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Theorem rolle 26028

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 11409	. . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
5		rolle.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
6	1, 2, 4, 5	evthicc 25494	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)))
7		reeanv 3229	. . 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 3111	. . . 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 726	. . . . . . . 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 726	. . . . . . . 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 726	. . . . . . . 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 726	. . . . . . . 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 726	. . . . . . . 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 3083	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢))
18		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (𝐹‘𝑦) = (𝐹‘𝑡))
19	18	breq1d 5153	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ (𝐹‘𝑡) ≤ (𝐹‘𝑢)))
20	19	cbvralvw 3237	. . . . . . . . . 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 728	. . . . . . . 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 777	. . . . . . . 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 773	. . . . . . . 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 26027	. . . . . . 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 726	. . . . . . . . 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 726	. . . . . . . . 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 726	. . . . . . . . 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 24919	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
31	5, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
32	31	ffvelcdmda 7104	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℝ)
33	32	renegcld 11690	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℝ)
34	33	fmpttd 7135	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ)
35		ax-resscn 11212	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
36		ssid 4006	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
37		cncfss 24925	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
38	35, 36, 37	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
39	38, 5	sselid 3981	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
40		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) = (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))
41	40	negfcncf 24950	. . . . . . . . . . . . 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 24924	. . . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
46	45	ad2antrr 726	. . . . . . . . 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 13469	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
49	1, 2, 48	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
50		fss 6752	. . . . . . . . . . . . . . . . 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	ffvelcdmda 7104	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℂ)
53	52	negcld 11607	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℂ)
54		tgioo4 24826	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
55		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
56		iccntr 24843	. . . . . . . . . . . . . . 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, 54, 55, 57	dvmptntr 26009	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))))
59		reelprrecn 11247	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ {ℝ, ℂ}
60	59	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
61		ioossicc 13473	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
62	61	sseli 3979	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ (𝐴[,]𝐵))
63	62, 52	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑢) ∈ ℂ)
64		fvexd 6921	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑢) ∈ V)
65	31	feqmptd 6977	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢)))
66	65	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))))
67		dvf 25942	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
68	14	feq2d 6722	. . . . . . . . . . . . . . . . 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 6977	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
71	47, 49, 52, 54, 55, 57	dvmptntr 26009	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))))
72	66, 70, 71	3eqtr3rd 2786	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
73	60, 63, 64, 72	dvmptneg 26004	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
74	58, 73	eqtrd 2777	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
75	74	dmeqd 5916	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
76		dmmptg 6262	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑢) ∈ V → dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵))
77		negex 11506	. . . . . . . . . . . . 13 ⊢ -((ℝ D 𝐹)‘𝑢) ∈ V
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑢) ∈ V)
79	76, 78	mprg 3067	. . . . . . . . . . 11 ⊢ dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵)
80	75, 79	eqtrdi 2793	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝐴(,)𝐵))
81	80	ad2antrr 726	. . . . . . . . 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 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))
85	83, 84	ffvelcdmd 7105	. . . . . . . . . . . . . . . 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 7104	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑦) ∈ ℝ)
88	85, 87	lenegd 11842	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ -(𝐹‘𝑦) ≤ -(𝐹‘𝑣)))
89		fveq2 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑦 → (𝐹‘𝑢) = (𝐹‘𝑦))
90	89	negeqd 11502	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑦 → -(𝐹‘𝑢) = -(𝐹‘𝑦))
91		negex 11506	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑦) ∈ V
92	90, 40, 91	fvmpt 7016	. . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑣 → (𝐹‘𝑢) = (𝐹‘𝑣))
95	94	negeqd 11502	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑣 → -(𝐹‘𝑢) = -(𝐹‘𝑣))
96		negex 11506	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑣) ∈ V
97	95, 40, 96	fvmpt 7016	. . . . . . . . . . . . . . . . 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 5156	. . . . . . . . . . . . . . 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 3167	. . . . . . . . . . . 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 6906	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑡))
105	104	breq1d 5153	. . . . . . . . . . . 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 3237	. . . . . . . . . . 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 717	. . . . . . . . 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 778	. . . . . . . . 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 773	. . . . . . . . 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 26027	. . . . . . . 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 6908	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥))
113		fveq2 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → ((ℝ D 𝐹)‘𝑢) = ((ℝ D 𝐹)‘𝑥))
114	113	negeqd 11502	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑥 → -((ℝ D 𝐹)‘𝑢) = -((ℝ D 𝐹)‘𝑥))
115		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))
116		negex 11506	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑥) ∈ V
117	114, 115, 116	fvmpt 7016	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
118	112, 117	sylan9eq 2797	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
119	118	eqeq1d 2739	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
120	14	eleq2d 2827	. . . . . . . . . . . . . 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 7103	. . . . . . . . . . . . 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 11612	. . . . . . . . . . 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 3177	. . . . . . . . 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 726	. . . . . . . 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 3484	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
131	130	elpr 4650	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝐴, 𝐵} ↔ (𝑢 = 𝐴 ∨ 𝑢 = 𝐵))
132		fveq2 6906	. . . . . . . . . . . 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 2743	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝐵) = (𝐹‘𝐴))
136		fveqeq2 6915	. . . . . . . . . . . 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 860	. . . . . . . . . 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 2824	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 ∈ {𝐴, 𝐵} ↔ 𝑣 ∈ {𝐴, 𝐵}))
141		fveqeq2 6915	. . . . . . . . . . . 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 609	. . . . . . . 8 ⊢ (𝜑 → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))))
146	145	ad2antrr 726	. . . . . . 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 11311	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
148	2	rexrd 11311	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
149		lbicc2 13504	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
150	147, 148, 4, 149	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
151	31, 150	ffvelcdmd 7105	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
152	151	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) ∈ ℝ)
153	87, 152	letri3d 11403	. . . . . . . . . . . . 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 5147	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑢) = (𝐹‘𝐴) → ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ (𝐹‘𝑦) ≤ (𝐹‘𝐴)))
155		breq1 5146	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑣) = (𝐹‘𝐴) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝐴) ≤ (𝐹‘𝑦)))
156	154, 155	bi2anan9 638	. . . . . . . . . . . . . 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 3176	. . . . . . . . 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 6737	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
163		fnconstg 6796	. . . . . . . . . . . . . 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 7051	. . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝐴) ∈ V
168	167	fvconst2 7224	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) = (𝐹‘𝐴))
169	168	eqeq2d 2748	. . . . . . . . . . . . 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 13413	. . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅)
175		fconstmpt 5747	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))
176	175	eqeq2i 2750	. . . . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))))
179	151	recnd 11289	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
180	179	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → (𝐹‘𝐴) ∈ ℂ)
181		0cnd 11254	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → 0 ∈ ℂ)
182	60, 179	dvmptc 25996	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℝ D (𝑢 ∈ ℝ ↦ (𝐹‘𝐴))) = (𝑢 ∈ ℝ ↦ 0))
183	60, 180, 181, 182, 49, 54, 55, 57	dvmptres2 26000	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
184	178, 183	sylan9eqr 2799	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
185	184	fveq1d 6908	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ((ℝ D 𝐹)‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥))
186		eqidd 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → 0 = 0)
187		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ 0) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0)
188		c0ex 11255	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
189	186, 187, 188	fvmpt 7016	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥) = 0)
190	185, 189	sylan9eq 2797	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) = 0)
191	190	ralrimiva 3146	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
192		r19.2z 4495	. . . . . . . . . . . . 13 ⊢ (((𝐴(,)𝐵) ≠ ∅ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
193	174, 191, 192	syl2an2r 685	. . . . . . . . . . . 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 726	. . . . . . . . 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 1036	. . . . 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 3213	. 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 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 {csn 4626 {cpr 4628 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 dom cdm 5685 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 -cneg 11493 (,)cioo 13387 [,]cicc 13390 TopOpenctopn 17466 topGenctg 17482 ℂ_fldccnfld 21364 intcnt 23025 –cn→ccncf 24902 D cdv 25898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902

This theorem is referenced by: cmvth 26029 cmvthOLD 26030 lhop1lem 26052

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