poimir - Mathbox for Brendan Leahy

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Theorem poimir 38227

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 38226	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
8		ovex 7444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1...𝑁) ∈ V
9		retopon 24889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
10	3	pttoponconst 23723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1...𝑁) ∈ V ∧ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) → 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))))
11	8, 9, 10	mp2an 704	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁)))
12	11	topontopi 23041	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑅 ∈ Top
13		reex 11191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℝ ∈ V
14		unitssre 13526	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]1) ⊆ ℝ
15		mapss 8887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁)))
16	13, 14, 15	mp2an 704	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁))
17	2, 16	eqsstri 3989	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
18	11	toponunii 23042	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
19	18	restuni 23288	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾_t 𝐼))
20	12, 17, 19	mp2an 704	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
21	20, 18	cnf 23372	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅) → 𝐹:𝐼⟶(ℝ ↑_m (1...𝑁)))
22	4, 21	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝐼⟶(ℝ ↑_m (1...𝑁)))
23	22	ffvelcdmda 7080	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) ∈ (ℝ ↑_m (1...𝑁)))
24		elmapi 8846	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑐) ∈ (ℝ ↑_m (1...𝑁)) → (𝐹‘𝑐):(1...𝑁)⟶ℝ)
25	23, 24	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐):(1...𝑁)⟶ℝ)
26	25	ffvelcdmda 7080	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) ∈ ℝ)
27		recn 11190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((𝐹‘𝑐)‘𝑛) ∈ ℂ)
28		absrpcl 15339	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℂ ∧ ((𝐹‘𝑐)‘𝑛) ≠ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺)
29	28	ex 417	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℂ → (((𝐹‘𝑐)‘𝑛) ≠ 0 → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺))
30	27, 29	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) ≠ 0 → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺))
31		ltsubrp 13054	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛))
32		ltaddrp 13055	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))
33	31, 32	jca 520	. . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . . 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 15490	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ)
37		resubcl 11522	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
38	36, 37	mpdan 699	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
39	38	rexrd 11259	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
40		readdcl 11183	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
41	36, 40	mpdan 699	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
42	41	rexrd 11259	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
43		rexr 11255	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((𝐹‘𝑐)‘𝑛) ∈ ℝ^*)
44		elioo5 13430	. . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . 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 262	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑐 → (𝐹‘𝑥) = (𝐹‘𝑐))
49	48	fveq1d 6884	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑐 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑐)‘𝑛))
50		eqid 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
51		fvex 6895	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑐)‘𝑛) ∈ V
52	49, 50, 51	fvmpt 6990	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) = ((𝐹‘𝑐)‘𝑛))
53	52	eleq1d 2854	. . . . . . . . . . . . . 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 739	. . . . . . . . . . . . 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 262	. . . . . . . . . . . 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 24891	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∈ (topGen‘ran (,))
57		resttopon 23287	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))) ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
58	11, 17, 57	mp2an 704	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼)
59	58	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
60	22	feqmptd 6950	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
61	60, 4	eqeltrrd 2870	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
62	61	adantr 485	. . . . . . . . . . . . . . . . . 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 24887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ Top
65	64	fconst6 6769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
66	18, 3	ptpjcn 23737	. . . . . . . . . . . . . . . . . . . . 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 1477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
68		fvex 6895	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ V
69	68	fvconst2 7203	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
70	69	oveq2d 7427	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) = (𝑅 Cn (topGen‘ran (,))))
71	67, 70	eleqtrd 2871	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (topGen‘ran (,))))
72	71	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (topGen‘ran (,))))
73		fveq1 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧‘𝑛) = ((𝐹‘𝑥)‘𝑛))
74	59, 62, 63, 72, 73	cnmpt11 23789	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (topGen‘ran (,))))
75	20	cncnpi 23404	. . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑐 ∈ 𝐼) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐))
77	76	an32s 664	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐))
78		iscnp 23363	. . . . . . . . . . . . . . . . 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 1477	. . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 500	. . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . . . . 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 641	. . . . . . . . . . . . . . . 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 3195	. . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . 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 69	. . . . . . . . . . . 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 48	. . . . . . . . . . 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 11210	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
92		letric 11310	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
93	26, 91, 92	sylancl 597	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
94	90, 93	jctird 535	. . . . . . . . . 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 3201	. . . . . . . . . . 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 473	. . . . . . . . . . . 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 3118	. . . . . . . . . . 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 280	. . . . . . . . . 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	imbitrdi 254	. . . . . . . . 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 23041	. . . . . . . . . . . . 13 ⊢ (𝑅 ↾_t 𝐼) ∈ Top
101	20	eltopss 23033	. . . . . . . . . . . . 13 ⊢ (((𝑅 ↾_t 𝐼) ∈ Top ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → 𝑣 ⊆ 𝐼)
102	100, 101	mpan 702	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) → 𝑣 ⊆ 𝐼)
103		fvex 6895	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥)‘𝑛) ∈ V
104	103, 50	dmmpti 6680	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = 𝐼
105	104	sseq2i 3972	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ⊆ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ↔ 𝑣 ⊆ 𝐼)
106		funmpt 6575	. . . . . . . . . . . . . . . . 17 ⊢ Fun (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
107		funimass4 6946	. . . . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . . . . 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 238	. . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ⊆ 𝐼 ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐼)
111		fveq2 6882	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
112	111	fveq1d 6884	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑧)‘𝑛))
113		fvex 6895	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑧)‘𝑛) ∈ V
114	112, 50, 113	fvmpt 6990	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) = ((𝐹‘𝑧)‘𝑛))
115	114	eleq1d 2854	. . . . . . . . . . . . . . . . . 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 13432	. . . . . . . . . . . . . . . . . 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	biimtrdi 256	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 3183	. . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 15349	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) = -((𝐹‘𝑐)‘𝑛))
123	122	oveq2d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11559	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + -((𝐹‘𝑐)‘𝑛)) = 0)
125	124	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + -((𝐹‘𝑐)‘𝑛)) = 0)
126	123, 125	eqtrd 2804	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
127	26, 126	sylan 591	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
128	127	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
129	128	breq2d 5123	. . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7080	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ (ℝ ↑_m (1...𝑁)))
131		elmapi 8846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹‘𝑧) ∈ (ℝ ↑_m (1...𝑁)) → (𝐹‘𝑧):(1...𝑁)⟶ℝ)
132	130, 131	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧):(1...𝑁)⟶ℝ)
133	132	ffvelcdmda 7080	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
134	133	an32s 664	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
135		0red 11211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → 0 ∈ ℝ)
136	134, 135	ltnled 11357	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
137	136	adantllr 731	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
138	137	adantlr 727	. . . . . . . . . . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . 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 3183	. . . . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . . . . 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 15347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (abs‘((𝐹‘𝑐)‘𝑛)) = ((𝐹‘𝑐)‘𝑛))
148	147	oveq2d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11557	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0)
150	149	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0)
151	148, 150	eqtrd 2804	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
152	26, 151	sylan 591	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
153	152	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
154	153	breq1d 5121	. . . . . . . . . . . . . . . . . . . . . . 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 11357	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (0 < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
156	155	adantllr 731	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (0 < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
157	156	adantlr 727	. . . . . . . . . . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . 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 3183	. . . . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . . . . 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 979	. . . . . . . . . . . . . 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 458	. . . . . . . . . . . . 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 614	. . . . . . . . . . . 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 604	. . . . . . . . . . 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 623	. . . . . . . . . 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 3184	. . . . . . . . 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 48	. . . . . . . 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 3097	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
174	173	rexbii 3118	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
175		letsr 18649	. . . . . . . . . . . . . . 15 ⊢ ≤ ∈ TosetRel
176	175	elexi 3483	. . . . . . . . . . . . . 14 ⊢ ≤ ∈ V
177	176	cnvex 7922	. . . . . . . . . . . . . 14 ⊢ ^◡ ≤ ∈ V
178		breq 5113	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
179	178	notbid 321	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
180	179	ralbidv 3194	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
181		breq 5113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0^◡ ≤ ((𝐹‘𝑧)‘𝑛)))
182		c0ex 11200	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
183	182, 113	brcnv 5869	. . . . . . . . . . . . . . . . 17 ⊢ (0^◡ ≤ ((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0)
184	181, 183	bitrdi 290	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0))
185	184	notbid 321	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ^◡ ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
186	185	ralbidv 3194	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ^◡ ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
187	176, 177, 180, 186	rexpr 4670	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
188		rexnal 3123	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
189	174, 187, 188	3bitr3i 304	. . . . . . . . . . . 12 ⊢ ((∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0) ↔ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
190	189	anbi2i 634	. . . . . . . . . . 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 408	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝑣 ∧ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) ↔ ¬ (𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
192	190, 191	bitri 278	. . . . . . . . . 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 3118	. . . . . . . . 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 3123	. . . . . . . . 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 278	. . . . . . . 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	imbitrdi 254	. . . . . . 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 2983	. . . . . 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 3183	. . . . 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	25	ffnd 6707	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) Fn (1...𝑁))
200	198, 199	jctild 534	. . . 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)))
201		fconstfv 7211	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ ((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0))
202	182	fconst2 7204	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
203	201, 202	bitr3i 280	. . . 4 ⊢ (((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0) ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
204	200, 203	imbitrdi 254	. . 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})))
205	204	reximdva 3184	. 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})))
206	7, 205	mpd 16	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 (𝐹‘𝑐) = ((1...𝑁) × {0}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 Vcvv 3461 ⊆ wss 3911 {csn 4592 {cpr 4594 ∪ cuni 4874 class class class wbr 5111 ↦ cmpt 5194 × cxp 5660 ^◡ccnv 5661 dom cdm 5662 ran crn 5663 “ cima 5665 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ↑_m cmap 8824 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 ℝ^*cxr 11242 < clt 11243 ≤ cle 11244 − cmin 11441 -cneg 11442 ℕcn 12233 ℝ⁺crp 13016 (,)cioo 13372 [,]cicc 13375 ...cfz 13535 abscabs 15285 ↾_t crest 17473 topGenctg 17490 ∏_tcpt 17491 TosetRel ctsr 18621 Topctop 23019 TopOnctopon 23036 Cn ccn 23350 CnP ccnp 23351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3375 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-omul 8458 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-dju 9887 df-card 9925 df-acn 9928 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-xnn0 12578 df-z 12592 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738 df-dvds 16311 df-rest 17475 df-topgen 17496 df-pt 17497 df-ps 18622 df-tsr 18623 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-top 23020 df-topon 23037 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-lp 23262 df-cn 23353 df-cnp 23354 df-t1 23440 df-haus 23441 df-cmp 23513 df-tx 23688 df-hmeo 23881 df-hmph 23882 df-ii 25005

This theorem is referenced by: broucube 38228

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