) ∈ ℝ)
153	87, 152	letri3d 11347	. . . . . . . . . . . . 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 5113	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑢) = (𝐹‘𝐴) → ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ (𝐹‘𝑦) ≤ (𝐹‘𝐴)))
155		breq1 5112	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑣) = (𝐹‘𝐴) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝐴) ≤ (𝐹‘𝑦)))
156	154, 155	bi2anan9 649	. . . . . . . . . . . . . 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 345	. . . . . . . . . . . . 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 250	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 411	. . . . . . . . . 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 3186	. . . . . . . . 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 6706	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
163		fnconstg 6766	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝐴) ∈ ℝ → ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) Fn (𝐴[,]𝐵))
164	151, 163	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) Fn (𝐴[,]𝐵))
165		eqfnfv 7025	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦)))
167		fvex 6894	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝐴) ∈ V
168	167	fvconst2 7202	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) = (𝐹‘𝐴))
169	168	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝐴)))
170	169	ralbiia 3109	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴))
171	166, 170	bitrdi 290	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴)))
172		ioon0 13393	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
173	147, 148, 172	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
174	3, 173	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅)
175		fconstmpt 5723	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))
176	175	eqeq2i 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴)))
177	176	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴)))
178	177	oveq2d 7426	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))))
179	151	recnd 11232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
180	179	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → (𝐹‘𝐴) ∈ ℂ)
181		0cnd 11194	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → 0 ∈ ℂ)
182	60, 179	dvmptc 26117	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℝ D (𝑢 ∈ ℝ ↦ (𝐹‘𝐴))) = (𝑢 ∈ ℝ ↦ 0))
183	60, 180, 181, 182, 49, 54, 55, 57	dvmptres2 26121	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
184	178, 183	sylan9eqr 2820	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
185	184	fveq1d 6883	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ((ℝ D 𝐹)‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥))
186		eqidd 2764	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → 0 = 0)
187		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ 0) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0)
188		c0ex 11195	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
189	186, 187, 188	fvmpt 6989	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥) = 0)
190	185, 189	sylan9eq 2818	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) = 0)
191	190	ralrimiva 3157	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
192		r19.2z 4460	. . . . . . . . . . . . 13 ⊢ (((𝐴(,)𝐵) ≠ ∅ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
193	174, 191, 192	syl2an2r 697	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
194	193	ex 417	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
195	171, 194	sylbird 263	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
196	195	ad2antrr 738	. . . . . . . . 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 263	. . . . . . . 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 456	. . . . . . 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 48	. . . . . 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 1051	. . . . 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 417	. . . 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	biimtrrid 246	. . 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 16	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 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ⊆ wss 3905 ∅c0 4286 {csn 4589 {cpr 4591 class class class wbr 5109 ↦ cmpt 5192 × cxp 5659 dom cdm 5661 ran crn 5662 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 -cneg 11437 (,)cioo 13367 [,]cicc 13370 TopOpenctopn 17469 topGenctg 17485 ℂ_fldccnfld 21522 intcnt 23174 –cn→ccncf 25035 D cdv 26022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-haus 23472 df-cmp 23544 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-limc 26025 df-dv 26026

This theorem is referenced by: cmvth 26150 lhop1lem 26172

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