poimir - Mathbox for Brendan Leahy

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Theorem poimir 33442

Description: Poincare-Miranda theorem. Theorem on [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimir.i	⊢ 𝐼 = ((0[,]1) ↑_𝑚 (1...𝑁))
poimir.r	⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
poimir.1	⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
poimir.2	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0)
poimir.3	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛))

**Assertion**
Ref	Expression
poimir	⊢ (𝜑 → ∃𝑐 ∈ 𝐼 (𝐹‘𝑐) = ((1...𝑁) × {0}))

Distinct variable groups: 𝑧,𝑛,𝜑 𝑛,𝐹 𝑛,𝑁 𝜑,𝑧 𝑧,𝐹 𝑧,𝑁 𝑛,𝑐,𝑧,𝜑 𝐹,𝑐 𝐼,𝑐,𝑛,𝑧 𝑁,𝑐 𝑅,𝑐,𝑛,𝑧

Proof of Theorem poimir Dummy variables 𝑥 𝑟 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.0	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		poimir.i	. . 3 ⊢ 𝐼 = ((0[,]1) ↑_𝑚 (1...𝑁))
3		poimir.r	. . 3 ⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
4		poimir.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
5		poimir.2	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0)
6		poimir.3	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛))
7	1, 2, 3, 4, 5, 6	poimirlem32 33441	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
8		ovex 6678	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1...𝑁) ∈ V
9		retopon 22567	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
10	3	pttoponconst 21400	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1...𝑁) ∈ V ∧ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) → 𝑅 ∈ (TopOn‘(ℝ ↑_𝑚 (1...𝑁))))
11	8, 9, 10	mp2an 708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅 ∈ (TopOn‘(ℝ ↑_𝑚 (1...𝑁)))
12	11	topontopi 20720	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑅 ∈ Top
13		reex 10027	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℝ ∈ V
14		unitssre 12319	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]1) ⊆ ℝ
15		mapss 7900	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_𝑚 (1...𝑁)) ⊆ (ℝ ↑_𝑚 (1...𝑁)))
16	13, 14, 15	mp2an 708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0[,]1) ↑_𝑚 (1...𝑁)) ⊆ (ℝ ↑_𝑚 (1...𝑁))
17	2, 16	eqsstri 3635	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐼 ⊆ (ℝ ↑_𝑚 (1...𝑁))
18	11	toponunii 20721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑_𝑚 (1...𝑁)) = ∪ 𝑅
19	18	restuni 20966	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑_𝑚 (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾_t 𝐼))
20	12, 17, 19	mp2an 708	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
21	20, 18	cnf 21050	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅) → 𝐹:𝐼⟶(ℝ ↑_𝑚 (1...𝑁)))
22	4, 21	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝐼⟶(ℝ ↑_𝑚 (1...𝑁)))
23	22	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) ∈ (ℝ ↑_𝑚 (1...𝑁)))
24		elmapi 7879	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑐) ∈ (ℝ ↑_𝑚 (1...𝑁)) → (𝐹‘𝑐):(1...𝑁)⟶ℝ)
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐):(1...𝑁)⟶ℝ)
26	25	ffvelrnda 6359	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) ∈ ℝ)
27		recn 10026	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((𝐹‘𝑐)‘𝑛) ∈ ℂ)
28		absrpcl 14028	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℂ ∧ ((𝐹‘𝑐)‘𝑛) ≠ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺)
29	28	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℂ → (((𝐹‘𝑐)‘𝑛) ≠ 0 → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺))
30	27, 29	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) ≠ 0 → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺))
31		ltsubrp 11866	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛))
32		ltaddrp 11867	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))
33	31, 32	jca 554	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛) ∧ ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))))
34	33	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺ → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛) ∧ ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
35	30, 34	syld 47	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) ≠ 0 → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛) ∧ ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
36	27	abscld 14175	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ)
37		resubcl 10345	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
38	36, 37	mpdan 702	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
39	38	rexrd 10089	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
40		readdcl 10019	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
41	36, 40	mpdan 702	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
42	41	rexrd 10089	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
43		rexr 10085	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((𝐹‘𝑐)‘𝑛) ∈ ℝ^*)
44		elioo5 12231	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^* ∧ (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^* ∧ ((𝐹‘𝑐)‘𝑛) ∈ ℝ^*) → (((𝐹‘𝑐)‘𝑛) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ↔ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛) ∧ ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
45	39, 42, 43, 44	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ↔ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛) ∧ ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
46	35, 45	sylibrd 249	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) ≠ 0 → ((𝐹‘𝑐)‘𝑛) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
47	26, 46	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≠ 0 → ((𝐹‘𝑐)‘𝑛) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
48		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑐 → (𝐹‘𝑥) = (𝐹‘𝑐))
49	48	fveq1d 6193	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑐 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑐)‘𝑛))
50		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
51		fvex 6201	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑐)‘𝑛) ∈ V
52	49, 50, 51	fvmpt 6282	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) = ((𝐹‘𝑐)‘𝑛))
53	52	eleq1d 2686	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝐼 → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ↔ ((𝐹‘𝑐)‘𝑛) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
54	53	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ↔ ((𝐹‘𝑐)‘𝑛) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
55	47, 54	sylibrd 249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≠ 0 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
56		iooretop 22569	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∈ (topGen‘ran (,))
57		resttopon 20965	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ (TopOn‘(ℝ ↑_𝑚 (1...𝑁))) ∧ 𝐼 ⊆ (ℝ ↑_𝑚 (1...𝑁))) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
58	11, 17, 57	mp2an 708	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼)
59	58	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
60	22	feqmptd 6249	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
61	60, 4	eqeltrrd 2702	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
62	61	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
63	11	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑅 ∈ (TopOn‘(ℝ ↑_𝑚 (1...𝑁))))
64		retop 22565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ Top
65	64	fconst6 6095	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
66	18, 3	ptpjcn 21414	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1...𝑁) ∈ V ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top ∧ 𝑛 ∈ (1...𝑁)) → (𝑧 ∈ (ℝ ↑_𝑚 (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
67	8, 65, 66	mp3an12 1414	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑧 ∈ (ℝ ↑_𝑚 (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
68		fvex 6201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ V
69	68	fvconst2 6469	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
70	69	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) = (𝑅 Cn (topGen‘ran (,))))
71	67, 70	eleqtrd 2703	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → (𝑧 ∈ (ℝ ↑_𝑚 (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (topGen‘ran (,))))
72	71	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑧 ∈ (ℝ ↑_𝑚 (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (topGen‘ran (,))))
73		fveq1 6190	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧‘𝑛) = ((𝐹‘𝑥)‘𝑛))
74	59, 62, 63, 72, 73	cnmpt11 21466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (topGen‘ran (,))))
75	20	cncnpi 21082	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (topGen‘ran (,))) ∧ 𝑐 ∈ 𝐼) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐))
76	74, 75	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑐 ∈ 𝐼) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐))
77	76	an32s 846	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐))
78		iscnp 21041	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼) ∧ (topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝑐 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐) ↔ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)):𝐼⟶ℝ ∧ ∀𝑧 ∈ (topGen‘ran (,))(((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ 𝑧 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧)))))
79	58, 9, 78	mp3an12 1414	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐) ↔ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)):𝐼⟶ℝ ∧ ∀𝑧 ∈ (topGen‘ran (,))(((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ 𝑧 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧)))))
80	79	ad2antlr 763	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐) ↔ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)):𝐼⟶ℝ ∧ ∀𝑧 ∈ (topGen‘ran (,))(((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ 𝑧 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧)))))
81	77, 80	mpbid 222	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)):𝐼⟶ℝ ∧ ∀𝑧 ∈ (topGen‘ran (,))(((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ 𝑧 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧))))
82	81	simprd 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ∀𝑧 ∈ (topGen‘ran (,))(((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ 𝑧 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧)))
83		eleq2 2690	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ 𝑧 ↔ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
84		sseq2 3627	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧 ↔ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
85	84	anbi2d 740	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ((𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧) ↔ (𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))))))
86	85	rexbidv 3052	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → (∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧) ↔ ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))))))
87	83, 86	imbi12d 334	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ((((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ 𝑧 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧)) ↔ (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))))
88	87	rspcv 3305	. . . . . . . . . . . . 13 ⊢ (((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∈ (topGen‘ran (,)) → (∀𝑧 ∈ (topGen‘ran (,))(((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ 𝑧 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ 𝑧)) → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))))
89	56, 82, 88	mpsyl 68	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))))))
90	55, 89	syld 47	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≠ 0 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))))))
91		0re 10040	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
92		letric 10137	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
93	26, 91, 92	sylancl 694	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
94	90, 93	jctird 567	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≠ 0 → (∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))))
95		r19.41v 3089	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ (𝑅 ↾_t 𝐼)((𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛))) ↔ (∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛))))
96		anass 681	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛))) ↔ (𝑐 ∈ 𝑣 ∧ (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))))
97	96	rexbii 3041	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ (𝑅 ↾_t 𝐼)((𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛))) ↔ ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))))
98	95, 97	bitr3i 266	. . . . . . . . . 10 ⊢ ((∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛))) ↔ ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))))
99	94, 98	syl6ib 241	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≠ 0 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛))))))
100	58	topontopi 20720	. . . . . . . . . . . . 13 ⊢ (𝑅 ↾_t 𝐼) ∈ Top
101	20	eltopss 20712	. . . . . . . . . . . . 13 ⊢ (((𝑅 ↾_t 𝐼) ∈ Top ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → 𝑣 ⊆ 𝐼)
102	100, 101	mpan 706	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) → 𝑣 ⊆ 𝐼)
103		fvex 6201	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥)‘𝑛) ∈ V
104	103, 50	dmmpti 6023	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = 𝐼
105	104	sseq2i 3630	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ⊆ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ↔ 𝑣 ⊆ 𝐼)
106		funmpt 5926	. . . . . . . . . . . . . . . . 17 ⊢ Fun (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
107		funimass4 6247	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∧ 𝑣 ⊆ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))) → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ↔ ∀𝑧 ∈ 𝑣 ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
108	106, 107	mpan 706	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ⊆ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ↔ ∀𝑧 ∈ 𝑣 ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
109	105, 108	sylbir 225	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ⊆ 𝐼 → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ↔ ∀𝑧 ∈ 𝑣 ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
110		ssel2 3598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ⊆ 𝐼 ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐼)
111		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
112	111	fveq1d 6193	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑧)‘𝑛))
113		fvex 6201	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑧)‘𝑛) ∈ V
114	112, 50, 113	fvmpt 6282	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) = ((𝐹‘𝑧)‘𝑛))
115	114	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐼 → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ↔ ((𝐹‘𝑧)‘𝑛) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
116		eliooord 12233	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑧)‘𝑛) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))))
117	115, 116	syl6bi 243	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐼 → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
118	110, 117	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ⊆ 𝐼 ∧ 𝑧 ∈ 𝑣) → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
119	118	ralimdva 2962	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ⊆ 𝐼 → (∀𝑧 ∈ 𝑣 ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) ∈ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
120	109, 119	sylbid 230	. . . . . . . . . . . . . 14 ⊢ (𝑣 ⊆ 𝐼 → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
121	120	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ⊆ 𝐼) → (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))))
122		absnid 14038	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) = -((𝐹‘𝑐)‘𝑛))
123	122	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = (((𝐹‘𝑐)‘𝑛) + -((𝐹‘𝑐)‘𝑛)))
124	27	negidd 10382	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + -((𝐹‘𝑐)‘𝑛)) = 0)
125	124	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + -((𝐹‘𝑐)‘𝑛)) = 0)
126	123, 125	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
127	26, 126	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
128	127	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
129	128	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ↔ ((𝐹‘𝑧)‘𝑛) < 0))
130	22	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ (ℝ ↑_𝑚 (1...𝑁)))
131		elmapi 7879	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹‘𝑧) ∈ (ℝ ↑_𝑚 (1...𝑁)) → (𝐹‘𝑧):(1...𝑁)⟶ℝ)
132	130, 131	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧):(1...𝑁)⟶ℝ)
133	132	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
134	133	an32s 846	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
135		0red 10041	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → 0 ∈ ℝ)
136	134, 135	ltnled 10184	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
137	136	adantllr 755	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
138	137	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
139	129, 138	bitrd 268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
140	139	biimpd 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) → ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
141	110, 140	sylan2 491	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ (𝑣 ⊆ 𝐼 ∧ 𝑧 ∈ 𝑣)) → (((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) → ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
142	141	anassrs 680	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑣 ⊆ 𝐼) ∧ 𝑧 ∈ 𝑣) → (((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) → ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
143	142	adantld 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑣 ⊆ 𝐼) ∧ 𝑧 ∈ 𝑣) → (((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
144	143	ralimdva 2962	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑣 ⊆ 𝐼) → (∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
145	144	an32s 846	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ⊆ 𝐼) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
146	145	impancom 456	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ⊆ 𝐼) ∧ ∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))) → (((𝐹‘𝑐)‘𝑛) ≤ 0 → ∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
147		absid 14036	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (abs‘((𝐹‘𝑐)‘𝑛)) = ((𝐹‘𝑐)‘𝑛))
148	147	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = (((𝐹‘𝑐)‘𝑛) − ((𝐹‘𝑐)‘𝑛)))
149	27	subidd 10380	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0)
150	149	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0)
151	148, 150	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
152	26, 151	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
153	152	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
154	153	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑧 ∈ 𝐼) → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ↔ 0 < ((𝐹‘𝑧)‘𝑛)))
155	135, 134	ltnled 10184	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (0 < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
156	155	adantllr 755	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (0 < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
157	156	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑧 ∈ 𝐼) → (0 < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
158	154, 157	bitrd 268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑧 ∈ 𝐼) → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
159	158	biimpd 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑧 ∈ 𝐼) → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) → ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
160	110, 159	sylan2 491	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ (𝑣 ⊆ 𝐼 ∧ 𝑧 ∈ 𝑣)) → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) → ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
161	160	anassrs 680	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑣 ⊆ 𝐼) ∧ 𝑧 ∈ 𝑣) → ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) → ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
162	161	adantrd 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑣 ⊆ 𝐼) ∧ 𝑧 ∈ 𝑣) → (((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
163	162	ralimdva 2962	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑣 ⊆ 𝐼) → (∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
164	163	an32s 846	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ⊆ 𝐼) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) → ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
165	164	impancom 456	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ⊆ 𝐼) ∧ ∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))) → (0 ≤ ((𝐹‘𝑐)‘𝑛) → ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
166	146, 165	orim12d 883	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ⊆ 𝐼) ∧ ∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))) → ((((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0)))
167	166	expimpd 629	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ⊆ 𝐼) → ((∀𝑧 ∈ 𝑣 ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑧)‘𝑛) ∧ ((𝐹‘𝑧)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛))) → (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0)))
168	121, 167	syland 498	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ⊆ 𝐼) → ((((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛))) → (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0)))
169	102, 168	sylan2 491	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → ((((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛))) → (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0)))
170	169	anim2d 589	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → ((𝑐 ∈ 𝑣 ∧ (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))) → (𝑐 ∈ 𝑣 ∧ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))))
171	170	reximdva 3017	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ (((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) “ 𝑣) ⊆ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∧ (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))) → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))))
172	99, 171	syld 47	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≠ 0 → ∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))))
173		ralnex 2992	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
174	173	rexbii 3041	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
175		letsr 17227	. . . . . . . . . . . . . . 15 ⊢ ≤ ∈ TosetRel
176	175	elexi 3213	. . . . . . . . . . . . . 14 ⊢ ≤ ∈ V
177	176	cnvex 7113	. . . . . . . . . . . . . 14 ⊢ ^◡ ≤ ∈ V
178		breq 4655	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
179	178	notbid 308	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
180	179	ralbidv 2986	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
181		breq 4655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0^◡ ≤ ((𝐹‘𝑧)‘𝑛)))
182		c0ex 10034	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
183	182, 113	brcnv 5305	. . . . . . . . . . . . . . . . 17 ⊢ (0^◡ ≤ ((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0)
184	181, 183	syl6bb 276	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0))
185	184	notbid 308	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ^◡ ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
186	185	ralbidv 2986	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ^◡ ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
187	176, 177, 180, 186	rexpr 4239	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
188		rexnal 2995	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
189	174, 187, 188	3bitr3i 290	. . . . . . . . . . . 12 ⊢ ((∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0) ↔ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
190	189	anbi2i 730	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝑣 ∧ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0)) ↔ (𝑐 ∈ 𝑣 ∧ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
191		annim 441	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝑣 ∧ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) ↔ ¬ (𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
192	190, 191	bitri 264	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝑣 ∧ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0)) ↔ ¬ (𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
193	192	rexbii 3041	. . . . . . . . 9 ⊢ (∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0)) ↔ ∃𝑣 ∈ (𝑅 ↾_t 𝐼) ¬ (𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
194		rexnal 2995	. . . . . . . . 9 ⊢ (∃𝑣 ∈ (𝑅 ↾_t 𝐼) ¬ (𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) ↔ ¬ ∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
195	193, 194	bitri 264	. . . . . . . 8 ⊢ (∃𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 ∧ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0)) ↔ ¬ ∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
196	172, 195	syl6ib 241	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≠ 0 → ¬ ∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
197	196	necon4ad 2813	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) → ((𝐹‘𝑐)‘𝑛) = 0))
198	197	ralimdva 2962	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) → ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0))
199		ffn 6045	. . . . . 6 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶ℝ → (𝐹‘𝑐) Fn (1...𝑁))
200	25, 199	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) Fn (1...𝑁))
201	198, 200	jctild 566	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) → ((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0)))
202		fconstfv 6476	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ ((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0))
203	182	fconst2 6470	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
204	202, 203	bitr3i 266	. . . 4 ⊢ (((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0) ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
205	201, 204	syl6ib 241	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) → (𝐹‘𝑐) = ((1...𝑁) × {0})))
206	205	reximdva 3017	. 2 ⊢ (𝜑 → (∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) → ∃𝑐 ∈ 𝐼 (𝐹‘𝑐) = ((1...𝑁) × {0})))
207	7, 206	mpd 15	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 (𝐹‘𝑐) = ((1...𝑁) × {0}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ⊆ wss 3574 {csn 4177 {cpr 4179 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 ℕcn 11020 ℝ⁺crp 11832 (,)cioo 12175 [,]cicc 12178 ...cfz 12326 abscabs 13974 ↾_t crest 16081 topGenctg 16098 ∏_tcpt 16099 TosetRel ctsr 17199 Topctop 20698 TopOnctopon 20715 Cn ccn 21028 CnP ccnp 21029

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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-disj 4621 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-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 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-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 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-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-rest 16083 df-topgen 16104 df-pt 16105 df-ps 17200 df-tsr 17201 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-lp 20940 df-cn 21031 df-cnp 21032 df-t1 21118 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-hmph 21559 df-ii 22680

This theorem is referenced by: broucube 33443

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