ioorrnopnlem - Mathbox for Glauco Siliprandi

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Theorem ioorrnopnlem 44535

Description: The a point in an indexed product of open intervals is contained in an open ball that is contained in the indexed product of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
ioorrnopnlem.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ioorrnopnlem.n	⊢ (𝜑 → 𝑋 ≠ ∅)
ioorrnopnlem.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
ioorrnopnlem.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
ioorrnopnlem.f	⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
ioorrnopnlem.h	⊢ 𝐻 = ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))
ioorrnopnlem.e	⊢ 𝐸 = inf(𝐻, ℝ, < )
ioorrnopnlem.v	⊢ 𝑉 = (𝐹(ball‘𝐷)𝐸)
ioorrnopnlem.d	⊢ 𝐷 = (𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))

**Assertion**
Ref	Expression
ioorrnopnlem	⊢ (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))

Distinct variable groups: 𝐴,𝑔 𝑣,𝐴 𝐵,𝑔 𝑣,𝐵 𝐷,𝑔,𝑖 𝑔,𝐸,𝑖 𝑔,𝐹,𝑖 𝑣,𝐹,𝑖 𝑣,𝑉 𝑓,𝑋,𝑔,𝑘 𝑖,𝑋,𝑣 𝜑,𝑓,𝑔,𝑘 𝜑,𝑖 Allowed substitution hints: 𝜑(𝑣) 𝐴(𝑓,𝑖,𝑘) 𝐵(𝑓,𝑖,𝑘) 𝐷(𝑣,𝑓,𝑘) 𝐸(𝑣,𝑓,𝑘) 𝐹(𝑓,𝑘) 𝐻(𝑣,𝑓,𝑔,𝑖,𝑘) 𝑉(𝑓,𝑔,𝑖,𝑘)

**Proof of Theorem ioorrnopnlem**
Step	Hyp	Ref	Expression
1		ioorrnopnlem.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		ioorrnopnlem.d	. . . . 5 ⊢ 𝐷 = (𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
3	1, 2	rrndsxmet 44534	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝑋)))
4		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑖𝜑
5		reex 11142	. . . . . . 7 ⊢ ℝ ∈ V
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
7		ioossre 13325	. . . . . . 7 ⊢ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ
8	7	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
9	4, 6, 8	ixpssmapc 43272	. . . . 5 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ (ℝ ↑_m 𝑋))
10		ioorrnopnlem.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
11	9, 10	sseldd 3945	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑_m 𝑋))
12		ioorrnopnlem.e	. . . . . 6 ⊢ 𝐸 = inf(𝐻, ℝ, < )
13		ioorrnopnlem.h	. . . . . . . . 9 ⊢ 𝐻 = ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))
14	13	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))))
15		ioorrnopnlem.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
16	15	ffvelcdmda 7035	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ)
17	10	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
18		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
19		fvixp2 43409	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
20	17, 18, 19	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
21	7, 20	sselid 3942	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
22	16, 21	resubcld 11583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
23		ioorrnopnlem.a	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
24	23	ffvelcdmda 7035	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ)
25	24	rexrd 11205	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
26	16	rexrd 11205	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
27		iooltub 43738	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐹‘𝑖) < (𝐵‘𝑖))
28	25, 26, 20, 27	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝐵‘𝑖))
29	21, 16	posdifd 11742	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) < (𝐵‘𝑖) ↔ 0 < ((𝐵‘𝑖) − (𝐹‘𝑖))))
30	28, 29	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐵‘𝑖) − (𝐹‘𝑖)))
31	22, 30	elrpd 12954	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ⁺)
32	21, 24	resubcld 11583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ)
33		ioogtlb 43723	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐴‘𝑖) < (𝐹‘𝑖))
34	25, 26, 20, 33	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝐹‘𝑖))
35	24, 21	posdifd 11742	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖) < (𝐹‘𝑖) ↔ 0 < ((𝐹‘𝑖) − (𝐴‘𝑖))))
36	34, 35	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐹‘𝑖) − (𝐴‘𝑖)))
37	32, 36	elrpd 12954	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ⁺)
38	31, 37	ifcld 4532	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ℝ⁺)
39	38	ralrimiva 3143	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ℝ⁺)
40		eqid 2736	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) = (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))
41	40	rnmptss 7070	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝑋 if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ℝ⁺ → ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ⊆ ℝ⁺)
42	39, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ⊆ ℝ⁺)
43	14, 42	eqsstrd 3982	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ ℝ⁺)
44		ltso 11235	. . . . . . . . 9 ⊢ < Or ℝ
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → < Or ℝ)
46	40	rnmptfi 43378	. . . . . . . . . 10 ⊢ (𝑋 ∈ Fin → ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ∈ Fin)
47	1, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ∈ Fin)
48	13, 47	eqeltrid 2842	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Fin)
49		ioorrnopnlem.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ ∅)
50	4, 38, 40, 49	rnmptn0 6196	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))) ≠ ∅)
51	14, 50	eqnetrd 3011	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ≠ ∅)
52		rpssre 12922	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ
53	52	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
54	43, 53	sstrd 3954	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
55		fiinfcl 9437	. . . . . . . 8 ⊢ (( < Or ℝ ∧ (𝐻 ∈ Fin ∧ 𝐻 ≠ ∅ ∧ 𝐻 ⊆ ℝ)) → inf(𝐻, ℝ, < ) ∈ 𝐻)
56	45, 48, 51, 54, 55	syl13anc 1372	. . . . . . 7 ⊢ (𝜑 → inf(𝐻, ℝ, < ) ∈ 𝐻)
57	43, 56	sseldd 3945	. . . . . 6 ⊢ (𝜑 → inf(𝐻, ℝ, < ) ∈ ℝ⁺)
58	12, 57	eqeltrid 2842	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
59		rpxr 12924	. . . . 5 ⊢ (𝐸 ∈ ℝ⁺ → 𝐸 ∈ ℝ^*)
60	58, 59	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
61		eqid 2736	. . . . 5 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
62	61	blopn 23856	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑_m 𝑋) ∧ 𝐸 ∈ ℝ^*) → (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷))
63	3, 11, 60, 62	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷))
64		ioorrnopnlem.v	. . . . 5 ⊢ 𝑉 = (𝐹(ball‘𝐷)𝐸)
65	64	a1i 11	. . . 4 ⊢ (𝜑 → 𝑉 = (𝐹(ball‘𝐷)𝐸))
66	1	rrxtopnfi 44518	. . . . 5 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))
67	2	eqcomi 2745	. . . . . . 7 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷
68	67	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷)
69	68	fveq2d 6846	. . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷))
70	66, 69	eqtrd 2776	. . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘𝐷))
71	65, 70	eleq12d 2832	. . 3 ⊢ (𝜑 → (𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)) ↔ (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷)))
72	63, 71	mpbird 256	. 2 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)))
73		xmetpsmet 23701	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘(ℝ ↑_m 𝑋)) → 𝐷 ∈ (PsMet‘(ℝ ↑_m 𝑋)))
74	3, 73	syl 17	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (PsMet‘(ℝ ↑_m 𝑋)))
75		blcntrps 23765	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘(ℝ ↑_m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑_m 𝑋) ∧ 𝐸 ∈ ℝ⁺) → 𝐹 ∈ (𝐹(ball‘𝐷)𝐸))
76	74, 11, 58, 75	syl3anc 1371	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐹(ball‘𝐷)𝐸))
77	65	eqcomd 2742	. . . 4 ⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) = 𝑉)
78	76, 77	eleqtrd 2840	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
79		nfv 1917	. . . . 5 ⊢ Ⅎ𝑔𝜑
80		elmapfn 8803	. . . . . . . 8 ⊢ (𝑔 ∈ (ℝ ↑_m 𝑋) → 𝑔 Fn 𝑋)
81	80	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → 𝑔 Fn 𝑋)
82	25	3ad2antl1 1185	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
83	26	3ad2antl1 1185	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
84		simpl2 1192	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝑔 ∈ (ℝ ↑_m 𝑋))
85		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
86		elmapi 8787	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑_m 𝑋) → 𝑔:𝑋⟶ℝ)
87	86	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑔:𝑋⟶ℝ)
88		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
89	87, 88	ffvelcdmd 7036	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ)
90	84, 85, 89	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ)
91	24	3ad2antl1 1185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ)
92	52, 58	sselid 3942	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
93	92	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
94	21, 93	resubcld 11583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) ∈ ℝ)
95	94	3ad2antl1 1185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) ∈ ℝ)
96	52, 38	sselid 3942	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ℝ)
97	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 = inf(𝐻, ℝ, < ))
98		infxrrefi 43606	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 ⊆ ℝ ∧ 𝐻 ∈ Fin ∧ 𝐻 ≠ ∅) → inf(𝐻, ℝ^*, < ) = inf(𝐻, ℝ, < ))
99	54, 48, 51, 98	syl3anc 1371	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → inf(𝐻, ℝ^*, < ) = inf(𝐻, ℝ, < ))
100	99	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → inf(𝐻, ℝ, < ) = inf(𝐻, ℝ^*, < ))
101	97, 100	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 = inf(𝐻, ℝ^*, < ))
102	101	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 = inf(𝐻, ℝ^*, < ))
103		ressxr 11199	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ ⊆ ℝ^*
104	103	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
105	54, 104	sstrd 3954	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐻 ⊆ ℝ^*)
106	105	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐻 ⊆ ℝ^*)
107	38	elexd 3465	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ V)
108	40	elrnmpt1 5913	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑋 ∧ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ V) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))))
109	18, 107, 108	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖)))))
110	109, 13	eleqtrrdi 2849	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ 𝐻)
111		infxrlb 13253	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 ⊆ ℝ^* ∧ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ 𝐻) → inf(𝐻, ℝ^*, < ) ≤ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))
112	106, 110, 111	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → inf(𝐻, ℝ^*, < ) ≤ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))
113	102, 112	eqbrtrd 5127	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))
114		min2 13109	. . . . . . . . . . . . . 14 ⊢ ((((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ ∧ ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)))
115	22, 32, 114	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)))
116	93, 96, 32, 113, 115	letrd 11312	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)))
117	93, 21, 24, 116	lesubd 11759	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ ((𝐹‘𝑖) − 𝐸))
118	117	3ad2antl1 1185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ ((𝐹‘𝑖) − 𝐸))
119	21	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
120	89	adantll 712	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ)
121	119, 120	resubcld 11583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℝ)
122	121	3adantl3 1168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℝ)
123	1, 2	rrndsmet 44533	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝑋)))
124	123	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐷 ∈ (Met‘(ℝ ↑_m 𝑋)))
125	11	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ (ℝ ↑_m 𝑋))
126		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑔 ∈ (ℝ ↑_m 𝑋))
127		metcl 23685	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘(ℝ ↑_m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑_m 𝑋) ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) → (𝐹𝐷𝑔) ∈ ℝ)
128	124, 125, 126, 127	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) ∈ ℝ)
129	128	3adantl3 1168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) ∈ ℝ)
130	93	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
131	130	3adantl3 1168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
132	121	recnd 11183	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℂ)
133	132	abscld 15321	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ∈ ℝ)
134	121	leabsd 15299	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
135	1	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑋 ∈ Fin)
136		ixpf 8858	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝐹:𝑋⟶∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
137	10, 136	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
138	8	ralrimiva 3143	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
139		iunss 5005	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ ↔ ∀𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
140	138, 139	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
141	137, 140	fssd 6686	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
142	141	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐹:𝑋⟶ℝ)
143	126, 86	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑔:𝑋⟶ℝ)
144		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
145		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
146	135, 142, 143, 144, 145	rrnprjdstle 44532	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹(dist‘(ℝ^‘𝑋))𝑔))
147		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
148		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ ↑_m 𝑋) = (ℝ ↑_m 𝑋)
149	147, 148	rrxdsfi 24775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ Fin → (dist‘(ℝ^‘𝑋)) = (𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
150	1, 149	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) = (𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
151	150, 68	eqtrd 2776	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) = 𝐷)
152	151	oveqd 7374	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹(dist‘(ℝ^‘𝑋))𝑔) = (𝐹𝐷𝑔))
153	152	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹(dist‘(ℝ^‘𝑋))𝑔) = (𝐹𝐷𝑔))
154	146, 153	breqtrd 5131	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹𝐷𝑔))
155	121, 133, 128, 134, 154	letrd 11312	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (𝐹𝐷𝑔))
156	155	3adantl3 1168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (𝐹𝐷𝑔))
157		simpl3 1193	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) < 𝐸)
158	122, 129, 131, 156, 157	lelttrd 11313	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸)
159		ltsub23 11635	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑖) ∈ ℝ ∧ (𝑔‘𝑖) ∈ ℝ ∧ 𝐸 ∈ ℝ) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖)))
160	119, 120, 130, 159	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖)))
161	160	3adantl3 1168	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖)))
162	158, 161	mpbid 231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖))
163	91, 95, 90, 118, 162	lelttrd 11313	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝑔‘𝑖))
164	21, 93	readdcld 11184	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ∈ ℝ)
165	164	3ad2antl1 1185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ∈ ℝ)
166	16	3ad2antl1 1185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ)
167	120, 119	resubcld 11583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
168	167	3adantl3 1168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
169	167	leabsd 15299	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝑔‘𝑖) − (𝐹‘𝑖))))
170	120	recnd 11183	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℂ)
171	119	recnd 11183	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℂ)
172	170, 171	abssubd 15338	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑔‘𝑖) − (𝐹‘𝑖))) = (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
173	169, 172	breqtrd 5131	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
174	167, 133, 128, 173, 154	letrd 11312	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (𝐹𝐷𝑔))
175	174	3adantl3 1168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (𝐹𝐷𝑔))
176	168, 129, 131, 175, 157	lelttrd 11313	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) < 𝐸)
177	119	3adantl3 1168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
178	90, 177, 131	ltsubadd2d 11753	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (((𝑔‘𝑖) − (𝐹‘𝑖)) < 𝐸 ↔ (𝑔‘𝑖) < ((𝐹‘𝑖) + 𝐸)))
179	176, 178	mpbid 231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) < ((𝐹‘𝑖) + 𝐸))
180		min1 13108	. . . . . . . . . . . . . 14 ⊢ ((((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ ∧ ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ≤ ((𝐵‘𝑖) − (𝐹‘𝑖)))
181	22, 32, 180	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ≤ ((𝐵‘𝑖) − (𝐹‘𝑖)))
182	93, 96, 22, 113, 181	letrd 11312	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑖) − (𝐹‘𝑖)))
183	21, 93, 16	leaddsub2d 11757	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖) ↔ 𝐸 ≤ ((𝐵‘𝑖) − (𝐹‘𝑖))))
184	182, 183	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖))
185	184	3ad2antl1 1185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖))
186	90, 165, 166, 179, 185	ltletrd 11315	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) < (𝐵‘𝑖))
187	82, 83, 90, 163, 186	eliood 43726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
188	187	ralrimiva 3143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
189	81, 188	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → (𝑔 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
190		vex 3449	. . . . . . 7 ⊢ 𝑔 ∈ V
191	190	elixp 8842	. . . . . 6 ⊢ (𝑔 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ (𝑔 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
192	189, 191	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → 𝑔 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
193	79, 74, 11, 60, 192	ballss3 43293	. . . 4 ⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
194	65, 193	eqsstrd 3982	. . 3 ⊢ (𝜑 → 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
195	78, 194	jca 512	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
196		eleq2 2826	. . . 4 ⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉))
197		sseq1 3969	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
198	196, 197	anbi12d 631	. . 3 ⊢ (𝑣 = 𝑉 → ((𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) ↔ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))))
199	198	rspcev 3581	. 2 ⊢ ((𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)) ∧ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
200	72, 195, 199	syl2anc 584	1 ⊢ (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 ifcif 4486 ∪ ciun 4954 class class class wbr 5105 ↦ cmpt 5188 Or wor 5544 ran crn 5634 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 ↑_m cmap 8765 Xcixp 8835 Fincfn 8883 infcinf 9377 ℝcr 11050 0cc0 11051 + caddc 11054 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 2c2 12208 ℝ⁺crp 12915 (,)cioo 13264 ↑cexp 13967 √csqrt 15118 abscabs 15119 Σcsu 15570 distcds 17142 TopOpenctopn 17303 PsMetcpsmet 20780 ∞Metcxmet 20781 Metcmet 20782 ballcbl 20783 MetOpencmopn 20786 ℝ^crrx 24747

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-prds 17329 df-pws 17331 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-field 20188 df-subrg 20220 df-staf 20304 df-srng 20305 df-lmod 20324 df-lss 20393 df-sra 20633 df-rgmod 20634 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-cnfld 20797 df-refld 21009 df-dsmm 21138 df-frlm 21153 df-top 22243 df-topon 22260 df-bases 22296 df-nm 23938 df-tng 23940 df-tcph 24533 df-rrx 24749

This theorem is referenced by: ioorrnopn 44536

W3C validator