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Theorem rolle 26048

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 11438	. . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
5		rolle.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
6	1, 2, 4, 5	evthicc 25513	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)))
7		reeanv 3235	. . 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 234	. 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 3117	. . . 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 725	. . . . . . . 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 725	. . . . . . . 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 725	. . . . . . . 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 725	. . . . . . . 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 725	. . . . . . . 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 482	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → (𝐹‘𝑦) ≤ (𝐹‘𝑢))
17	16	ralimi 3089	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢))
18		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (𝐹‘𝑦) = (𝐹‘𝑡))
19	18	breq1d 5176	. . . . . . . . . . 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 218	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹‘𝑡) ≤ (𝐹‘𝑢))
22	21	ad2antrl 727	. . . . . . . 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 776	. . . . . . . 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 772	. . . . . . . 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 26047	. . . . . . 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 456	. . . . . 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 725	. . . . . . . . 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 725	. . . . . . . . 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 725	. . . . . . . . 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 24938	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
31	5, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
32	31	ffvelcdmda 7118	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℝ)
33	32	renegcld 11717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℝ)
34	33	fmpttd 7149	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ)
35		ax-resscn 11241	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
36		ssid 4031	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
37		cncfss 24944	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
38	35, 36, 37	mp2an 691	. . . . . . . . . . . . . 14 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
39	38, 5	sselid 4006	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
40		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) = (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))
41	40	negfcncf 24969	. . . . . . . . . . . . 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		cncfcdm 24943	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ))
45	34, 44	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
46	45	ad2antrr 725	. . . . . . . . 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 13489	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
49	1, 2, 48	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
50		fss 6763	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
51	31, 35, 50	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
52	51	ffvelcdmda 7118	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℂ)
53	52	negcld 11634	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℂ)
54		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
55	54	tgioo2 24844	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
56		iccntr 24862	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
57	1, 2, 56	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
58	47, 49, 53, 55, 54, 57	dvmptntr 26029	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))))
59		reelprrecn 11276	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ {ℝ, ℂ}
60	59	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
61		ioossicc 13493	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
62	61	sseli 4004	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ (𝐴[,]𝐵))
63	62, 52	sylan2 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑢) ∈ ℂ)
64		fvexd 6935	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑢) ∈ V)
65	31	feqmptd 6990	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢)))
66	65	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))))
67		dvf 25962	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
68	14	feq2d 6733	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
69	67, 68	mpbii 233	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
70	69	feqmptd 6990	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
71	47, 49, 52, 55, 54, 57	dvmptntr 26029	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))))
72	66, 70, 71	3eqtr3rd 2789	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
73	60, 63, 64, 72	dvmptneg 26024	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
74	58, 73	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
75	74	dmeqd 5930	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
76		dmmptg 6273	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑢) ∈ V → dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵))
77		negex 11534	. . . . . . . . . . . . 13 ⊢ -((ℝ D 𝐹)‘𝑢) ∈ V
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑢) ∈ V)
79	76, 78	mprg 3073	. . . . . . . . . . 11 ⊢ dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵)
80	75, 79	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝐴(,)𝐵))
81	80	ad2antrr 725	. . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → (𝐹‘𝑣) ≤ (𝐹‘𝑦))
83	31	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
84		simplrr 777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑣 ∈ (𝐴[,]𝐵))
85	83, 84	ffvelcdmd 7119	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑣) ∈ ℝ)
86	31	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
87	86	ffvelcdmda 7118	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑦) ∈ ℝ)
88	85, 87	lenegd 11869	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ -(𝐹‘𝑦) ≤ -(𝐹‘𝑣)))
89		fveq2 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑦 → (𝐹‘𝑢) = (𝐹‘𝑦))
90	89	negeqd 11530	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑦 → -(𝐹‘𝑢) = -(𝐹‘𝑦))
91		negex 11534	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑦) ∈ V
92	90, 40, 91	fvmpt 7029	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = -(𝐹‘𝑦))
93	92	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = -(𝐹‘𝑦))
94		fveq2 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑣 → (𝐹‘𝑢) = (𝐹‘𝑣))
95	94	negeqd 11530	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑣 → -(𝐹‘𝑢) = -(𝐹‘𝑣))
96		negex 11534	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑣) ∈ V
97	95, 40, 96	fvmpt 7029	. . . . . . . . . . . . . . . . 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 5179	. . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . 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 244	. . . . . . . . . . . . 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 3173	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 6920	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑡))
105	104	breq1d 5176	. . . . . . . . . . . 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 218	. . . . . . . . . 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 716	. . . . . . . . 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 777	. . . . . . . . 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 772	. . . . . . . . 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 26047	. . . . . . . 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 6922	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥))
113		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → ((ℝ D 𝐹)‘𝑢) = ((ℝ D 𝐹)‘𝑥))
114	113	negeqd 11530	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑥 → -((ℝ D 𝐹)‘𝑢) = -((ℝ D 𝐹)‘𝑥))
115		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))
116		negex 11534	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑥) ∈ V
117	114, 115, 116	fvmpt 7029	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
118	112, 117	sylan9eq 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
119	118	eqeq1d 2742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
120	14	eleq2d 2830	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ dom (ℝ D 𝐹) ↔ 𝑥 ∈ (𝐴(,)𝐵)))
121	120	biimpar 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ dom (ℝ D 𝐹))
122	67	ffvelcdmi 7117	. . . . . . . . . . . . 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 11639	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
125	119, 124	bitr4d 282	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑥) = 0))
126	125	rexbidva 3183	. . . . . . . . 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 725	. . . . . . . 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 232	. . . . . . 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 456	. . . . . 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 3492	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
131	130	elpr 4672	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝐴, 𝐵} ↔ (𝑢 = 𝐴 ∨ 𝑢 = 𝐵))
132		fveq2 6920	. . . . . . . . . . . 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 2746	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝐵) = (𝐹‘𝐴))
136		fveqeq2 6929	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → ((𝐹‘𝑢) = (𝐹‘𝐴) ↔ (𝐹‘𝐵) = (𝐹‘𝐴)))
137	135, 136	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 = 𝐵 → (𝐹‘𝑢) = (𝐹‘𝐴)))
138	133, 137	jaod 858	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 = 𝐴 ∨ 𝑢 = 𝐵) → (𝐹‘𝑢) = (𝐹‘𝐴)))
139	131, 138	biimtrid 242	. . . . . . . . 9 ⊢ (𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴)))
140		eleq1w 2827	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 ∈ {𝐴, 𝐵} ↔ 𝑣 ∈ {𝐴, 𝐵}))
141		fveqeq2 6929	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → ((𝐹‘𝑢) = (𝐹‘𝐴) ↔ (𝐹‘𝑣) = (𝐹‘𝐴)))
142	140, 141	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → ((𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴)) ↔ (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴))))
143	142	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → ((𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴))) ↔ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴)))))
144	143, 139	chvarvv 1998	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴)))
145	139, 144	anim12d 608	. . . . . . . 8 ⊢ (𝜑 → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))))
146	145	ad2antrr 725	. . . . . . 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 11340	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
148	2	rexrd 11340	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
149		lbicc2 13524	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
150	147, 148, 4, 149	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
151	31, 150	ffvelcdmd 7119	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
152	151	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) ∈ ℝ)
153	87, 152	letri3d 11432	. . . . . . . . . . . . 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 5170	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑢) = (𝐹‘𝐴) → ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ (𝐹‘𝑦) ≤ (𝐹‘𝐴)))
155		breq1 5169	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑣) = (𝐹‘𝐴) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝐴) ≤ (𝐹‘𝑦)))
156	154, 155	bi2anan9 637	. . . . . . . . . . . . . 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 342	. . . . . . . . . . . . 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 247	. . . . . . . . . . . 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 451	. . . . . . . . . . 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 406	. . . . . . . . . 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 3182	. . . . . . . . 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 6748	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
163		fnconstg 6809	. . . . . . . . . . . . . 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 7064	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦)))
167		fvex 6933	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝐴) ∈ V
168	167	fvconst2 7241	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) = (𝐹‘𝐴))
169	168	eqeq2d 2751	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝐴)))
170	169	ralbiia 3097	. . . . . . . . . . . 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 13433	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
173	147, 148, 172	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
174	3, 173	mpbird 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅)
175		fconstmpt 5762	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))
176	175	eqeq2i 2753	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴)))
177	176	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴)))
178	177	oveq2d 7464	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))))
179	151	recnd 11318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
180	179	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → (𝐹‘𝐴) ∈ ℂ)
181		0cnd 11283	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → 0 ∈ ℂ)
182	60, 179	dvmptc 26016	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℝ D (𝑢 ∈ ℝ ↦ (𝐹‘𝐴))) = (𝑢 ∈ ℝ ↦ 0))
183	60, 180, 181, 182, 49, 55, 54, 57	dvmptres2 26020	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
184	178, 183	sylan9eqr 2802	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
185	184	fveq1d 6922	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ((ℝ D 𝐹)‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥))
186		eqidd 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → 0 = 0)
187		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ 0) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0)
188		c0ex 11284	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
189	186, 187, 188	fvmpt 7029	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥) = 0)
190	185, 189	sylan9eq 2800	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) = 0)
191	190	ralrimiva 3152	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
192		r19.2z 4518	. . . . . . . . . . . . 13 ⊢ (((𝐴(,)𝐵) ≠ ∅ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
193	174, 191, 192	syl2an2r 684	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
194	193	ex 412	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
195	171, 194	sylbird 260	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
196	195	ad2antrr 725	. . . . . . . . 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 260	. . . . . . . 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 451	. . . . . . 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 1035	. . . . 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 412	. . . 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 243	. . 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 3219	. 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 206 ∧ wa 395 ∨ wo 846 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ⊆ wss 3976 ∅c0 4352 {csn 4648 {cpr 4650 class class class wbr 5166 ↦ cmpt 5249 × cxp 5698 dom cdm 5700 ran crn 5701 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 -cneg 11521 (,)cioo 13407 [,]cicc 13410 TopOpenctopn 17481 topGenctg 17497 ℂ_fldccnfld 21387 intcnt 23046 –cn→ccncf 24921 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: cmvth 26049 cmvthOLD 26050 lhop1lem 26072

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