ioorrnopnlem - Mathbox for Glauco Siliprandi

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Theorem ioorrnopnlem 47001

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 47000	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝑋)))
4		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑖𝜑
5		reex 11192	. . . . . . 7 ⊢ ℝ ∈ V
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
7		ioossre 13435	. . . . . . 7 ⊢ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ
8	7	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
9	4, 6, 8	ixpssmapc 45776	. . . . 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 3939	. . . 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 7081	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ)
17	10	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
18		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
19		fvixp2 45899	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
20	17, 18, 19	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
21	7, 20	sselid 3936	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
22	16, 21	resubcld 11643	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
23		ioorrnopnlem.a	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
24	23	ffvelcdmda 7081	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ)
25	24	rexrd 11260	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
26	16	rexrd 11260	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
27		iooltub 46209	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐹‘𝑖) < (𝐵‘𝑖))
28	25, 26, 20, 27	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝐵‘𝑖))
29	21, 16	posdifd 11802	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) < (𝐵‘𝑖) ↔ 0 < ((𝐵‘𝑖) − (𝐹‘𝑖))))
30	28, 29	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐵‘𝑖) − (𝐹‘𝑖)))
31	22, 30	elrpd 13058	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ⁺)
32	21, 24	resubcld 11643	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ)
33		ioogtlb 46194	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐴‘𝑖) < (𝐹‘𝑖))
34	25, 26, 20, 33	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝐹‘𝑖))
35	24, 21	posdifd 11802	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖) < (𝐹‘𝑖) ↔ 0 < ((𝐹‘𝑖) − (𝐴‘𝑖))))
36	34, 35	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐹‘𝑖) − (𝐴‘𝑖)))
37	32, 36	elrpd 13058	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ⁺)
38	31, 37	ifcld 4535	. . . . . . . . . 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 3157	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ℝ⁺)
40		eqid 2763	. . . . . . . . . 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 7120	. . . . . . . . 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 18	. . . . . . . 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 3972	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ ℝ⁺)
44		ltso 11291	. . . . . . . . 9 ⊢ < Or ℝ
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → < Or ℝ)
46	40	rnmptfi 45872	. . . . . . . . . 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 18	. . . . . . . . 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 2867	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Fin)
49		ioorrnopnlem.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ ∅)
50	4, 38, 40, 49	rnmptn0 6247	. . . . . . . . 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 3025	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ≠ ∅)
52		rpssre 13025	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ
53	52	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
54	43, 53	sstrd 3948	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
55		fiinfcl 9464	. . . . . . . 8 ⊢ (( < Or ℝ ∧ (𝐻 ∈ Fin ∧ 𝐻 ≠ ∅ ∧ 𝐻 ⊆ ℝ)) → inf(𝐻, ℝ, < ) ∈ 𝐻)
56	45, 48, 51, 54, 55	syl13anc 1399	. . . . . . 7 ⊢ (𝜑 → inf(𝐻, ℝ, < ) ∈ 𝐻)
57	43, 56	sseldd 3939	. . . . . 6 ⊢ (𝜑 → inf(𝐻, ℝ, < ) ∈ ℝ⁺)
58	12, 57	eqeltrid 2867	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
59		rpxr 13027	. . . . 5 ⊢ (𝐸 ∈ ℝ⁺ → 𝐸 ∈ ℝ^*)
60	58, 59	syl 18	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
61		eqid 2763	. . . . 5 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
62	61	blopn 24638	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑_m 𝑋) ∧ 𝐸 ∈ ℝ^*) → (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷))
63	3, 11, 60, 62	syl3anc 1398	. . 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 46984	. . . . 5 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))
67	2	eqcomi 2772	. . . . . . 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 6887	. . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷))
70	66, 69	eqtrd 2798	. . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘𝐷))
71	65, 70	eleq12d 2857	. . 3 ⊢ (𝜑 → (𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)) ↔ (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷)))
72	63, 71	mpbird 260	. 2 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)))
73		xmetpsmet 24486	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘(ℝ ↑_m 𝑋)) → 𝐷 ∈ (PsMet‘(ℝ ↑_m 𝑋)))
74	3, 73	syl 18	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (PsMet‘(ℝ ↑_m 𝑋)))
75		blcntrps 24550	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘(ℝ ↑_m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑_m 𝑋) ∧ 𝐸 ∈ ℝ⁺) → 𝐹 ∈ (𝐹(ball‘𝐷)𝐸))
76	74, 11, 58, 75	syl3anc 1398	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐹(ball‘𝐷)𝐸))
77	65	eqcomd 2769	. . . 4 ⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) = 𝑉)
78	76, 77	eleqtrd 2865	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
79		nfv 1944	. . . . 5 ⊢ Ⅎ𝑔𝜑
80		elmapfn 8863	. . . . . . . 8 ⊢ (𝑔 ∈ (ℝ ↑_m 𝑋) → 𝑔 Fn 𝑋)
81	80	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → 𝑔 Fn 𝑋)
82	25	3ad2antl1 1204	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
83	26	3ad2antl1 1204	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
84		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝑔 ∈ (ℝ ↑_m 𝑋))
85		simpr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
86		elmapi 8847	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑_m 𝑋) → 𝑔:𝑋⟶ℝ)
87	86	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑔:𝑋⟶ℝ)
88		simpr 489	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
89	87, 88	ffvelcdmd 7082	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ)
90	84, 85, 89	syl2anc 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ)
91	24	3ad2antl1 1204	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ)
92	52, 58	sselid 3936	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
93	92	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
94	21, 93	resubcld 11643	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) ∈ ℝ)
95	94	3ad2antl1 1204	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) ∈ ℝ)
96	52, 38	sselid 3936	. . . . . . . . . . . . 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 46080	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 ⊆ ℝ ∧ 𝐻 ∈ Fin ∧ 𝐻 ≠ ∅) → inf(𝐻, ℝ^*, < ) = inf(𝐻, ℝ, < ))
99	54, 48, 51, 98	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → inf(𝐻, ℝ^*, < ) = inf(𝐻, ℝ, < ))
100	99	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → inf(𝐻, ℝ, < ) = inf(𝐻, ℝ^*, < ))
101	97, 100	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 = inf(𝐻, ℝ^*, < ))
102	101	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 = inf(𝐻, ℝ^*, < ))
103		ressxr 11254	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ ⊆ ℝ^*
104	103	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
105	54, 104	sstrd 3948	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐻 ⊆ ℝ^*)
106	105	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐻 ⊆ ℝ^*)
107	38	elexd 3478	. . . . . . . . . . . . . . . . 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 5952	. . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . 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 2874	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ 𝐻)
111		infxrlb 13362	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 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 5134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))
114		min2 13217	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 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 11368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)))
117	93, 21, 24, 116	lesubd 11819	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ ((𝐹‘𝑖) − 𝐸))
118	117	3ad2antl1 1204	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ ((𝐹‘𝑖) − 𝐸))
119	21	adantlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
120	89	adantll 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ)
121	119, 120	resubcld 11643	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℝ)
122	121	3adantl3 1187	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℝ)
123	1, 2	rrndsmet 46999	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝑋)))
124	123	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐷 ∈ (Met‘(ℝ ↑_m 𝑋)))
125	11	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ (ℝ ↑_m 𝑋))
126		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑔 ∈ (ℝ ↑_m 𝑋))
127		metcl 24470	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘(ℝ ↑_m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑_m 𝑋) ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) → (𝐹𝐷𝑔) ∈ ℝ)
128	124, 125, 126, 127	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) ∈ ℝ)
129	128	3adantl3 1187	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) ∈ ℝ)
130	93	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
131	130	3adantl3 1187	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
132	121	recnd 11238	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℂ)
133	132	abscld 15492	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ∈ ℝ)
134	121	leabsd 15468	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
135	1	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑋 ∈ Fin)
136		ixpf 8919	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝐹:𝑋⟶∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
137	10, 136	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
138	8	ralrimiva 3157	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
139		iunss 5010	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ ↔ ∀𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
140	138, 139	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
141	137, 140	fssd 6725	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
142	141	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐹:𝑋⟶ℝ)
143	126, 86	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑔:𝑋⟶ℝ)
144		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
145		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
146	135, 142, 143, 144, 145	rrnprjdstle 46998	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹(dist‘(ℝ^‘𝑋))𝑔))
147		eqid 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
148		eqid 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ ↑_m 𝑋) = (ℝ ↑_m 𝑋)
149	147, 148	rrxdsfi 25551	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) = (𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
151	150, 68	eqtrd 2798	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) = 𝐷)
152	151	oveqd 7429	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹(dist‘(ℝ^‘𝑋))𝑔) = (𝐹𝐷𝑔))
153	152	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹(dist‘(ℝ^‘𝑋))𝑔) = (𝐹𝐷𝑔))
154	146, 153	breqtrd 5138	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹𝐷𝑔))
155	121, 133, 128, 134, 154	letrd 11368	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (𝐹𝐷𝑔))
156	155	3adantl3 1187	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (𝐹𝐷𝑔))
157		simpl3 1212	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) < 𝐸)
158	122, 129, 131, 156, 157	lelttrd 11369	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸)
159		ltsub23 11695	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖)))
161	160	3adantl3 1187	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖)))
162	158, 161	mpbid 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖))
163	91, 95, 90, 118, 162	lelttrd 11369	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝑔‘𝑖))
164	21, 93	readdcld 11239	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ∈ ℝ)
165	164	3ad2antl1 1204	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ∈ ℝ)
166	16	3ad2antl1 1204	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ)
167	120, 119	resubcld 11643	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
168	167	3adantl3 1187	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
169	167	leabsd 15468	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝑔‘𝑖) − (𝐹‘𝑖))))
170	120	recnd 11238	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℂ)
171	119	recnd 11238	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℂ)
172	170, 171	abssubd 15509	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑔‘𝑖) − (𝐹‘𝑖))) = (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
173	169, 172	breqtrd 5138	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
174	167, 133, 128, 173, 154	letrd 11368	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (𝐹𝐷𝑔))
175	174	3adantl3 1187	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (𝐹𝐷𝑔))
176	168, 129, 131, 175, 157	lelttrd 11369	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) < 𝐸)
177	119	3adantl3 1187	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
178	90, 177, 131	ltsubadd2d 11813	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (((𝑔‘𝑖) − (𝐹‘𝑖)) < 𝐸 ↔ (𝑔‘𝑖) < ((𝐹‘𝑖) + 𝐸)))
179	176, 178	mpbid 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) < ((𝐹‘𝑖) + 𝐸))
180		min1 13216	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 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 11368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑖) − (𝐹‘𝑖)))
183	21, 93, 16	leaddsub2d 11817	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖) ↔ 𝐸 ≤ ((𝐵‘𝑖) − (𝐹‘𝑖))))
184	182, 183	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖))
185	184	3ad2antl1 1204	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖))
186	90, 165, 166, 179, 185	ltletrd 11371	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) < (𝐵‘𝑖))
187	82, 83, 90, 163, 186	eliood 46197	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
188	187	ralrimiva 3157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
189	81, 188	jca 520	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → (𝑔 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
190		vex 3459	. . . . . . 7 ⊢ 𝑔 ∈ V
191	190	elixp 8903	. . . . . 6 ⊢ (𝑔 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ (𝑔 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
192	189, 191	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → 𝑔 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
193	79, 74, 11, 60, 192	ballss3 45794	. . . 4 ⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
194	65, 193	eqsstrd 3972	. . 3 ⊢ (𝜑 → 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
195	78, 194	jca 520	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
196		eleq2 2852	. . . 4 ⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉))
197		sseq1 3963	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
198	196, 197	anbi12d 643	. . 3 ⊢ (𝑣 = 𝑉 → ((𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) ↔ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))))
199	198	rspcev 3582	. 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 595	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 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ⊆ wss 3906 ∅c0 4287 ifcif 4488 ∪ ciun 4957 class class class wbr 5110 ↦ cmpt 5193 Or wor 5570 ran crn 5664 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 ∈ cmpo 7414 ↑_m cmap 8825 Xcixp 8896 Fincfn 8944 infcinf 9402 ℝcr 11100 0cc0 11101 + caddc 11104 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11442 2c2 12296 ℝ⁺crp 13017 (,)cioo 13373 ↑cexp 14099 √csqrt 15286 abscabs 15287 Σcsu 15739 distcds 17320 TopOpenctopn 17475 PsMetcpsmet 21487 ∞Metcxmet 21488 Metcmet 21489 ballcbl 21490 MetOpencmopn 21493 ℝ^crrx 25523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180 ax-mulf 11181

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-tpos 8223 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ico 13379 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-prds 17501 df-pws 17503 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-mhm 18842 df-grp 19004 df-minusg 19005 df-sbg 19006 df-subg 19190 df-ghm 19285 df-cntz 19388 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-cring 20319 df-oppr 20420 df-dvdsr 20440 df-unit 20441 df-invr 20471 df-dvr 20484 df-rhm 20555 df-subrng 20632 df-subrg 20656 df-drng 20816 df-field 20817 df-staf 20923 df-srng 20924 df-lmod 20964 df-lss 21034 df-sra 21275 df-rgmod 21276 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-cnfld 21504 df-refld 21736 df-dsmm 21863 df-frlm 21878 df-top 23032 df-topon 23049 df-bases 23084 df-nm 24720 df-tng 24722 df-tcph 25309 df-rrx 25525

This theorem is referenced by: ioorrnopn 47002

W3C validator