broucube - Mathbox for Brendan Leahy

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

Theorem broucube 37197

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 8882	. . . . . . . 8 ⊢ (𝑥 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑥 Fn (1...𝑁))
5	4, 2	eleq2s 2843	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 → 𝑥 Fn (1...𝑁))
6	5	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 Fn (1...𝑁))
7		broucube.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn (𝑅 ↾_t 𝐼)))
8		ovex 7450	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ∈ V
9		retopon 24710	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
10	3	pttoponconst 23531	. . . . . . . . . . . . 13 ⊢ (((1...𝑁) ∈ V ∧ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) → 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))))
11	8, 9, 10	mp2an 690	. . . . . . . . . . . 12 ⊢ 𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁)))
12		reex 11229	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
13		unitssre 13508	. . . . . . . . . . . . . 14 ⊢ (0[,]1) ⊆ ℝ
14		mapss 8906	. . . . . . . . . . . . . 14 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁)))
15	12, 13, 14	mp2an 690	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ↑_m (1...𝑁)) ⊆ (ℝ ↑_m (1...𝑁))
16	2, 15	eqsstri 4012	. . . . . . . . . . . 12 ⊢ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))
17		resttopon 23095	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ (TopOn‘(ℝ ↑_m (1...𝑁))) ∧ 𝐼 ⊆ (ℝ ↑_m (1...𝑁))) → (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼))
18	11, 16, 17	mp2an 690	. . . . . . . . . . 11 ⊢ (𝑅 ↾_t 𝐼) ∈ (TopOn‘𝐼)
19	18	toponunii 22848	. . . . . . . . . 10 ⊢ 𝐼 = ∪ (𝑅 ↾_t 𝐼)
20	19, 19	cnf 23180	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝑅 ↾_t 𝐼) Cn (𝑅 ↾_t 𝐼)) → 𝐹:𝐼⟶𝐼)
21	7, 20	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐼⟶𝐼)
22	21	ffvelcdmda 7091	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ 𝐼)
23		elmapfn 8882	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑥) Fn (1...𝑁))
24	23, 2	eleq2s 2843	. . . . . . 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 7452	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (1...𝑁) ∈ V)
27		inidm 4218	. . . . . 6 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
28		eqidd 2726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑥‘𝑛) = (𝑥‘𝑛))
29		eqidd 2726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑥)‘𝑛) = ((𝐹‘𝑥)‘𝑛))
30	6, 25, 26, 26, 27, 28, 29	offval 7692	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑥 ∘_f − (𝐹‘𝑥)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))))
31	30	mpteq2dva 5248	. . . 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 7452	. . . . 5 ⊢ (𝜑 → (1...𝑁) ∈ V)
34		retop 24708	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ Top
35	34	fconst6 6785	. . . . . 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 2725	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
39	38	cnfldtop 24730	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
40		cnrest2r 23221	. . . . . . . . . . 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 6041	. . . . . . . . . . . . 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 22848	. . . . . . . . . . . . . . 15 ⊢ (ℝ ↑_m (1...𝑁)) = ∪ 𝑅
45	44, 3	ptpjcn 23545	. . . . . . . . . . . . . 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 1447	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → (𝑥 ∈ (ℝ ↑_m (1...𝑁)) ↦ (𝑥‘𝑛)) ∈ (𝑅 Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
47	44	cnrest 23219	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 2830	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)))
50		fvex 6907	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ V
51	50	fvconst2 7214	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
52	38	tgioo2 24749	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
53	51, 52	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = ((TopOpen‘ℂ_fld) ↾_t ℝ))
54	53	oveq2d 7433	. . . . . . . . . . 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 2827	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
56	41, 55	sselid 3975	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) → (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
57	56	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
58	21	feqmptd 6964	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
59	58, 7	eqeltrrd 2826	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ ((𝑅 ↾_t 𝐼) Cn (𝑅 ↾_t 𝐼)))
60	59	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) ∈ ((𝑅 ↾_t 𝐼) Cn (𝑅 ↾_t 𝐼)))
61		fveq1 6893	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥‘𝑛) = (𝑧‘𝑛))
62	61	cbvmptv 5261	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑥‘𝑛)) = (𝑧 ∈ 𝐼 ↦ (𝑧‘𝑛))
63	62, 57	eqeltrrid 2830	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑧 ∈ 𝐼 ↦ (𝑧‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
64		fveq1 6893	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧‘𝑛) = ((𝐹‘𝑥)‘𝑛))
65	37, 60, 37, 63, 64	cnmpt11 23597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑛)) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
66	38	subcn 24812	. . . . . . . . 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 23600	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ∈ ((𝑅 ↾_t 𝐼) Cn (TopOpen‘ℂ_fld)))
69		elmapi 8866	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑥:(1...𝑁)⟶(0[,]1))
70	69, 2	eleq2s 2843	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐼 → 𝑥:(1...𝑁)⟶(0[,]1))
71	70	ffvelcdmda 7091	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥‘𝑛) ∈ (0[,]1))
72	13, 71	sselid 3975	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥‘𝑛) ∈ ℝ)
73	72	adantll 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑥‘𝑛) ∈ ℝ)
74		elmapi 8866	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑥):(1...𝑁)⟶(0[,]1))
75	74, 2	eleq2s 2843	. . . . . . . . . . . . . 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	ffvelcdmda 7091	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑥)‘𝑛) ∈ (0[,]1))
78	13, 77	sselid 3975	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑥)‘𝑛) ∈ ℝ)
79	73, 78	resubcld 11672	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛)) ∈ ℝ)
80	79	an32s 650	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑥 ∈ 𝐼) → ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛)) ∈ ℝ)
81	80	fmpttd 7122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))):𝐼⟶ℝ)
82		frn 6728	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))):𝐼⟶ℝ → ran (𝑥 ∈ 𝐼 ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛))) ⊆ ℝ)
83	38	cnfldtopon 24729	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
84		ax-resscn 11195	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
85		cnrest2 23220	. . . . . . . . 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 1448	. . . . . . . 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 480	. . . . . 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 2828	. . . . 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 23561	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑥‘𝑛) − ((𝐹‘𝑥)‘𝑛)))) ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
92	31, 91	eqeltrd 2825	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥))) ∈ ((𝑅 ↾_t 𝐼) Cn 𝑅))
93		simpr2 1192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → 𝑧 ∈ 𝐼)
94		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
95		fveq2 6894	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
96	94, 95	oveq12d 7435	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ∘_f − (𝐹‘𝑥)) = (𝑧 ∘_f − (𝐹‘𝑧)))
97		eqid 2725	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))
98		ovex 7450	. . . . . . . 8 ⊢ (𝑧 ∘_f − (𝐹‘𝑧)) ∈ V
99	96, 97, 98	fvmpt 7002	. . . . . . 7 ⊢ (𝑧 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧) = (𝑧 ∘_f − (𝐹‘𝑧)))
100	99	fveq1d 6896	. . . . . 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 8882	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑧 Fn (1...𝑁))
103	102, 2	eleq2s 2843	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐼 → 𝑧 Fn (1...𝑁))
104	103	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) → 𝑧 Fn (1...𝑁))
105	21	ffvelcdmda 7091	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) ∈ 𝐼)
106		elmapfn 8882	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑧) Fn (1...𝑁))
107	106, 2	eleq2s 2843	. . . . . . . . . . . 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 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) Fn (1...𝑁))
110		ovexd 7452	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) → (1...𝑁) ∈ V)
111		simpllr 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑧‘𝑛) = 0)
112		eqidd 2726	. . . . . . . . . 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 7694	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = (0 − ((𝐹‘𝑧)‘𝑛)))
114		df-neg 11477	. . . . . . . . 9 ⊢ -((𝐹‘𝑧)‘𝑛) = (0 − ((𝐹‘𝑧)‘𝑛))
115	113, 114	eqtr4di 2783	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 0) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = -((𝐹‘𝑧)‘𝑛))
116	115	exp41 433	. . . . . . 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 1346	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = -((𝐹‘𝑧)‘𝑛))
119	101, 118	eqtrd 2765	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧)‘𝑛) = -((𝐹‘𝑧)‘𝑛))
120		elmapi 8866	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑧):(1...𝑁)⟶(0[,]1))
121	120, 2	eleq2s 2843	. . . . . . . . . . 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	ffvelcdmda 7091	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) ∈ (0[,]1))
124		0xr 11291	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
125		1xr 11303	. . . . . . . . . 10 ⊢ 1 ∈ ℝ^*
126		iccgelb 13412	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ ((𝐹‘𝑧)‘𝑛) ∈ (0[,]1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛))
127	124, 125, 126	mp3an12 1447	. . . . . . . . 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 3975	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑧)‘𝑛) ∈ ℝ)
130	129	le0neg2d 11816	. . . . . . . 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 650	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → -((𝐹‘𝑧)‘𝑛) ≤ 0)
133	132	anasss 465	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼)) → -((𝐹‘𝑧)‘𝑛) ≤ 0)
134	133	3adantr3 1168	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → -((𝐹‘𝑧)‘𝑛) ≤ 0)
135	119, 134	eqbrtrd 5170	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑧)‘𝑛) ≤ 0)
136		iccleub 13411	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ ((𝐹‘𝑧)‘𝑛) ∈ (0[,]1)) → ((𝐹‘𝑧)‘𝑛) ≤ 1)
137	124, 125, 136	mp3an12 1447	. . . . . . . . 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 11245	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → 1 ∈ ℝ)
140	139, 129	subge0d 11834	. . . . . . . 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 650	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝐼) → 0 ≤ (1 − ((𝐹‘𝑧)‘𝑛)))
143	142	anasss 465	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼)) → 0 ≤ (1 − ((𝐹‘𝑧)‘𝑛)))
144	143	3adantr3 1168	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ (1 − ((𝐹‘𝑧)‘𝑛)))
145		simpr2 1192	. . . . . 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 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) → 𝑧 Fn (1...𝑁))
148	108	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) Fn (1...𝑁))
149		ovexd 7452	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) → (1...𝑁) ∈ V)
150		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑧‘𝑛) = 1)
151		eqidd 2726	. . . . . . . . 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 7694	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑧‘𝑛) = 1) ∧ 𝑧 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = (1 − ((𝐹‘𝑧)‘𝑛)))
153	152	exp41 433	. . . . . . 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 1346	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → ((𝑧 ∘_f − (𝐹‘𝑧))‘𝑛) = (1 − ((𝐹‘𝑧)‘𝑛)))
156	146, 155	eqtrd 2765	. . . 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 5176	. . 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 37196	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = ((1...𝑁) × {0}))
159		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑐 → 𝑥 = 𝑐)
160		fveq2 6894	. . . . . . . 8 ⊢ (𝑥 = 𝑐 → (𝐹‘𝑥) = (𝐹‘𝑐))
161	159, 160	oveq12d 7435	. . . . . . 7 ⊢ (𝑥 = 𝑐 → (𝑥 ∘_f − (𝐹‘𝑥)) = (𝑐 ∘_f − (𝐹‘𝑐)))
162		ovex 7450	. . . . . . 7 ⊢ (𝑐 ∘_f − (𝐹‘𝑐)) ∈ V
163	161, 97, 162	fvmpt 7002	. . . . . 6 ⊢ (𝑐 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = (𝑐 ∘_f − (𝐹‘𝑐)))
164	163	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = (𝑐 ∘_f − (𝐹‘𝑐)))
165	164	eqeq1d 2727	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (((𝑥 ∈ 𝐼 ↦ (𝑥 ∘_f − (𝐹‘𝑥)))‘𝑐) = ((1...𝑁) × {0}) ↔ (𝑐 ∘_f − (𝐹‘𝑐)) = ((1...𝑁) × {0})))
166		elmapfn 8882	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑐 Fn (1...𝑁))
167	166, 2	eleq2s 2843	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐼 → 𝑐 Fn (1...𝑁))
168	167	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → 𝑐 Fn (1...𝑁))
169	21	ffvelcdmda 7091	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝐹‘𝑐) ∈ 𝐼)
170		elmapfn 8882	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑐) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑐) Fn (1...𝑁))
171	170, 2	eleq2s 2843	. . . . . . . . . 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 7452	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (1...𝑁) ∈ V)
174		eqidd 2726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑐‘𝑛) = (𝑐‘𝑛))
175		eqidd 2726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) = ((𝐹‘𝑐)‘𝑛))
176	168, 172, 173, 173, 27, 174, 175	ofval 7694	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑐 ∘_f − (𝐹‘𝑐))‘𝑛) = ((𝑐‘𝑛) − ((𝐹‘𝑐)‘𝑛)))
177		c0ex 11238	. . . . . . . . . 10 ⊢ 0 ∈ V
178	177	fvconst2 7214	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
179	178	adantl 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
180	176, 179	eqeq12d 2741	. . . . . . 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 3987	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℂ
182		elmapi 8866	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ((0[,]1) ↑_m (1...𝑁)) → 𝑐:(1...𝑁)⟶(0[,]1))
183	182, 2	eleq2s 2843	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐼 → 𝑐:(1...𝑁)⟶(0[,]1))
184	183	ffvelcdmda 7091	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑛 ∈ (1...𝑁)) → (𝑐‘𝑛) ∈ (0[,]1))
185	181, 184	sselid 3975	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑛 ∈ (1...𝑁)) → (𝑐‘𝑛) ∈ ℂ)
186	185	adantll 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → (𝑐‘𝑛) ∈ ℂ)
187		elmapi 8866	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑐) ∈ ((0[,]1) ↑_m (1...𝑁)) → (𝐹‘𝑐):(1...𝑁)⟶(0[,]1))
188	187, 2	eleq2s 2843	. . . . . . . . . . 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	ffvelcdmda 7091	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) ∈ (0[,]1))
191	181, 190	sselid 3975	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑛 ∈ (1...𝑁)) → ((𝐹‘𝑐)‘𝑛) ∈ ℂ)
192	186, 191	subeq0ad 11611	. . . . . . 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 3166	. . . . 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 7696	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∘_f − (𝐹‘𝑐)) Fn (1...𝑁))
196		fnconstg 6783	. . . . . . 7 ⊢ (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
197	177, 196	ax-mp 5	. . . . . 6 ⊢ ((1...𝑁) × {0}) Fn (1...𝑁)
198		eqfnfv 7037	. . . . . 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 584	. . . . 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 7037	. . . . . 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 582	. . . . 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 3167	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 Vcvv 3463 ⊆ wss 3945 {csn 4629 class class class wbr 5148 ↦ cmpt 5231 × cxp 5675 ran crn 5678 ↾ cres 5679 Fn wfn 6542 ⟶wf 6543 ‘cfv 6547 (class class class)co 7417 ∘_f cof 7681 ↑_m cmap 8843 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 ℝ^*cxr 11277 ≤ cle 11279 − cmin 11474 -cneg 11475 ℕcn 12242 (,)cioo 13356 [,]cicc 13359 ...cfz 13516 ↾_t crest 17401 TopOpenctopn 17402 topGenctg 17418 ∏_tcpt 17419 ℂ_fldccnfld 21283 Topctop 22825 TopOnctopon 22842 Cn ccn 23158 ×_t ctx 23494

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 df-dvds 16231 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-ps 18557 df-tsr 18558 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-cnfld 21284 df-top 22826 df-topon 22843 df-topsp 22865 df-bases 22879 df-cld 22953 df-ntr 22954 df-cls 22955 df-lp 23070 df-cn 23161 df-cnp 23162 df-t1 23248 df-haus 23249 df-cmp 23321 df-tx 23496 df-hmeo 23689 df-hmph 23690 df-xms 24256 df-ms 24257 df-tms 24258 df-ii 24827

This theorem is referenced by: (None)

W3C validator