poimir - Mathbox for Brendan Leahy

	Mathbox for Brendan Leahy		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > poimir		Structured version Visualization version GIF version

Theorem poimir 37823

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 37822	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾_t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
8		ovex 7391	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1...𝑁) ∈ V
9		retopon 24709	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
10	3	pttoponconst 23543	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1...𝑁) ∈ V ∧ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) → 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))))
11	8, 9, 10	mp2an 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁)))
12	11	topontopi 22861	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑅 ∈ Top
13		reex 11119	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℝ ∈ V
14		unitssre 13417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]1) ⊆ ℝ
15		mapss 8829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁)))
16	13, 14, 15	mp2an 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁))
17	2, 16	eqsstri 3979	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
18	11	toponunii 22862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
19	18	restuni 23108	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾_t 𝐼))
20	12, 17, 19	mp2an 693	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
21	20, 18	cnf 23192	. . . . . . . . . . . . . . . . . 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 7029	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) ∈ (ℝ ↑_m (1...𝑁)))
24		elmapi 8788	. . . . . . . . . . . . . . . 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 7029	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) ∈ ℝ)
27		recn 11118	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → ((𝐹‘𝑐)‘𝑛) ∈ ℂ)
28		absrpcl 15213	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℂ ∧ ((𝐹‘𝑐)‘𝑛) ≠ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺)
29	28	ex 412	. . . . . . . . . . . . . . . . 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 12945	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) < ((𝐹‘𝑐)‘𝑛))
32		ltaddrp 12946	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ⁺) → ((𝐹‘𝑐)‘𝑛) < (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))))
33	31, 32	jca 511	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . 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 15364	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ)
37		resubcl 11447	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
38	36, 37	mpdan 688	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
39	38	rexrd 11184	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
40		readdcl 11111	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ (abs‘((𝐹‘𝑐)‘𝑛)) ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
41	36, 40	mpdan 688	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ)
42	41	rexrd 11184	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) ∈ ℝ^*)
43		rexr 11180	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . 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 6833	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑐 → (𝐹‘𝑥) = (𝐹‘𝑐))
49	48	fveq1d 6835	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑐 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑐)‘𝑛))
50		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
51		fvex 6846	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑐)‘𝑛) ∈ V
52	49, 50, 51	fvmpt 6940	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑐) = ((𝐹‘𝑐)‘𝑛))
53	52	eleq1d 2820	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . . 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 259	. . . . . . . . . . . 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 24711	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛)))(,)(((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛)))) ∈ (topGen‘ran (,))
57		resttopon 23107	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))) ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
58	11, 17, 57	mp2an 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼)
59	58	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
60	22	feqmptd 6901	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
61	60, 4	eqeltrrd 2836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
62	61	adantr 480	. . . . . . . . . . . . . . . . . 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 24707	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ Top
65	64	fconst6 6723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
66	18, 3	ptpjcn 23557	. . . . . . . . . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
68		fvex 6846	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (topGen‘ran (,)) ∈ V
69	68	fvconst2 7150	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
70	69	oveq2d 7374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) = (𝑅 Cn (topGen‘ran (,))))
71	67, 70	eleqtrd 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (topGen‘ran (,))))
72	71	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑧 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑧‘𝑛)) ∈ (𝑅 Cn (topGen‘ran (,))))
73		fveq1 6832	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧‘𝑛) = ((𝐹‘𝑥)‘𝑛))
74	59, 62, 63, 72, 73	cnmpt11 23609	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (topGen‘ran (,))))
75	20	cncnpi 23224	. . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑐 ∈ 𝐼) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐))
77	76	an32s 653	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ (((𝑅 ↾_t 𝐼) CnP (topGen‘ran (,)))‘𝑐))
78		iscnp 23183	. . . . . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . 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 3959	. . . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . . 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 3571	. . . . . . . . . . . . 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 11136	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
92		letric 11235	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
93	26, 91, 92	sylancl 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘𝑐)‘𝑛) ≤ 0 ∨ 0 ≤ ((𝐹‘𝑐)‘𝑛)))
94	90, 93	jctird 526	. . . . . . . . . 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 3165	. . . . . . . . . . 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 468	. . . . . . . . . . . 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 3082	. . . . . . . . . . 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 277	. . . . . . . . . 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 251	. . . . . . . . 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 22861	. . . . . . . . . . . . 13 ⊢ (𝑅 ↾_t 𝐼) ∈ Top
101	20	eltopss 22853	. . . . . . . . . . . . 13 ⊢ (((𝑅 ↾_t 𝐼) ∈ Top ∧ 𝑣 ∈ (𝑅 ↾_t 𝐼)) → 𝑣 ⊆ 𝐼)
102	100, 101	mpan 691	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (𝑅 ↾_t 𝐼) → 𝑣 ⊆ 𝐼)
103		fvex 6846	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥)‘𝑛) ∈ V
104	103, 50	dmmpti 6635	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) = 𝐼
105	104	sseq2i 3962	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ⊆ dom (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ↔ 𝑣 ⊆ 𝐼)
106		funmpt 6529	. . . . . . . . . . . . . . . . 17 ⊢ Fun (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))
107		funimass4 6897	. . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 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 3927	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ⊆ 𝐼 ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐼)
111		fveq2 6833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
112	111	fveq1d 6835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑧)‘𝑛))
113		fvex 6846	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑧)‘𝑛) ∈ V
114	112, 50, 113	fvmpt 6940	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛))‘𝑧) = ((𝐹‘𝑧)‘𝑛))
115	114	eleq1d 2820	. . . . . . . . . . . . . . . . . 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	biimtrdi 253	. . . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 15223	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (abs‘((𝐹‘𝑐)‘𝑛)) = -((𝐹‘𝑐)‘𝑛))
123	122	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) + -((𝐹‘𝑐)‘𝑛)) = 0)
125	124	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + -((𝐹‘𝑐)‘𝑛)) = 0)
126	123, 125	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
127	26, 126	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
128	127	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ ((𝐹‘𝑐)‘𝑛) ≤ 0) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑐)‘𝑛) + (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
129	128	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . 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 7029	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ (ℝ ↑_m (1...𝑁)))
131		elmapi 8788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7029	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
134	133	an32s 653	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
135		0red 11137	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → 0 ∈ ℝ)
136	134, 135	ltnled 11282	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
137	136	adantllr 720	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑧)‘𝑛) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
138	137	adantlr 716	. . . . . . . . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . . . . 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 653	. . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . . 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 15221	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (abs‘((𝐹‘𝑐)‘𝑛)) = ((𝐹‘𝑐)‘𝑛))
148	147	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑐)‘𝑛) ∈ ℝ → (((𝐹‘𝑐)‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0)
150	149	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0)
151	148, 150	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝑐)‘𝑛) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
152	26, 151	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
153	152	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 0 ≤ ((𝐹‘𝑐)‘𝑛)) ∧ 𝑧 ∈ 𝐼) → (((𝐹‘𝑐)‘𝑛) − (abs‘((𝐹‘𝑐)‘𝑛))) = 0)
154	153	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . . 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 11282	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (0 < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
156	155	adantllr 720	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → (0 < ((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
157	156	adantlr 716	. . . . . . . . . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . . . . 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 653	. . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . . 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 967	. . . . . . . . . . . . . 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 453	. . . . . . . . . . . . 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 3148	. . . . . . . . 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 3061	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
174	173	rexbii 3082	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
175		letsr 18518	. . . . . . . . . . . . . . 15 ⊢ ≤ ∈ TosetRel
176	175	elexi 3462	. . . . . . . . . . . . . 14 ⊢ ≤ ∈ V
177	176	cnvex 7867	. . . . . . . . . . . . . 14 ⊢ ^◡ ≤ ∈ V
178		breq 5099	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
179	178	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
180	179	ralbidv 3158	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛)))
181		breq 5099	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ 0^◡ ≤ ((𝐹‘𝑧)‘𝑛)))
182		c0ex 11128	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
183	182, 113	brcnv 5830	. . . . . . . . . . . . . . . . 17 ⊢ (0^◡ ≤ ((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0)
184	181, 183	bitrdi 287	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = ^◡ ≤ → (0𝑟((𝐹‘𝑧)‘𝑛) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 0))
185	184	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = ^◡ ≤ → (¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
186	185	ralbidv 3158	. . . . . . . . . . . . . 14 ⊢ (𝑟 = ^◡ ≤ → (∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
187	176, 177, 180, 186	rexpr 4657	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ }∀𝑧 ∈ 𝑣 ¬ 0𝑟((𝐹‘𝑧)‘𝑛) ↔ (∀𝑧 ∈ 𝑣 ¬ 0 ≤ ((𝐹‘𝑧)‘𝑛) ∨ ∀𝑧 ∈ 𝑣 ¬ ((𝐹‘𝑧)‘𝑛) ≤ 0))
188		rexnal 3087	. . . . . . . . . . . . 13 ⊢ (∃𝑟 ∈ { ≤ , ^◡ ≤ } ¬ ∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛) ↔ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))
189	174, 187, 188	3bitr3i 301	. . . . . . . . . . . 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 403	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝑣 ∧ ¬ ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)) ↔ ¬ (𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ^◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
192	190, 191	bitri 275	. . . . . . . . . 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 3082	. . . . . . . . 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 3087	. . . . . . . . 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 275	. . . . . . . 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 251	. . . . . . 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 2950	. . . . . 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 3147	. . . . 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 6662	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) Fn (1...𝑁))
200	198, 199	jctild 525	. . . 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 7158	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ ((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0))
202	182	fconst2 7151	. . . . 5 ⊢ ((𝐹‘𝑐):(1...𝑁)⟶{0} ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
203	201, 202	bitr3i 277	. . . 4 ⊢ (((𝐹‘𝑐) Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)((𝐹‘𝑐)‘𝑛) = 0) ↔ (𝐹‘𝑐) = ((1...𝑁) × {0}))
204	200, 203	imbitrdi 251	. . 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 3148	. 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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2931 ∀wral 3050 ∃wrex 3059 Vcvv 3439 ⊆ wss 3900 {csn 4579 {cpr 4581 ∪ cuni 4862 class class class wbr 5097 ↦ cmpt 5178 × cxp 5621 ^◡ccnv 5622 dom cdm 5623 ran crn 5624 “ cima 5626 Fun wfun 6485 Fn wfn 6486 ⟶wf 6487 ‘cfv 6491 (class class class)co 7358 ↑_m cmap 8765 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 − cmin 11366 -cneg 11367 ℕcn 12147 ℝ⁺crp 12907 (,)cioo 13263 [,]cicc 13266 ...cfz 13425 abscabs 15159 ↾_t crest 17342 topGenctg 17359 ∏_tcpt 17360 TosetRel ctsr 18490 Topctop 22839 TopOnctopon 22856 Cn ccn 23170 CnP ccnp 23171

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-disj 5065 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-dju 9815 df-card 9853 df-acn 9856 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-xnn0 12477 df-z 12491 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-dvds 16182 df-rest 17344 df-topgen 17365 df-pt 17366 df-ps 18491 df-tsr 18492 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-top 22840 df-topon 22857 df-bases 22892 df-cld 22965 df-ntr 22966 df-cls 22967 df-lp 23082 df-cn 23173 df-cnp 23174 df-t1 23260 df-haus 23261 df-cmp 23333 df-tx 23508 df-hmeo 23701 df-hmph 23702 df-ii 24828

This theorem is referenced by: broucube 37824

W3C validator