rolle - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rolle		Structured version Visualization version GIF version

Theorem rolle 23753

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 10185	. . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
5		rolle.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
6	1, 2, 4, 5	evthicc 23228	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ ∃𝑣 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑣) ≤ (𝐹‘𝑦)))
7		reeanv 3107	. . 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 224	. 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 3064	. . . 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 762	. . . . . . . 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 762	. . . . . . . 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 762	. . . . . . . 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 762	. . . . . . . 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 762	. . . . . . . 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 473	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → (𝐹‘𝑦) ≤ (𝐹‘𝑢))
17	16	ralimi 2952	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢))
18		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → (𝐹‘𝑦) = (𝐹‘𝑡))
19	18	breq1d 4663	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ (𝐹‘𝑡) ≤ (𝐹‘𝑢)))
20	19	cbvralv 3171	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹‘𝑡) ≤ (𝐹‘𝑢))
21	17, 20	sylib 208	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → ∀𝑡 ∈ (𝐴[,]𝐵)(𝐹‘𝑡) ≤ (𝐹‘𝑢))
22	21	ad2antrl 764	. . . . . . . 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 800	. . . . . . . 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 796	. . . . . . . 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 23752	. . . . . . 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 643	. . . . . 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 762	. . . . . . . . 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 762	. . . . . . . . 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 762	. . . . . . . . 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 22696	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
31	5, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
32	31	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℝ)
33	32	renegcld 10457	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℝ)
34		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) = (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))
35	33, 34	fmptd 6385	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ)
36		ax-resscn 9993	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
37		ssid 3624	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
38		cncfss 22702	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
39	36, 37, 38	mp2an 708	. . . . . . . . . . . . . 14 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
40	39, 5	sseldi 3601	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
41	34	negfcncf 22722	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
42	40, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
43		cncffvrn 22701	. . . . . . . . . . . 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	36, 42, 43	sylancr 695	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ) ↔ (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)):(𝐴[,]𝐵)⟶ℝ))
45	35, 44	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
46	45	ad2antrr 762	. . . . . . . . 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	36	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ⊆ ℂ)
48		iccssre 12255	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
49	1, 2, 48	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
50		fss 6056	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
51	31, 36, 50	sylancl 694	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
52	51	ffvelrnda 6359	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑢) ∈ ℂ)
53	52	negcld 10379	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → -(𝐹‘𝑢) ∈ ℂ)
54		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
55	54	tgioo2 22606	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
56		iccntr 22624	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
57	1, 2, 56	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
58	47, 49, 53, 55, 54, 57	dvmptntr 23734	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))))
59		reelprrecn 10028	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ {ℝ, ℂ}
60	59	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
61		ioossicc 12259	. . . . . . . . . . . . . . . 16 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
62	61	sseli 3599	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ (𝐴[,]𝐵))
63	62, 52	sylan2 491	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑢) ∈ ℂ)
64		fvexd 6203	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑢) ∈ V)
65	31	feqmptd 6249	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢)))
66	65	oveq2d 6666	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))))
67		dvf 23671	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
68	14	feq2d 6031	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
69	67, 68	mpbii 223	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
70	69	feqmptd 6249	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
71	47, 49, 52, 55, 54, 57	dvmptntr 23734	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑢))) = (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))))
72	66, 70, 71	3eqtr3rd 2665	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑢)))
73	60, 63, 64, 72	dvmptneg 23729	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
74	58, 73	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
75	74	dmeqd 5326	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)))
76		dmmptg 5632	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑢) ∈ V → dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵))
77		negex 10279	. . . . . . . . . . . . 13 ⊢ -((ℝ D 𝐹)‘𝑢) ∈ V
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑢) ∈ V)
79	76, 78	mprg 2926	. . . . . . . . . . 11 ⊢ dom (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝐴(,)𝐵)
80	75, 79	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))) = (𝐴(,)𝐵))
81	80	ad2antrr 762	. . . . . . . . 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 477	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑦) ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑣) ≤ (𝐹‘𝑦)) → (𝐹‘𝑣) ≤ (𝐹‘𝑦))
83	31	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
84		simplrr 801	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑣 ∈ (𝐴[,]𝐵))
85	83, 84	ffvelrnd 6360	. . . . . . . . . . . . . . . 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 6359	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑦) ∈ ℝ)
88	85, 87	lenegd 10606	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ -(𝐹‘𝑦) ≤ -(𝐹‘𝑣)))
89		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑦 → (𝐹‘𝑢) = (𝐹‘𝑦))
90	89	negeqd 10275	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑦 → -(𝐹‘𝑢) = -(𝐹‘𝑦))
91		negex 10279	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑦) ∈ V
92	90, 34, 91	fvmpt 6282	. . . . . . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑣 → (𝐹‘𝑢) = (𝐹‘𝑣))
95	94	negeqd 10275	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑣 → -(𝐹‘𝑢) = -(𝐹‘𝑣))
96		negex 10279	. . . . . . . . . . . . . . . . . 18 ⊢ -(𝐹‘𝑣) ∈ V
97	95, 34, 96	fvmpt 6282	. . . . . . . . . . . . . . . . 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 4666	. . . . . . . . . . . . . . 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 271	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 2962	. . . . . . . . . . . 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 445	. . . . . . . . . . 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 6191	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑦) = ((𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢))‘𝑡))
105	104	breq1d 4663	. . . . . . . . . . . 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	cbvralv 3171	. . . . . . . . . . 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 208	. . . . . . . . . 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 753	. . . . . . . . 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 801	. . . . . . . . 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 796	. . . . . . . . 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 23752	. . . . . . . 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 6193	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥))
113		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → ((ℝ D 𝐹)‘𝑢) = ((ℝ D 𝐹)‘𝑥))
114	113	negeqd 10275	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑥 → -((ℝ D 𝐹)‘𝑢) = -((ℝ D 𝐹)‘𝑥))
115		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢)) = (𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))
116		negex 10279	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑥) ∈ V
117	114, 115, 116	fvmpt 6282	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑢))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
118	112, 117	sylan9eq 2676	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
119	118	eqeq1d 2624	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
120	14	eleq2d 2687	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ dom (ℝ D 𝐹) ↔ 𝑥 ∈ (𝐴(,)𝐵)))
121	120	biimpar 502	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ dom (ℝ D 𝐹))
122	67	ffvelrni 6358	. . . . . . . . . . . . 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 10384	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) = 0 ↔ -((ℝ D 𝐹)‘𝑥) = 0))
125	119, 124	bitr4d 271	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ -(𝐹‘𝑢)))‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑥) = 0))
126	125	rexbidva 3049	. . . . . . . . 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 762	. . . . . . . 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 222	. . . . . . 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 643	. . . . . 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 3203	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
131	130	elpr 4198	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝐴, 𝐵} ↔ (𝑢 = 𝐴 ∨ 𝑢 = 𝐵))
132		fveq2 6191	. . . . . . . . . . . 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 2628	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝐵) = (𝐹‘𝐴))
136		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝐵 → (𝐹‘𝑢) = (𝐹‘𝐵))
137	136	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → ((𝐹‘𝑢) = (𝐹‘𝐴) ↔ (𝐹‘𝐵) = (𝐹‘𝐴)))
138	135, 137	syl5ibrcom 237	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 = 𝐵 → (𝐹‘𝑢) = (𝐹‘𝐴)))
139	133, 138	jaod 395	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 = 𝐴 ∨ 𝑢 = 𝐵) → (𝐹‘𝑢) = (𝐹‘𝐴)))
140	131, 139	syl5bi 232	. . . . . . . . 9 ⊢ (𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴)))
141		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 ∈ {𝐴, 𝐵} ↔ 𝑣 ∈ {𝐴, 𝐵}))
142	94	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → ((𝐹‘𝑢) = (𝐹‘𝐴) ↔ (𝐹‘𝑣) = (𝐹‘𝐴)))
143	141, 142	imbi12d 334	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → ((𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴)) ↔ (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴))))
144	143	imbi2d 330	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → ((𝜑 → (𝑢 ∈ {𝐴, 𝐵} → (𝐹‘𝑢) = (𝐹‘𝐴))) ↔ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴)))))
145	144, 140	chvarv 2263	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝐴, 𝐵} → (𝐹‘𝑣) = (𝐹‘𝐴)))
146	140, 145	anim12d 586	. . . . . . . 8 ⊢ (𝜑 → ((𝑢 ∈ {𝐴, 𝐵} ∧ 𝑣 ∈ {𝐴, 𝐵}) → ((𝐹‘𝑢) = (𝐹‘𝐴) ∧ (𝐹‘𝑣) = (𝐹‘𝐴))))
147	146	ad2antrr 762	. . . . . . 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))))
148	1	rexrd 10089	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
149	2	rexrd 10089	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
150		lbicc2 12288	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
151	148, 149, 4, 150	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
152	31, 151	ffvelrnd 6360	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
153	152	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) ∈ ℝ)
154	87, 153	letri3d 10179	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑣 ∈ (𝐴[,]𝐵))) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑦) = (𝐹‘𝐴) ↔ ((𝐹‘𝑦) ≤ (𝐹‘𝐴) ∧ (𝐹‘𝐴) ≤ (𝐹‘𝑦))))
155		breq2 4657	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑢) = (𝐹‘𝐴) → ((𝐹‘𝑦) ≤ (𝐹‘𝑢) ↔ (𝐹‘𝑦) ≤ (𝐹‘𝐴)))
156		breq1 4656	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑣) = (𝐹‘𝐴) → ((𝐹‘𝑣) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝐴) ≤ (𝐹‘𝑦)))
157	155, 156	bi2anan9 917	. . . . . . . . . . . . . 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))))
158	157	bibi2d 332	. . . . . . . . . . . . 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)))))
159	154, 158	syl5ibrcom 237	. . . . . . . . . . . 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)))))
160	159	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)))))
161	160	imp 445	. . . . . . . . . 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))))
162	161	ralbidva 2985	. . . . . . . . 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))))
163		ffn 6045	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
164	31, 163	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
165		fnconstg 6093	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝐴) ∈ ℝ → ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) Fn (𝐴[,]𝐵))
166	152, 165	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) Fn (𝐴[,]𝐵))
167		eqfnfv 6311	. . . . . . . . . . . . 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)))
168	164, 166, 167	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦)))
169		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝐴) ∈ V
170	169	fvconst2 6469	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) = (𝐹‘𝐴))
171	170	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → ((𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝐴)))
172	171	ralbiia 2979	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑦) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴))
173	168, 172	syl6bb 276	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴)))
174		fconstmpt 5163	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))
175	174	eqeq2i 2634	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) ↔ 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴)))
176	175	biimpi 206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → 𝐹 = (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴)))
177	176	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → (ℝ D 𝐹) = (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))))
178	152	recnd 10068	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
179	178	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → (𝐹‘𝐴) ∈ ℂ)
180		0cnd 10033	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ℝ) → 0 ∈ ℂ)
181	60, 178	dvmptc 23721	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℝ D (𝑢 ∈ ℝ ↦ (𝐹‘𝐴))) = (𝑢 ∈ ℝ ↦ 0))
182	60, 179, 180, 181, 49, 55, 54, 57	dvmptres2 23725	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℝ D (𝑢 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝐴))) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
183	177, 182	sylan9eqr 2678	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → (ℝ D 𝐹) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0))
184	183	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ((ℝ D 𝐹)‘𝑥) = ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥))
185		eqidd 2623	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → 0 = 0)
186		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝐴(,)𝐵) ↦ 0) = (𝑢 ∈ (𝐴(,)𝐵) ↦ 0)
187		c0ex 10034	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
188	185, 186, 187	fvmpt 6282	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑢 ∈ (𝐴(,)𝐵) ↦ 0)‘𝑥) = 0)
189	184, 188	sylan9eq 2676	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) = 0)
190	189	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
191		ioon0 12201	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
192	148, 149, 191	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))
193	3, 192	mpbird 247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅)
194		r19.2z 4060	. . . . . . . . . . . . . 14 ⊢ (((𝐴(,)𝐵) ≠ ∅ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
195	193, 194	sylan 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
196	190, 195	syldan 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
197	196	ex 450	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
198	173, 197	sylbird 250	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) = (𝐹‘𝐴) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0))
199	198	ad2antrr 762	. . . . . . . . 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))
200	162, 199	sylbird 250	. . . . . . . 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))
201	200	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))
202	147, 201	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))
203	26, 129, 202	ecased 985	. . . . 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)
204	203	ex 450	. . . 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))
205	9, 204	syl5bir 233	. . 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))
206	205	rexlimdvva 3038	. 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))
207	8, 206	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 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 {csn 4177 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ran crn 5115 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 -cneg 10267 (,)cioo 12175 [,]cicc 12178 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 intcnt 20821 –cn→ccncf 22679 D cdv 23627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: cmvth 23754 lhop1lem 23776

W3C validator