ioorrnopnlem - Mathbox for Glauco Siliprandi

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Theorem ioorrnopnlem 46302

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 46301	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝑋)))
4		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑖𝜑
5		reex 11159	. . . . . . 7 ⊢ ℝ ∈ V
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
7		ioossre 13368	. . . . . . 7 ⊢ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ
8	7	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ)
9	4, 6, 8	ixpssmapc 45067	. . . . 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 3947	. . . 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 7056	. . . . . . . . . . . . 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 45193	. . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
22	16, 21	resubcld 11606	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ)
23		ioorrnopnlem.a	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
24	23	ffvelcdmda 7056	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ)
25	24	rexrd 11224	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
26	16	rexrd 11224	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
27		iooltub 45508	. . . . . . . . . . . . . 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 11765	. . . . . . . . . . . . 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 12992	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐵‘𝑖) − (𝐹‘𝑖)) ∈ ℝ⁺)
32	21, 24	resubcld 11606	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝐴‘𝑖)) ∈ ℝ)
33		ioogtlb 45493	. . . . . . . . . . . . . 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 11765	. . . . . . . . . . . . 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 12992	. . . . . . . . . . 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 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 2729	. . . . . . . . . 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 7095	. . . . . . . . 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 3981	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ ℝ⁺)
44		ltso 11254	. . . . . . . . 9 ⊢ < Or ℝ
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → < Or ℝ)
46	40	rnmptfi 45165	. . . . . . . . . 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 2832	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Fin)
49		ioorrnopnlem.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ ∅)
50	4, 38, 40, 49	rnmptn0 6217	. . . . . . . . 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 2992	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ≠ ∅)
52		rpssre 12959	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ
53	52	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
54	43, 53	sstrd 3957	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
55		fiinfcl 9454	. . . . . . . 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 3947	. . . . . 6 ⊢ (𝜑 → inf(𝐻, ℝ, < ) ∈ ℝ⁺)
58	12, 57	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
59		rpxr 12961	. . . . 5 ⊢ (𝐸 ∈ ℝ⁺ → 𝐸 ∈ ℝ^*)
60	58, 59	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
61		eqid 2729	. . . . 5 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
62	61	blopn 24388	. . . 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 46285	. . . . 5 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))
67	2	eqcomi 2738	. . . . . . 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 6862	. . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝑋), 𝑔 ∈ (ℝ ↑_m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷))
70	66, 69	eqtrd 2764	. . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘𝐷))
71	65, 70	eleq12d 2822	. . 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 24236	. . . . . 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 24300	. . . . 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 2735	. . . 4 ⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) = 𝑉)
78	76, 77	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
79		nfv 1914	. . . . 5 ⊢ Ⅎ𝑔𝜑
80		elmapfn 8838	. . . . . . . 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 8822	. . . . . . . . . . . 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 7057	. . . . . . . . . 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 3944	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
93	92	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ∈ ℝ)
94	21, 93	resubcld 11606	. . . . . . . . . . 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 3944	. . . . . . . . . . . . 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 45378	. . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → inf(𝐻, ℝ, < ) = inf(𝐻, ℝ^*, < ))
101	97, 100	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 = inf(𝐻, ℝ^*, < ))
102	101	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 = inf(𝐻, ℝ^*, < ))
103		ressxr 11218	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ ⊆ ℝ^*
104	103	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
105	54, 104	sstrd 3957	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐻 ⊆ ℝ^*)
106	105	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐻 ⊆ ℝ^*)
107	38	elexd 3471	. . . . . . . . . . . . . . . . 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 5924	. . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))) ∈ 𝐻)
111		infxrlb 13295	. . . . . . . . . . . . . . 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 5129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))
114		min2 13150	. . . . . . . . . . . . . 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 11331	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)))
117	93, 21, 24, 116	lesubd 11782	. . . . . . . . . . 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 11606	. . . . . . . . . . . . 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 46300	. . . . . . . . . . . . . . 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 24220	. . . . . . . . . . . . . 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 11202	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) ∈ ℂ)
133	132	abscld 15405	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ∈ ℝ)
134	121	leabsd 15381	. . . . . . . . . . . . . 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 8893	. . . . . . . . . . . . . . . . . . 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 5009	. . . . . . . . . . . . . . . . . . 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 6705	. . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
146	135, 142, 143, 144, 145	rrnprjdstle 46299	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹(dist‘(ℝ^‘𝑋))𝑔))
147		eqid 2729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
148		eqid 2729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ ↑_m 𝑋) = (ℝ ↑_m 𝑋)
149	147, 148	rrxdsfi 25311	. . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) = 𝐷)
152	151	oveqd 7404	. . . . . . . . . . . . . . . 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 5133	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))) ≤ (𝐹𝐷𝑔))
155	121, 133, 128, 134, 154	letrd 11331	. . . . . . . . . . . . 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 11332	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − (𝑔‘𝑖)) < 𝐸)
159		ltsub23 11658	. . . . . . . . . . . . 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 11332	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝑔‘𝑖))
164	21, 93	readdcld 11203	. . . . . . . . . . 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 11606	. . . . . . . . . . . . 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 15381	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝑔‘𝑖) − (𝐹‘𝑖))))
170	120	recnd 11202	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) ∈ ℂ)
171	119	recnd 11202	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℂ)
172	170, 171	abssubd 15422	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑔‘𝑖) − (𝐹‘𝑖))) = (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
173	169, 172	breqtrd 5133	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ((𝑔‘𝑖) − (𝐹‘𝑖)) ≤ (abs‘((𝐹‘𝑖) − (𝑔‘𝑖))))
174	167, 133, 128, 173, 154	letrd 11331	. . . . . . . . . . . . 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 11332	. . . . . . . . . . 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 11776	. . . . . . . . . . 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 13149	. . . . . . . . . . . . . 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 11331	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑖) − (𝐹‘𝑖)))
183	21, 93, 16	leaddsub2d 11780	. . . . . . . . . . . 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 11334	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (ℝ ↑_m 𝑋) ∧ (𝐹𝐷𝑔) < 𝐸) ∧ 𝑖 ∈ 𝑋) → (𝑔‘𝑖) < (𝐵‘𝑖))
187	82, 83, 90, 163, 186	eliood 45496	. . . . . . . 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 3451	. . . . . . 7 ⊢ 𝑔 ∈ V
191	190	elixp 8877	. . . . . 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 45087	. . . 4 ⊢ (𝜑 → (𝐹(ball‘𝐷)𝐸) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
194	65, 193	eqsstrd 3981	. . 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 2817	. . . 4 ⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉))
197		sseq1 3972	. . . 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 3588	. 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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ⊆ wss 3914 ∅c0 4296 ifcif 4488 ∪ ciun 4955 class class class wbr 5107 ↦ cmpt 5188 Or wor 5545 ran crn 5639 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 ↑_m cmap 8799 Xcixp 8870 Fincfn 8918 infcinf 9392 ℝcr 11067 0cc0 11068 + caddc 11071 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 2c2 12241 ℝ⁺crp 12951 (,)cioo 13306 ↑cexp 14026 √csqrt 15199 abscabs 15200 Σcsu 15652 distcds 17229 TopOpenctopn 17384 PsMetcpsmet 21248 ∞Metcxmet 21249 Metcmet 21250 ballcbl 21251 MetOpencmopn 21254 ℝ^crrx 25283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ico 13312 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-prds 17410 df-pws 17412 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-drng 20640 df-field 20641 df-staf 20748 df-srng 20749 df-lmod 20768 df-lss 20838 df-sra 21080 df-rgmod 21081 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-cnfld 21265 df-refld 21514 df-dsmm 21641 df-frlm 21656 df-top 22781 df-topon 22798 df-bases 22833 df-nm 24470 df-tng 24472 df-tcph 25069 df-rrx 25285

This theorem is referenced by: ioorrnopn 46303

W3C validator