poimir - Mathbox for Brendan Leahy

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Theorem poimir 36111

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 36110	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
8		ovex 7390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1...𝑁) ∈ V
9		retopon 24127	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
10	3	pttoponconst 22948	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1...𝑁) ∈ V ∧ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) → 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))))
11	8, 9, 10	mp2an 690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁)))
12	11	topontopi 22264	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑅 ∈ Top
13		reex 11142	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℝ ∈ V
14		unitssre 13416	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]1) ⊆ ℝ
15		mapss 8827	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁)))
16	13, 14, 15	mp2an 690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁))
17	2, 16	eqsstri 3978	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
18	11	toponunii 22265	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
19	18	restuni 22513	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾_t 𝐼))
20	12, 17, 19	mp2an 690	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
21	20, 18	cnf 22597	. . . . . . . . . . . . . . . . . 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	ffvelcdmda 7035	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) ∈ (ℝ ↑_m (1...𝑁)))
24		elmapi 8787	. . . . . . . . . . . . . . . 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	ffvelcdmda 7035	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) ∈ ℝ)
27		recn 11141	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((𝐹‘𝑐)‘𝑛) ∈ ℂ)
28		absrpcl 15173	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℂ ∧ ((𝐹‘𝑐)‘𝑛) ≠ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺)
29	28	ex 413	. . . . . . . . . . . . . . . . 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 12951	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛))
32		ltaddrp 12952	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))
33	31, 32	jca 512	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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 15321	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ)
37		resubcl 11465	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
38	36, 37	mpdan 685	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
39	38	rexrd 11205	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
40		readdcl 11134	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
41	36, 40	mpdan 685	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
42	41	rexrd 11205	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
43		rexr 11201	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((𝐹‘𝑐)‘𝑛) ∈ ℝ^*)
44		elioo5 13321	. . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑐 → (𝐹‘𝑥) = (𝐹‘𝑐))
49	48	fveq1d 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑐 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑐)‘𝑛))
50		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
51		fvex 6855	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑐)‘𝑛) ∈ V
52	49, 50, 51	fvmpt 6948	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) = ((𝐹‘𝑐)‘𝑛))
53	52	eleq1d 2822	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . . 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 258	. . . . . . . . . . . 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 24129	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∈ (topGen‘ran (,))
57		resttopon 22512	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))) ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
58	11, 17, 57	mp2an 690	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼)
59	58	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
60	22	feqmptd 6910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
61	60, 4	eqeltrrd 2839	. . . . . . . . . . . . . . . . . . 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 24125	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ Top
65	64	fconst6 6732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
66	18, 3	ptpjcn 22962	. . . . . . . . . . . . . . . . . . . . 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 1451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
68		fvex 6855	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ V
69	68	fvconst2 7153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
70	69	oveq2d 7373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) = (𝑅 Cn (topGen‘ran (,))))
71	67, 70	eleqtrd 2840	. . . . . . . . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧‘𝑛) = ((𝐹‘𝑥)‘𝑛))
74	59, 62, 63, 72, 73	cnmpt11 23014	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (topGen‘ran (,))))
75	20	cncnpi 22629	. . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑐 ∈ 𝐼) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐))
77	76	an32s 650	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐))
78		iscnp 22588	. . . . . . . . . . . . . . . . 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 1451	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . 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 11157	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
92		letric 11255	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
93	26, 91, 92	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
94	90, 93	jctird 527	. . . . . . . . . 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 3185	. . . . . . . . . . 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 469	. . . . . . . . . . . 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 3097	. . . . . . . . . . 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 276	. . . . . . . . . 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 250	. . . . . . . . 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 22264	. . . . . . . . . . . . 13 ⊢ (𝑅 ↾_t 𝐼) ∈ Top
101	20	eltopss 22256	. . . . . . . . . . . . 13 ⊢ (((𝑅 ↾_t 𝐼) ∈ Top ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → 𝑣 ⊆ 𝐼)
102	100, 101	mpan 688	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) → 𝑣 ⊆ 𝐼)
103		fvex 6855	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥)‘𝑛) ∈ V
104	103, 50	dmmpti 6645	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = 𝐼
105	104	sseq2i 3973	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ⊆ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ↔ 𝑣 ⊆ 𝐼)
106		funmpt 6539	. . . . . . . . . . . . . . . . 17 ⊢ Fun (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
107		funimass4 6907	. . . . . . . . . . . . . . . . 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 688	. . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ⊆ 𝐼 ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐼)
111		fveq2 6842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
112	111	fveq1d 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑧)‘𝑛))
113		fvex 6855	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑧)‘𝑛) ∈ V
114	112, 50, 113	fvmpt 6948	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) = ((𝐹‘𝑧)‘𝑛))
115	114	eleq1d 2822	. . . . . . . . . . . . . . . . . 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 13323	. . . . . . . . . . . . . . . . . 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 252	. . . . . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . 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 15183	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) = -((𝐹‘𝑐)‘𝑛))
123	122	oveq2d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11502	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
127	26, 126	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . . . . . . . . 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 7035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ (ℝ ↑_m (1...𝑁)))
131		elmapi 8787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7035	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
134	133	an32s 650	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
135		0red 11158	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → 0 ∈ ℝ)
136	134, 135	ltnled 11302	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
137	136	adantllr 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
138	137	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . . . . . 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 650	. . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . 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 15181	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (abs‘((𝐹‘𝑐)‘𝑛)) = ((𝐹‘𝑐)‘𝑛))
148	147	oveq2d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11500	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
152	26, 151	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . 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 5115	. . . . . . . . . . . . . . . . . . . . . . 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 11302	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (0 < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
156	155	adantllr 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (0 < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
157	156	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . . . . . 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 650	. . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . 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 963	. . . . . . . . . . . . . 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 454	. . . . . . . . . . . . 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 603	. . . . . . . . . . . 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 593	. . . . . . . . . . 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 612	. . . . . . . . . 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 3165	. . . . . . . . 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 3075	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
174	173	rexbii 3097	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
175		letsr 18482	. . . . . . . . . . . . . . 15 ⊢ ≤ ∈ TosetRel
176	175	elexi 3464	. . . . . . . . . . . . . 14 ⊢ ≤ ∈ V
177	176	cnvex 7862	. . . . . . . . . . . . . 14 ⊢ ^◡ ≤ ∈ V
178		breq 5107	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
179	178	notbid 317	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
180	179	ralbidv 3174	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
181		breq 5107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0^◡ ≤ ((𝐹‘𝑧)‘𝑛)))
182		c0ex 11149	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
183	182, 113	brcnv 5838	. . . . . . . . . . . . . . . . 17 ⊢ (0^◡ ≤ ((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0)
184	181, 183	bitrdi 286	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0))
185	184	notbid 317	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ^◡ ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
186	185	ralbidv 3174	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ^◡ ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
187	176, 177, 180, 186	rexpr 4662	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
188		rexnal 3103	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
189	174, 187, 188	3bitr3i 300	. . . . . . . . . . . 12 ⊢ ((∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0) ↔ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
190	189	anbi2i 623	. . . . . . . . . . 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 404	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝑣 ∧ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) ↔ ¬ (𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
192	190, 191	bitri 274	. . . . . . . . . 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 3097	. . . . . . . . 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 3103	. . . . . . . . 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 274	. . . . . . . 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 250	. . . . . . 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 2962	. . . . . 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 3164	. . . . 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 6669	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) Fn (1...𝑁))
200	198, 199	jctild 526	. . . 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 7162	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ ((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0))
202	182	fconst2 7154	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
203	201, 202	bitr3i 276	. . . 4 ⊢ (((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0) ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
204	200, 203	syl6ib 250	. . 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 3165	. 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 15	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 (𝐹‘𝑐) = ((1...𝑁) × {0}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ⊆ wss 3910 {csn 4586 {cpr 4588 ∪ cuni 4865 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ^◡ccnv 5632 dom cdm 5633 ran crn 5634 “ cima 5636 Fun wfun 6490 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 -cneg 11386 ℕcn 12153 ℝ⁺crp 12915 (,)cioo 13264 [,]cicc 13267 ...cfz 13424 abscabs 15119 ↾_t crest 17302 topGenctg 17319 ∏_tcpt 17320 TosetRel ctsr 18454 Topctop 22242 TopOnctopon 22259 Cn ccn 22575 CnP ccnp 22576

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-dvds 16137 df-rest 17304 df-topgen 17325 df-pt 17326 df-ps 18455 df-tsr 18456 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-top 22243 df-topon 22260 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-lp 22487 df-cn 22578 df-cnp 22579 df-t1 22665 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-hmph 23107 df-ii 24240

This theorem is referenced by: broucube 36112

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