ioorrnopnlem - Mathbox for Glauco Siliprandi

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Theorem ioorrnopnlem 46429

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 46428	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝑋)))
4		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑖𝜑
5		reex 11106	. . . . . . 7 ⊢ ℝ ∈ V
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
7		ioossre 13311	. . . . . . 7 ⊢ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ
8	7	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
9	4, 6, 8	ixpssmapc 45197	. . . . 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 3931	. . . 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 7025	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ)
17	10	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
18		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
19		fvixp2 45323	. . . . . . . . . . . . . . 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 3928	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
22	16, 21	resubcld 11554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
23		ioorrnopnlem.a	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
24	23	ffvelcdmda 7025	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ)
25	24	rexrd 11171	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
26	16	rexrd 11171	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
27		iooltub 45637	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐹‘𝑖) < (𝐵‘𝑖))
28	25, 26, 20, 27	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝐵‘𝑖))
29	21, 16	posdifd 11713	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) < (𝐵‘𝑖) ↔ 0 < ((𝐵‘𝑖) − (𝐹‘𝑖))))
30	28, 29	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐵‘𝑖) − (𝐹‘𝑖)))
31	22, 30	elrpd 12935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ⁺)
32	21, 24	resubcld 11554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ)
33		ioogtlb 45622	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐴‘𝑖) < (𝐹‘𝑖))
34	25, 26, 20, 33	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝐹‘𝑖))
35	24, 21	posdifd 11713	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖) < (𝐹‘𝑖) ↔ 0 < ((𝐹‘𝑖) − (𝐴‘𝑖))))
36	34, 35	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐹‘𝑖) − (𝐴‘𝑖)))
37	32, 36	elrpd 12935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ⁺)
38	31, 37	ifcld 4523	. . . . . . . . . 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 3125	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ ℝ⁺)
40		eqid 2733	. . . . . . . . . 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 7064	. . . . . . . . 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 3965	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ ℝ⁺)
44		ltso 11202	. . . . . . . . 9 ⊢ < Or ℝ
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → < Or ℝ)
46	40	rnmptfi 45295	. . . . . . . . . 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 2837	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Fin)
49		ioorrnopnlem.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ ∅)
50	4, 38, 40, 49	rnmptn0 6198	. . . . . . . . 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 2996	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ≠ ∅)
52		rpssre 12902	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ
53	52	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
54	43, 53	sstrd 3941	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
55		fiinfcl 9396	. . . . . . . 8 ⊢ (( < Or ℝ ∧ (𝐻 ∈ Fin ∧ 𝐻 ≠ ∅ ∧ 𝐻 ⊆ ℝ)) → inf(𝐻, ℝ, < ) ∈ 𝐻)
56	45, 48, 51, 54, 55	syl13anc 1374	. . . . . . 7 ⊢ (𝜑 → inf(𝐻, ℝ, < ) ∈ 𝐻)
57	43, 56	sseldd 3931	. . . . . 6 ⊢ (𝜑 → inf(𝐻, ℝ, < ) ∈ ℝ⁺)
58	12, 57	eqeltrid 2837	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
59		rpxr 12904	. . . . 5 ⊢ (𝐸 ∈ ℝ⁺ → 𝐸 ∈ ℝ^*)
60	58, 59	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
61		eqid 2733	. . . . 5 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
62	61	blopn 24418	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑_m 𝑋) ∧ 𝐸 ∈ ℝ^*) → (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷))
63	3, 11, 60, 62	syl3anc 1373	. . 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 46412	. . . . 5 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))
67	2	eqcomi 2742	. . . . . . 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 6834	. . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷))
70	66, 69	eqtrd 2768	. . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘𝐷))
71	65, 70	eleq12d 2827	. . 3 ⊢ (𝜑 → (𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)) ↔ (𝐹(ball‘𝐷)𝐸) ∈ (MetOpen‘𝐷)))
72	63, 71	mpbird 257	. 2 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)))
73		xmetpsmet 24266	. . . . . 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 24330	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘(ℝ ↑_m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑_m 𝑋) ∧ 𝐸 ∈ ℝ⁺) → 𝐹 ∈ (𝐹(ball‘𝐷)𝐸))
76	74, 11, 58, 75	syl3anc 1373	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐹(ball‘𝐷)𝐸))
77	65	eqcomd 2739	. . . 4 ⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) = 𝑉)
78	76, 77	eleqtrd 2835	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
79		nfv 1915	. . . . 5 ⊢ Ⅎ𝑔𝜑
80		elmapfn 8797	. . . . . . . 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 1186	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
83	26	3ad2antl1 1186	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
84		simpl2 1193	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝑔 ∈ (ℝ ↑_m 𝑋))
85		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
86		elmapi 8781	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑_m 𝑋) → 𝑔:𝑋⟶ℝ)
87	86	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑔:𝑋⟶ℝ)
88		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
89	87, 88	ffvelcdmd 7026	. . . . . . . . . 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 1186	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ)
92	52, 58	sselid 3928	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
93	92	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
94	21, 93	resubcld 11554	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) ∈ ℝ)
95	94	3ad2antl1 1186	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) ∈ ℝ)
96	52, 38	sselid 3928	. . . . . . . . . . . . 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 45507	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 ⊆ ℝ ∧ 𝐻 ∈ Fin ∧ 𝐻 ≠ ∅) → inf(𝐻, ℝ^*, < ) = inf(𝐻, ℝ, < ))
99	54, 48, 51, 98	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → inf(𝐻, ℝ^*, < ) = inf(𝐻, ℝ, < ))
100	99	eqcomd 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → inf(𝐻, ℝ, < ) = inf(𝐻, ℝ^*, < ))
101	97, 100	eqtrd 2768	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 = inf(𝐻, ℝ^*, < ))
102	101	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 = inf(𝐻, ℝ^*, < ))
103		ressxr 11165	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ ⊆ ℝ^*
104	103	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
105	54, 104	sstrd 3941	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐻 ⊆ ℝ^*)
106	105	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐻 ⊆ ℝ^*)
107	38	elexd 3461	. . . . . . . . . . . . . . . . 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 5906	. . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ 𝐻)
111		infxrlb 13238	. . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))
114		min2 13093	. . . . . . . . . . . . . 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 11279	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)))
117	93, 21, 24, 116	lesubd 11730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ ((𝐹‘𝑖) − 𝐸))
118	117	3ad2antl1 1186	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ ((𝐹‘𝑖) − 𝐸))
119	21	adantlr 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
120	89	adantll 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℝ)
121	119, 120	resubcld 11554	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℝ)
122	121	3adantl3 1169	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℝ)
123	1, 2	rrndsmet 46427	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝑋)))
124	123	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐷 ∈ (Met‘(ℝ ↑_m 𝑋)))
125	11	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ (ℝ ↑_m 𝑋))
126		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑔 ∈ (ℝ ↑_m 𝑋))
127		metcl 24250	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘(ℝ ↑_m 𝑋)) ∧ 𝐹 ∈ (ℝ ↑_m 𝑋) ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) → (𝐹𝐷𝑔) ∈ ℝ)
128	124, 125, 126, 127	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) ∈ ℝ)
129	128	3adantl3 1169	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) ∈ ℝ)
130	93	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
131	130	3adantl3 1169	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
132	121	recnd 11149	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℂ)
133	132	abscld 15350	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ∈ ℝ)
134	121	leabsd 15326	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
135	1	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑋 ∈ Fin)
136		ixpf 8852	. . . . . . . . . . . . . . . . . . 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 3125	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
139		iunss 4997	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ ↔ ∀𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
140	138, 139	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∪ 𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
141	137, 140	fssd 6675	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
142	141	ad2antrr 726	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
145		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
146	135, 142, 143, 144, 145	rrnprjdstle 46426	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹(dist‘(ℝ^‘𝑋))𝑔))
147		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
148		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ ↑_m 𝑋) = (ℝ ↑_m 𝑋)
149	147, 148	rrxdsfi 25341	. . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) = 𝐷)
152	151	oveqd 7371	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹(dist‘(ℝ^‘𝑋))𝑔) = (𝐹𝐷𝑔))
153	152	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹(dist‘(ℝ^‘𝑋))𝑔) = (𝐹𝐷𝑔))
154	146, 153	breqtrd 5121	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹𝐷𝑔))
155	121, 133, 128, 134, 154	letrd 11279	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (𝐹𝐷𝑔))
156	155	3adantl3 1169	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ≤ (𝐹𝐷𝑔))
157		simpl3 1194	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹𝐷𝑔) < 𝐸)
158	122, 129, 131, 156, 157	lelttrd 11280	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸)
159		ltsub23 11606	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖)))
161	160	3adantl3 1169	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸 ↔ ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖)))
162	158, 161	mpbid 232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 𝐸) < (𝑔‘𝑖))
163	91, 95, 90, 118, 162	lelttrd 11280	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝑔‘𝑖))
164	21, 93	readdcld 11150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ∈ ℝ)
165	164	3ad2antl1 1186	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ∈ ℝ)
166	16	3ad2antl1 1186	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ)
167	120, 119	resubcld 11554	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
168	167	3adantl3 1169	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
169	167	leabsd 15326	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝑔‘𝑖) − (𝐹‘𝑖))))
170	120	recnd 11149	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℂ)
171	119	recnd 11149	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℂ)
172	170, 171	abssubd 15367	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑔‘𝑖) − (𝐹‘𝑖))) = (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
173	169, 172	breqtrd 5121	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
174	167, 133, 128, 173, 154	letrd 11279	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (𝐹𝐷𝑔))
175	174	3adantl3 1169	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (𝐹𝐷𝑔))
176	168, 129, 131, 175, 157	lelttrd 11280	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) < 𝐸)
177	119	3adantl3 1169	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
178	90, 177, 131	ltsubadd2d 11724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (((𝑔‘𝑖) − (𝐹‘𝑖)) < 𝐸 ↔ (𝑔‘𝑖) < ((𝐹‘𝑖) + 𝐸)))
179	176, 178	mpbid 232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) < ((𝐹‘𝑖) + 𝐸))
180		min1 13092	. . . . . . . . . . . . . 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 11279	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑖) − (𝐹‘𝑖)))
183	21, 93, 16	leaddsub2d 11728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖) ↔ 𝐸 ≤ ((𝐵‘𝑖) − (𝐹‘𝑖))))
184	182, 183	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖))
185	184	3ad2antl1 1186	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 𝐸) ≤ (𝐵‘𝑖))
186	90, 165, 166, 179, 185	ltletrd 11282	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) < (𝐵‘𝑖))
187	82, 83, 90, 163, 186	eliood 45625	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
188	187	ralrimiva 3125	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
189	81, 188	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → (𝑔 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
190		vex 3441	. . . . . . 7 ⊢ 𝑔 ∈ V
191	190	elixp 8836	. . . . . 6 ⊢ (𝑔 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ (𝑔 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑔‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
192	189, 191	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) → 𝑔 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
193	79, 74, 11, 60, 192	ballss3 45217	. . . 4 ⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
194	65, 193	eqsstrd 3965	. . 3 ⊢ (𝜑 → 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
195	78, 194	jca 511	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
196		eleq2 2822	. . . 4 ⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉))
197		sseq1 3956	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
198	196, 197	anbi12d 632	. . 3 ⊢ (𝑣 = 𝑉 → ((𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) ↔ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))))
199	198	rspcev 3573	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 ∃wrex 3057 Vcvv 3437 ⊆ wss 3898 ∅c0 4282 ifcif 4476 ∪ ciun 4943 class class class wbr 5095 ↦ cmpt 5176 Or wor 5528 ran crn 5622 Fn wfn 6483 ⟶wf 6484 ‘cfv 6488 (class class class)co 7354 ∈ cmpo 7356 ↑_m cmap 8758 Xcixp 8829 Fincfn 8877 infcinf 9334 ℝcr 11014 0cc0 11015 + caddc 11018 ℝ^*cxr 11154 < clt 11155 ≤ cle 11156 − cmin 11353 2c2 12189 ℝ⁺crp 12894 (,)cioo 13249 ↑cexp 13972 √csqrt 15144 abscabs 15145 Σcsu 15597 distcds 17174 TopOpenctopn 17329 PsMetcpsmet 21279 ∞Metcxmet 21280 Metcmet 21281 ballcbl 21282 MetOpencmopn 21285 ℝ^crrx 25313

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-inf2 9540 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092 ax-pre-sup 11093 ax-addf 11094 ax-mulf 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7618 df-om 7805 df-1st 7929 df-2nd 7930 df-supp 8099 df-tpos 8164 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-er 8630 df-map 8760 df-ixp 8830 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-fsupp 9255 df-sup 9335 df-inf 9336 df-oi 9405 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-div 11784 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-5 12200 df-6 12201 df-7 12202 df-8 12203 df-9 12204 df-n0 12391 df-z 12478 df-dec 12597 df-uz 12741 df-q 12851 df-rp 12895 df-xneg 13015 df-xadd 13016 df-xmul 13017 df-ioo 13253 df-ico 13255 df-fz 13412 df-fzo 13559 df-seq 13913 df-exp 13973 df-hash 14242 df-cj 15010 df-re 15011 df-im 15012 df-sqrt 15146 df-abs 15147 df-clim 15399 df-sum 15598 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17125 df-ress 17146 df-plusg 17178 df-mulr 17179 df-starv 17180 df-sca 17181 df-vsca 17182 df-ip 17183 df-tset 17184 df-ple 17185 df-ds 17187 df-unif 17188 df-hom 17189 df-cco 17190 df-rest 17330 df-topn 17331 df-0g 17349 df-gsum 17350 df-topgen 17351 df-prds 17355 df-pws 17357 df-mgm 18552 df-sgrp 18631 df-mnd 18647 df-mhm 18695 df-grp 18853 df-minusg 18854 df-sbg 18855 df-subg 19040 df-ghm 19129 df-cntz 19233 df-cmn 19698 df-abl 19699 df-mgp 20063 df-rng 20075 df-ur 20104 df-ring 20157 df-cring 20158 df-oppr 20259 df-dvdsr 20279 df-unit 20280 df-invr 20310 df-dvr 20323 df-rhm 20394 df-subrng 20465 df-subrg 20489 df-drng 20650 df-field 20651 df-staf 20758 df-srng 20759 df-lmod 20799 df-lss 20869 df-sra 21111 df-rgmod 21112 df-psmet 21287 df-xmet 21288 df-met 21289 df-bl 21290 df-mopn 21291 df-cnfld 21296 df-refld 21546 df-dsmm 21673 df-frlm 21688 df-top 22812 df-topon 22829 df-bases 22864 df-nm 24500 df-tng 24502 df-tcph 25099 df-rrx 25315

This theorem is referenced by: ioorrnopn 46430

W3C validator