broucube - Mathbox for Brendan Leahy

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Theorem broucube 35738

Description: Brouwer - or as Kulpa calls it, "Bohl-Brouwer" - fixed point theorem for the unit cube. Theorem on [Kulpa] p. 548. (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 (,))}))
broucube.1	⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn (𝑅 ↾_t 𝐼)))

**Assertion**
Ref	Expression
broucube	⊢ (𝜑 → ∃𝑐 ∈ 𝐼 𝑐 = (𝐹‘𝑐))

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

Proof of Theorem broucube Dummy variables 𝑛 𝑥 𝑧 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		elmapfn 8611	. . . . . . . 8 ⊢ (𝑥 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑥 Fn (1...𝑁))
5	4, 2	eleq2s 2857	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 → 𝑥 Fn (1...𝑁))
6	5	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 Fn (1...𝑁))
7		broucube.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn (𝑅 ↾_t 𝐼)))
8		ovex 7288	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ∈ V
9		retopon 23833	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
10	3	pttoponconst 22656	. . . . . . . . . . . . 13 ⊢ (((1...𝑁) ∈ V ∧ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) → 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))))
11	8, 9, 10	mp2an 688	. . . . . . . . . . . 12 ⊢ 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁)))
12		reex 10893	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
13		unitssre 13160	. . . . . . . . . . . . . 14 ⊢ (0[,]1) ⊆ ℝ
14		mapss 8635	. . . . . . . . . . . . . 14 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁)))
15	12, 13, 14	mp2an 688	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁))
16	2, 15	eqsstri 3951	. . . . . . . . . . . 12 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
17		resttopon 22220	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))) ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
18	11, 16, 17	mp2an 688	. . . . . . . . . . 11 ⊢ (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼)
19	18	toponunii 21973	. . . . . . . . . 10 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
20	19, 19	cnf 22305	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn (𝑅 ↾_t 𝐼)) → 𝐹:𝐼⟶𝐼)
21	7, 20	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐼⟶𝐼)
22	21	ffvelrnda 6943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ 𝐼)
23		elmapfn 8611	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑥) Fn (1...𝑁))
24	23, 2	eleq2s 2857	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ 𝐼 → (𝐹‘𝑥) Fn (1...𝑁))
25	22, 24	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) Fn (1...𝑁))
26		ovexd 7290	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (1...𝑁) ∈ V)
27		inidm 4149	. . . . . 6 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
28		eqidd 2739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑥‘𝑛) = (𝑥‘𝑛))
29		eqidd 2739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑥)‘𝑛))
30	6, 25, 26, 26, 27, 28, 29	offval 7520	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑥 ∘_f − (𝐹‘𝑥)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))))
31	30	mpteq2dva 5170	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛)))))
32	18	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
33		ovexd 7290	. . . . 5 ⊢ (𝜑 → (1...𝑁) ∈ V)
34		retop 23831	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ Top
35	34	fconst6 6648	. . . . . 6 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
36	35	a1i 11	. . . . 5 ⊢ (𝜑 → ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top)
37	18	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
38		eqid 2738	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
39	38	cnfldtop 23853	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
40		cnrest2r 22346	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)) ⊆ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
41	39, 40	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)) ⊆ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld))
42		resmpt 5934	. . . . . . . . . . . . 13 ⊢ (𝐼 ⊆ (ℝ ↑_m (1...𝑁)) → ((𝑥 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑥‘𝑛)) ↾ 𝐼) = (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)))
43	16, 42	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑥‘𝑛)) ↾ 𝐼) = (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛))
44	11	toponunii 21973	. . . . . . . . . . . . . . 15 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
45	44, 3	ptpjcn 22670	. . . . . . . . . . . . . 14 ⊢ (((1...𝑁) ∈ V ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑥‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
46	8, 35, 45	mp3an12 1449	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → (𝑥 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑥‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
47	44	cnrest 22344	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑥‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → ((𝑥 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑥‘𝑛)) ↾ 𝐼) ∈ ((𝑅 ↾_t 𝐼) Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
48	46, 16, 47	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑥 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑥‘𝑛)) ↾ 𝐼) ∈ ((𝑅 ↾_t 𝐼) Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
49	43, 48	eqeltrrid 2844	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
50		fvex 6769	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ V
51	50	fvconst2 7061	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
52	38	tgioo2 23872	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
53	51, 52	eqtrdi 2795	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = ((TopOpen‘ℂ_fld) ↾_t ℝ))
54	53	oveq2d 7271	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑅 ↾_t 𝐼) Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) = ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
55	49, 54	eleqtrd 2841	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
56	41, 55	sselid 3915	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) → (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
57	56	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
58	21	feqmptd 6819	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
59	58, 7	eqeltrrd 2840	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ ((𝑅 ↾_t 𝐼) Cn (𝑅 ↾_t 𝐼)))
60	59	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ ((𝑅 ↾_t 𝐼) Cn (𝑅 ↾_t 𝐼)))
61		fveq1 6755	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥‘𝑛) = (𝑧‘𝑛))
62	61	cbvmptv 5183	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) = (𝑧 ∈ 𝐼 ↦ (𝑧‘𝑛))
63	62, 57	eqeltrrid 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑧 ∈ 𝐼 ↦ (𝑧‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
64		fveq1 6755	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧‘𝑛) = ((𝐹‘𝑥)‘𝑛))
65	37, 60, 37, 63, 64	cnmpt11 22722	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
66	38	subcn 23935	. . . . . . . . 9 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
67	66	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
68	37, 57, 65, 67	cnmpt12f 22725	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
69		elmapi 8595	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑥:(1...𝑁)⟶(0[,]1))
70	69, 2	eleq2s 2857	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐼 → 𝑥:(1...𝑁)⟶(0[,]1))
71	70	ffvelrnda 6943	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥‘𝑛) ∈ (0[,]1))
72	13, 71	sselid 3915	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥‘𝑛) ∈ ℝ)
73	72	adantll 710	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑥‘𝑛) ∈ ℝ)
74		elmapi 8595	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑥):(1...𝑁)⟶(0[,]1))
75	74, 2	eleq2s 2857	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑥) ∈ 𝐼 → (𝐹‘𝑥):(1...𝑁)⟶(0[,]1))
76	22, 75	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥):(1...𝑁)⟶(0[,]1))
77	76	ffvelrnda 6943	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑥)‘𝑛) ∈ (0[,]1))
78	13, 77	sselid 3915	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑥)‘𝑛) ∈ ℝ)
79	73, 78	resubcld 11333	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛)) ∈ ℝ)
80	79	an32s 648	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑥 ∈ 𝐼) → ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛)) ∈ ℝ)
81	80	fmpttd 6971	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))):𝐼⟶ℝ)
82		frn 6591	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))):𝐼⟶ℝ → ran (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ⊆ ℝ)
83	38	cnfldtopon 23852	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
84		ax-resscn 10859	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
85		cnrest2 22345	. . . . . . . . 9 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ran (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ⊆ ℝ ∧ ℝ ⊆ ℂ) → ((𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)) ↔ (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))))
86	83, 84, 85	mp3an13 1450	. . . . . . . 8 ⊢ (ran (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ⊆ ℝ → ((𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)) ↔ (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))))
87	81, 82, 86	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)) ↔ (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))))
88	68, 87	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
89	54	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝑅 ↾_t 𝐼) Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) = ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
90	88, 89	eleqtrrd 2842	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
91	3, 32, 33, 36, 90	ptcn 22686	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛)))) ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
92	31, 91	eqeltrd 2839	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥))) ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
93		simpr2 1193	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → 𝑧 ∈ 𝐼)
94		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
95		fveq2 6756	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
96	94, 95	oveq12d 7273	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ∘_f − (𝐹‘𝑥)) = (𝑧 ∘_f − (𝐹‘𝑧)))
97		eqid 2738	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))
98		ovex 7288	. . . . . . . 8 ⊢ (𝑧 ∘_f − (𝐹‘𝑧)) ∈ V
99	96, 97, 98	fvmpt 6857	. . . . . . 7 ⊢ (𝑧 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧) = (𝑧 ∘_f − (𝐹‘𝑧)))
100	99	fveq1d 6758	. . . . . 6 ⊢ (𝑧 ∈ 𝐼 → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧)‘𝑛) = ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛))
101	93, 100	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧)‘𝑛) = ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛))
102		elmapfn 8611	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑧 Fn (1...𝑁))
103	102, 2	eleq2s 2857	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐼 → 𝑧 Fn (1...𝑁))
104	103	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) → 𝑧 Fn (1...𝑁))
105	21	ffvelrnda 6943	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ 𝐼)
106		elmapfn 8611	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑧) Fn (1...𝑁))
107	106, 2	eleq2s 2857	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ 𝐼 → (𝐹‘𝑧) Fn (1...𝑁))
108	105, 107	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) Fn (1...𝑁))
109	108	adantlr 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) Fn (1...𝑁))
110		ovexd 7290	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) → (1...𝑁) ∈ V)
111		simpllr 772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑧‘𝑛) = 0)
112		eqidd 2739	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) = ((𝐹‘𝑧)‘𝑛))
113	104, 109, 110, 110, 27, 111, 112	ofval 7522	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = (0 − ((𝐹‘𝑧)‘𝑛)))
114		df-neg 11138	. . . . . . . . 9 ⊢ -((𝐹‘𝑧)‘𝑛) = (0 − ((𝐹‘𝑧)‘𝑛))
115	113, 114	eqtr4di 2797	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = -((𝐹‘𝑧)‘𝑛))
116	115	exp41 434	. . . . . . 7 ⊢ (𝜑 → ((𝑧‘𝑛) = 0 → (𝑧 ∈ 𝐼 → (𝑛 ∈ (1...𝑁) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = -((𝐹‘𝑧)‘𝑛)))))
117	116	com24 95	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑧 ∈ 𝐼 → ((𝑧‘𝑛) = 0 → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = -((𝐹‘𝑧)‘𝑛)))))
118	117	3imp2 1347	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = -((𝐹‘𝑧)‘𝑛))
119	101, 118	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧)‘𝑛) = -((𝐹‘𝑧)‘𝑛))
120		elmapi 8595	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑧):(1...𝑁)⟶(0[,]1))
121	120, 2	eleq2s 2857	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) ∈ 𝐼 → (𝐹‘𝑧):(1...𝑁)⟶(0[,]1))
122	105, 121	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧):(1...𝑁)⟶(0[,]1))
123	122	ffvelrnda 6943	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) ∈ (0[,]1))
124		0xr 10953	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
125		1xr 10965	. . . . . . . . . 10 ⊢ 1 ∈ ℝ^*
126		iccgelb 13064	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ ((𝐹‘𝑧)‘𝑛) ∈ (0[,]1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛))
127	124, 125, 126	mp3an12 1449	. . . . . . . . 9 ⊢ (((𝐹‘𝑧)‘𝑛) ∈ (0[,]1) → 0 ≤ ((𝐹‘𝑧)‘𝑛))
128	123, 127	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → 0 ≤ ((𝐹‘𝑧)‘𝑛))
129	13, 123	sselid 3915	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
130	129	le0neg2d 11477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (0 ≤ ((𝐹‘𝑧)‘𝑛) ↔ -((𝐹‘𝑧)‘𝑛) ≤ 0))
131	128, 130	mpbid 231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → -((𝐹‘𝑧)‘𝑛) ≤ 0)
132	131	an32s 648	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → -((𝐹‘𝑧)‘𝑛) ≤ 0)
133	132	anasss 466	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼)) → -((𝐹‘𝑧)‘𝑛) ≤ 0)
134	133	3adantr3 1169	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → -((𝐹‘𝑧)‘𝑛) ≤ 0)
135	119, 134	eqbrtrd 5092	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧)‘𝑛) ≤ 0)
136		iccleub 13063	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ ((𝐹‘𝑧)‘𝑛) ∈ (0[,]1)) → ((𝐹‘𝑧)‘𝑛) ≤ 1)
137	124, 125, 136	mp3an12 1449	. . . . . . . . 9 ⊢ (((𝐹‘𝑧)‘𝑛) ∈ (0[,]1) → ((𝐹‘𝑧)‘𝑛) ≤ 1)
138	123, 137	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) ≤ 1)
139		1red 10907	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → 1 ∈ ℝ)
140	139, 129	subge0d 11495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (0 ≤ (1 − ((𝐹‘𝑧)‘𝑛)) ↔ ((𝐹‘𝑧)‘𝑛) ≤ 1))
141	138, 140	mpbird 256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → 0 ≤ (1 − ((𝐹‘𝑧)‘𝑛)))
142	141	an32s 648	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → 0 ≤ (1 − ((𝐹‘𝑧)‘𝑛)))
143	142	anasss 466	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼)) → 0 ≤ (1 − ((𝐹‘𝑧)‘𝑛)))
144	143	3adantr3 1169	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ (1 − ((𝐹‘𝑧)‘𝑛)))
145		simpr2 1193	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 𝑧 ∈ 𝐼)
146	145, 100	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧)‘𝑛) = ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛))
147	103	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) → 𝑧 Fn (1...𝑁))
148	108	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) Fn (1...𝑁))
149		ovexd 7290	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) → (1...𝑁) ∈ V)
150		simpllr 772	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑧‘𝑛) = 1)
151		eqidd 2739	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) = ((𝐹‘𝑧)‘𝑛))
152	147, 148, 149, 149, 27, 150, 151	ofval 7522	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = (1 − ((𝐹‘𝑧)‘𝑛)))
153	152	exp41 434	. . . . . . 7 ⊢ (𝜑 → ((𝑧‘𝑛) = 1 → (𝑧 ∈ 𝐼 → (𝑛 ∈ (1...𝑁) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = (1 − ((𝐹‘𝑧)‘𝑛))))))
154	153	com24 95	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑧 ∈ 𝐼 → ((𝑧‘𝑛) = 1 → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = (1 − ((𝐹‘𝑧)‘𝑛))))))
155	154	3imp2 1347	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = (1 − ((𝐹‘𝑧)‘𝑛)))
156	146, 155	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧)‘𝑛) = (1 − ((𝐹‘𝑧)‘𝑛)))
157	144, 156	breqtrrd 5098	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧)‘𝑛))
158	1, 2, 3, 92, 135, 157	poimir 35737	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = ((1...𝑁) × {0}))
159		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑐 → 𝑥 = 𝑐)
160		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = 𝑐 → (𝐹‘𝑥) = (𝐹‘𝑐))
161	159, 160	oveq12d 7273	. . . . . . 7 ⊢ (𝑥 = 𝑐 → (𝑥 ∘_f − (𝐹‘𝑥)) = (𝑐 ∘_f − (𝐹‘𝑐)))
162		ovex 7288	. . . . . . 7 ⊢ (𝑐 ∘_f − (𝐹‘𝑐)) ∈ V
163	161, 97, 162	fvmpt 6857	. . . . . 6 ⊢ (𝑐 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = (𝑐 ∘_f − (𝐹‘𝑐)))
164	163	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = (𝑐 ∘_f − (𝐹‘𝑐)))
165	164	eqeq1d 2740	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = ((1...𝑁) × {0}) ↔ (𝑐 ∘_f − (𝐹‘𝑐)) = ((1...𝑁) × {0})))
166		elmapfn 8611	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑐 Fn (1...𝑁))
167	166, 2	eleq2s 2857	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐼 → 𝑐 Fn (1...𝑁))
168	167	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → 𝑐 Fn (1...𝑁))
169	21	ffvelrnda 6943	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) ∈ 𝐼)
170		elmapfn 8611	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑐) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑐) Fn (1...𝑁))
171	170, 2	eleq2s 2857	. . . . . . . . . 10 ⊢ ((𝐹‘𝑐) ∈ 𝐼 → (𝐹‘𝑐) Fn (1...𝑁))
172	169, 171	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) Fn (1...𝑁))
173		ovexd 7290	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (1...𝑁) ∈ V)
174		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑐‘𝑛) = (𝑐‘𝑛))
175		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) = ((𝐹‘𝑐)‘𝑛))
176	168, 172, 173, 173, 27, 174, 175	ofval 7522	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑐 ∘_f − (𝐹‘𝑐))‘𝑛) = ((𝑐‘𝑛) − ((𝐹‘𝑐)‘𝑛)))
177		c0ex 10900	. . . . . . . . . 10 ⊢ 0 ∈ V
178	177	fvconst2 7061	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
179	178	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
180	176, 179	eqeq12d 2754	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑐 ∘_f − (𝐹‘𝑐))‘𝑛) = (((1...𝑁) × {0})‘𝑛) ↔ ((𝑐‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0))
181	13, 84	sstri 3926	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℂ
182		elmapi 8595	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑐:(1...𝑁)⟶(0[,]1))
183	182, 2	eleq2s 2857	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐼 → 𝑐:(1...𝑁)⟶(0[,]1))
184	183	ffvelrnda 6943	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑛 ∈ (1...𝑁)) → (𝑐‘𝑛) ∈ (0[,]1))
185	181, 184	sselid 3915	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑛 ∈ (1...𝑁)) → (𝑐‘𝑛) ∈ ℂ)
186	185	adantll 710	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑐‘𝑛) ∈ ℂ)
187		elmapi 8595	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑐) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑐):(1...𝑁)⟶(0[,]1))
188	187, 2	eleq2s 2857	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑐) ∈ 𝐼 → (𝐹‘𝑐):(1...𝑁)⟶(0[,]1))
189	169, 188	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐):(1...𝑁)⟶(0[,]1))
190	189	ffvelrnda 6943	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) ∈ (0[,]1))
191	181, 190	sselid 3915	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) ∈ ℂ)
192	186, 191	subeq0ad 11272	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑐‘𝑛) − ((𝐹‘𝑐)‘𝑛)) = 0 ↔ (𝑐‘𝑛) = ((𝐹‘𝑐)‘𝑛)))
193	180, 192	bitrd 278	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑐 ∘_f − (𝐹‘𝑐))‘𝑛) = (((1...𝑁) × {0})‘𝑛) ↔ (𝑐‘𝑛) = ((𝐹‘𝑐)‘𝑛)))
194	193	ralbidva 3119	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (∀𝑛 ∈ (1...𝑁)((𝑐 ∘_f − (𝐹‘𝑐))‘𝑛) = (((1...𝑁) × {0})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑐‘𝑛) = ((𝐹‘𝑐)‘𝑛)))
195	168, 172, 173, 173, 27	offn 7524	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∘_f − (𝐹‘𝑐)) Fn (1...𝑁))
196		fnconstg 6646	. . . . . . 7 ⊢ (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
197	177, 196	ax-mp 5	. . . . . 6 ⊢ ((1...𝑁) × {0}) Fn (1...𝑁)
198		eqfnfv 6891	. . . . . 6 ⊢ (((𝑐 ∘_f − (𝐹‘𝑐)) Fn (1...𝑁) ∧ ((1...𝑁) × {0}) Fn (1...𝑁)) → ((𝑐 ∘_f − (𝐹‘𝑐)) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁)((𝑐 ∘_f − (𝐹‘𝑐))‘𝑛) = (((1...𝑁) × {0})‘𝑛)))
199	195, 197, 198	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → ((𝑐 ∘_f − (𝐹‘𝑐)) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁)((𝑐 ∘_f − (𝐹‘𝑐))‘𝑛) = (((1...𝑁) × {0})‘𝑛)))
200		eqfnfv 6891	. . . . . 6 ⊢ ((𝑐 Fn (1...𝑁) ∧ (𝐹‘𝑐) Fn (1...𝑁)) → (𝑐 = (𝐹‘𝑐) ↔ ∀𝑛 ∈ (1...𝑁)(𝑐‘𝑛) = ((𝐹‘𝑐)‘𝑛)))
201	168, 172, 200	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑐 = (𝐹‘𝑐) ↔ ∀𝑛 ∈ (1...𝑁)(𝑐‘𝑛) = ((𝐹‘𝑐)‘𝑛)))
202	194, 199, 201	3bitr4d 310	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → ((𝑐 ∘_f − (𝐹‘𝑐)) = ((1...𝑁) × {0}) ↔ 𝑐 = (𝐹‘𝑐)))
203	165, 202	bitrd 278	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = ((1...𝑁) × {0}) ↔ 𝑐 = (𝐹‘𝑐)))
204	203	rexbidva 3224	. 2 ⊢ (𝜑 → (∃𝑐 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = ((1...𝑁) × {0}) ↔ ∃𝑐 ∈ 𝐼 𝑐 = (𝐹‘𝑐)))
205	158, 204	mpbid 231	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 𝑐 = (𝐹‘𝑐))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ⊆ wss 3883 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 ran crn 5581 ↾ cres 5582 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 ↑_m cmap 8573 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 ℝ^*cxr 10939 ≤ cle 10941 − cmin 11135 -cneg 11136 ℕcn 11903 (,)cioo 13008 [,]cicc 13011 ...cfz 13168 ↾_t crest 17048 TopOpenctopn 17049 topGenctg 17065 ∏_tcpt 17066 ℂ_fldccnfld 20510 Topctop 21950 TopOnctopon 21967 Cn ccn 22283 ×_t ctx 22619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-dvds 15892 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-ps 18199 df-tsr 18200 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-lp 22195 df-cn 22286 df-cnp 22287 df-t1 22373 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-hmph 22815 df-xms 23381 df-ms 23382 df-tms 23383 df-ii 23946

This theorem is referenced by: (None)

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