poimir - Mathbox for Brendan Leahy

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Theorem poimir 34919

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 34918	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
8		ovex 7183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1...𝑁) ∈ V
9		retopon 23366	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
10	3	pttoponconst 22199	. . . . . . . . . . . . . . . . . . . . . 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 21517	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑅 ∈ Top
13		reex 10622	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℝ ∈ V
14		unitssre 12879	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]1) ⊆ ℝ
15		mapss 8447	. . . . . . . . . . . . . . . . . . . . . 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 4000	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
18	11	toponunii 21518	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
19	18	restuni 21764	. . . . . . . . . . . . . . . . . . . 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 21848	. . . . . . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) ∈ (ℝ ↑_m (1...𝑁)))
24		elmapi 8422	. . . . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) ∈ ℝ)
27		recn 10621	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((𝐹‘𝑐)‘𝑛) ∈ ℂ)
28		absrpcl 14642	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℂ ∧ ((𝐹‘𝑐)‘𝑛) ≠ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺)
29	28	ex 415	. . . . . . . . . . . . . . . . 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 12419	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛))
32		ltaddrp 12420	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))
33	31, 32	jca 514	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . 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 14790	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ)
37		resubcl 10944	. . . . . . . . . . . . . . . . . 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 10685	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
40		readdcl 10614	. . . . . . . . . . . . . . . . . 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 10685	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
43		rexr 10681	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((𝐹‘𝑐)‘𝑛) ∈ ℝ^*)
44		elioo5 12788	. . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . 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 261	. . . . . . . . . . . . . 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 6664	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑐 → (𝐹‘𝑥) = (𝐹‘𝑐))
49	48	fveq1d 6666	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑐 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑐)‘𝑛))
50		eqid 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
51		fvex 6677	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑐)‘𝑛) ∈ V
52	49, 50, 51	fvmpt 6762	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) = ((𝐹‘𝑐)‘𝑛))
53	52	eleq1d 2897	. . . . . . . . . . . . . 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 261	. . . . . . . . . . . 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 23368	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∈ (topGen‘ran (,))
57		resttopon 21763	. . . . . . . . . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
61	60, 4	eqeltrrd 2914	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
62	61	adantr 483	. . . . . . . . . . . . . . . . . 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 23364	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ Top
65	64	fconst6 6563	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
66	18, 3	ptpjcn 22213	. . . . . . . . . . . . . . . . . . . . 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 1447	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
68		fvex 6677	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ V
69	68	fvconst2 6960	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
70	69	oveq2d 7166	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) = (𝑅 Cn (topGen‘ran (,))))
71	67, 70	eleqtrd 2915	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (topGen‘ran (,))))
72	71	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (topGen‘ran (,))))
73		fveq1 6663	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧‘𝑛) = ((𝐹‘𝑥)‘𝑛))
74	59, 62, 63, 72, 73	cnmpt11 22265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (topGen‘ran (,))))
75	20	cncnpi 21880	. . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . 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 21839	. . . . . . . . . . . . . . . . 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 1447	. . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 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 498	. . . . . . . . . . . . 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 2901	. . . . . . . . . . . . . . 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 3992	. . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . 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 3297	. . . . . . . . . . . . . . 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 3617	. . . . . . . . . . . . 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 10637	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
92		letric 10734	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
93	26, 91, 92	sylancl 588	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
94	90, 93	jctird 529	. . . . . . . . . 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 3347	. . . . . . . . . . 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 471	. . . . . . . . . . . 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 3247	. . . . . . . . . . 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 279	. . . . . . . . . 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 253	. . . . . . . . 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 21517	. . . . . . . . . . . . 13 ⊢ (𝑅 ↾_t 𝐼) ∈ Top
101	20	eltopss 21509	. . . . . . . . . . . . 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 6677	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥)‘𝑛) ∈ V
104	103, 50	dmmpti 6486	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = 𝐼
105	104	sseq2i 3995	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ⊆ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ↔ 𝑣 ⊆ 𝐼)
106		funmpt 6387	. . . . . . . . . . . . . . . . 17 ⊢ Fun (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
107		funimass4 6724	. . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ⊆ 𝐼 ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐼)
111		fveq2 6664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
112	111	fveq1d 6666	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑧)‘𝑛))
113		fvex 6677	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑧)‘𝑛) ∈ V
114	112, 50, 113	fvmpt 6762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) = ((𝐹‘𝑧)‘𝑛))
115	114	eleq1d 2897	. . . . . . . . . . . . . . . . . 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 12790	. . . . . . . . . . . . . . . . . 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 255	. . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 14652	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) = -((𝐹‘𝑐)‘𝑛))
123	122	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 10981	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + -((𝐹‘𝑐)‘𝑛)) = 0)
125	124	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + -((𝐹‘𝑐)‘𝑛)) = 0)
126	123, 125	eqtrd 2856	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
127	26, 126	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
128	127	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
129	128	breq2d 5070	. . . . . . . . . . . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ (ℝ ↑_m (1...𝑁)))
131		elmapi 8422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 10638	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → 0 ∈ ℝ)
136	134, 135	ltnled 10781	. . . . . . . . . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . . . 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 14650	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (abs‘((𝐹‘𝑐)‘𝑛)) = ((𝐹‘𝑐)‘𝑛))
148	147	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 10979	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0)
150	149	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0)
151	148, 150	eqtrd 2856	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
152	26, 151	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
153	152	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
154	153	breq1d 5068	. . . . . . . . . . . . . . . . . . . . . . 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 10781	. . . . . . . . . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . . . 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 961	. . . . . . . . . . . . . 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 456	. . . . . . . . . . . . 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 604	. . . . . . . . . . . 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 594	. . . . . . . . . . 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 613	. . . . . . . . . 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 3274	. . . . . . . . 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 3236	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
174	173	rexbii 3247	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
175		letsr 17831	. . . . . . . . . . . . . . 15 ⊢ ≤ ∈ TosetRel
176	175	elexi 3513	. . . . . . . . . . . . . 14 ⊢ ≤ ∈ V
177	176	cnvex 7624	. . . . . . . . . . . . . 14 ⊢ ^◡ ≤ ∈ V
178		breq 5060	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
179	178	notbid 320	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
180	179	ralbidv 3197	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
181		breq 5060	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0^◡ ≤ ((𝐹‘𝑧)‘𝑛)))
182		c0ex 10629	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
183	182, 113	brcnv 5747	. . . . . . . . . . . . . . . . 17 ⊢ (0^◡ ≤ ((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0)
184	181, 183	syl6bb 289	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0))
185	184	notbid 320	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ^◡ ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
186	185	ralbidv 3197	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ^◡ ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
187	176, 177, 180, 186	rexpr 4630	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
188		rexnal 3238	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
189	174, 187, 188	3bitr3i 303	. . . . . . . . . . . 12 ⊢ ((∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0) ↔ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
190	189	anbi2i 624	. . . . . . . . . . 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 406	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝑣 ∧ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) ↔ ¬ (𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
192	190, 191	bitri 277	. . . . . . . . . 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 3247	. . . . . . . . 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 3238	. . . . . . . . 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 277	. . . . . . . 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 253	. . . . . . 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 3035	. . . . . 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 3177	. . . . 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 6509	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) Fn (1...𝑁))
200	198, 199	jctild 528	. . . 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 6969	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ ((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0))
202	182	fconst2 6961	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
203	201, 202	bitr3i 279	. . . 4 ⊢ (((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0) ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
204	200, 203	syl6ib 253	. . 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 3274	. 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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3935 {csn 4560 {cpr 4562 ∪ cuni 4831 class class class wbr 5058 ↦ cmpt 5138 × cxp 5547 ^◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 Fun wfun 6343 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑_m cmap 8400 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 − cmin 10864 -cneg 10865 ℕcn 11632 ℝ⁺crp 12383 (,)cioo 12732 [,]cicc 12735 ...cfz 12886 abscabs 14587 ↾_t crest 16688 topGenctg 16705 ∏_tcpt 16706 TosetRel ctsr 17803 Topctop 21495 TopOnctopon 21512 Cn ccn 21826 CnP ccnp 21827

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-dvds 15602 df-rest 16690 df-topgen 16711 df-pt 16712 df-ps 17804 df-tsr 17805 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-lp 21738 df-cn 21829 df-cnp 21830 df-t1 21916 df-haus 21917 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-hmph 22358 df-ii 23479

This theorem is referenced by: broucube 34920

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