Theorem prdsbl 24469

Description: A ball in the product metric for finite index set is the Cartesian product of balls in all coordinates. For infinite index set this is no longer true; instead the correct statement is that a *closed ball* is the product of closed balls in each coordinate (where closed ball means a set of the form in blcld 24483) - for a counterexample the point 𝑝 in ℝ↑ℕ whose 𝑛-th coordinate is 1 − 1 / 𝑛 is in X𝑛 ∈ ℕball(0, 1) but is not in the 1-ball of the product (since 𝑑(0, 𝑝) = 1).

The last assumption, 0 < 𝐴, is needed only in the case 𝐼 = ∅, when the right side evaluates to {∅} and the left evaluates to ∅ if 𝐴 ≤ 0 and {∅} if 0 < 𝐴. (Contributed by Mario Carneiro, 28-Aug-2015.)

Hypotheses

Ref	Expression
prdsbl.y	⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅))
prdsbl.b	⊢ 𝐵 = (Base‘𝑌)
prdsbl.v	⊢ 𝑉 = (Base‘𝑅)
prdsbl.e	⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
prdsbl.d	⊢ 𝐷 = (dist‘𝑌)
prdsbl.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
prdsbl.i	⊢ (𝜑 → 𝐼 ∈ Fin)
prdsbl.r	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)
prdsbl.m	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
prdsbl.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
prdsbl.a	⊢ (𝜑 → 𝐴 ∈ ℝ*)
prdsbl.g	⊢ (𝜑 → 0 < 𝐴)

Assertion

Ref	Expression
prdsbl	⊢ (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐼   𝑥,𝑃   𝜑,𝑥

Allowed substitution hints:   𝑅(𝑥)   𝑆(𝑥)   𝐸(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑌(𝑥)   𝑍(𝑥)

Proof of Theorem prdsbl

Dummy variables 𝑓 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsbl.y	. . . . . . . . 9 ⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅))
2		prdsbl.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑌)
3		prdsbl.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
4		prdsbl.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ Fin)
5		prdsbl.r	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)
6	5	ralrimiva 3130	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍)
7		prdsbl.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑅)
8	1, 2, 3, 4, 6, 7	prdsbas3 17438	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝑉)
9	8	eleq2d 2823	. . . . . . 7 ⊢ (𝜑 → (𝑓 ∈ 𝐵 ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉))
10	9	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉)
11		ixpfn 8845	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐼 𝑉 → 𝑓 Fn 𝐼)
12		vex 3434	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	elixp 8846	. . . . . . 7 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ (𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
14	13	baib 535	. . . . . 6 ⊢ (𝑓 Fn 𝐼 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
15	10, 11, 14	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
16		prdsbl.m	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
17	16	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
18		prdsbl.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
19	18	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ ℝ*)
20		prdsbl.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
21	1, 2, 3, 4, 6, 7, 20	prdsbascl 17440	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉)
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉)
23	22	r19.21bi 3230	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) ∈ 𝑉)
24	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑆 ∈ 𝑊)
25	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝐼 ∈ Fin)
26	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍)
27		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵)
28	1, 2, 24, 25, 26, 7, 27	prdsbascl 17440	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
29	28	r19.21bi 3230	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉)
30		elbl2 24368	. . . . . . 7 ⊢ (((𝐸 ∈ (∞Met‘𝑉) ∧ 𝐴 ∈ ℝ*) ∧ ((𝑃‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉)) → ((𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
31	17, 19, 23, 29, 30	syl22anc 839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
32	31	ralbidva 3159	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
33		xmetcl 24309	. . . . . . . . . 10 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ*)
34	17, 23, 29, 33	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ*)
35	34	ralrimiva 3130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ*)
36		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))
37		breq1 5089	. . . . . . . . 9 ⊢ (𝑧 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) → (𝑧 < 𝐴 ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
38	36, 37	ralrnmptw 7041	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ* → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
39	35, 38	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
40		prdsbl.g	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐴)
41	40	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 0 < 𝐴)
42		c0ex 11132	. . . . . . . . . 10 ⊢ 0 ∈ V
43		breq1 5089	. . . . . . . . . 10 ⊢ (𝑧 = 0 → (𝑧 < 𝐴 ↔ 0 < 𝐴))
44	42, 43	ralsn 4626	. . . . . . . . 9 ⊢ (∀𝑧 ∈ {0}𝑧 < 𝐴 ↔ 0 < 𝐴)
45	41, 44	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑧 ∈ {0}𝑧 < 𝐴)
46		ralunb 4138	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})𝑧 < 𝐴 ↔ (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 ∧ ∀𝑧 ∈ {0}𝑧 < 𝐴))
47	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑃 ∈ 𝐵)
48		prdsbl.e	. . . . . . . . . . . 12 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
49		prdsbl.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘𝑌)
50	1, 2, 24, 25, 26, 47, 27, 7, 48, 49	prdsdsval3 17442	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < ))
51		xrltso 13086	. . . . . . . . . . . . 13 ⊢ < Or ℝ*
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → < Or ℝ*)
53	36	rnmpt 5907	. . . . . . . . . . . . . . 15 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))}
54		abrexfi 9256	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ Fin → {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))} ∈ Fin)
55	53, 54	eqeltrid 2841	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ Fin → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∈ Fin)
56	25, 55	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∈ Fin)
57		snfi 8984	. . . . . . . . . . . . 13 ⊢ {0} ∈ Fin
58		unfi 9099	. . . . . . . . . . . . 13 ⊢ ((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∈ Fin ∧ {0} ∈ Fin) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ∈ Fin)
59	56, 57, 58	sylancl 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ∈ Fin)
60		ssun2 4120	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
61	42	snss 4729	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ↔ {0} ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
62	60, 61	mpbir 231	. . . . . . . . . . . . 13 ⊢ 0 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
63		ne0i 4282	. . . . . . . . . . . . 13 ⊢ (0 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ≠ ∅)
64	62, 63	mp1i 13	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ≠ ∅)
65	34	fmpttd 7062	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))):𝐼⟶ℝ*)
66	65	frnd 6671	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆ ℝ*)
67		0xr 11186	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ*
68	67	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 0 ∈ ℝ*)
69	68	snssd 4753	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → {0} ⊆ ℝ*)
70	66, 69	unssd 4133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ*)
71		fisupcl 9377	. . . . . . . . . . . 12 ⊢ (( < Or ℝ* ∧ ((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ∈ Fin ∧ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ≠ ∅ ∧ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ*)) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < ) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
72	52, 59, 64, 70, 71	syl13anc 1375	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < ) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
73	50, 72	eqeltrd 2837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
74		breq1 5089	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑃𝐷𝑓) → (𝑧 < 𝐴 ↔ (𝑃𝐷𝑓) < 𝐴))
75	74	rspcv 3561	. . . . . . . . . 10 ⊢ ((𝑃𝐷𝑓) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) → (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴))
76	73, 75	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴))
77	46, 76	biimtrrid 243	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 ∧ ∀𝑧 ∈ {0}𝑧 < 𝐴) → (𝑃𝐷𝑓) < 𝐴))
78	45, 77	mpan2d 695	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴))
79	39, 78	sylbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴 → (𝑃𝐷𝑓) < 𝐴))
80		ssun1 4119	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
81		ovex 7394	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ V
82	81	elabrex 7191	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐼 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))})
83	82	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))})
84	83, 53	eleqtrrdi 2848	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))))
85	80, 84	sselid 3920	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
86		supxrub 13270	. . . . . . . . . 10 ⊢ (((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ* ∧ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < ))
87	70, 85, 86	syl2an2r 686	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < ))
88	50	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < ))
89	87, 88	breqtrrd 5114	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓))
90	1, 2, 7, 48, 49, 3, 4, 5, 16	prdsxmet 24347	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
91	90	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐷 ∈ (∞Met‘𝐵))
92	20	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ 𝐵)
93	27	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑓 ∈ 𝐵)
94		xmetcl 24309	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈ ℝ*)
95	91, 92, 93, 94	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) ∈ ℝ*)
96		xrlelttr 13101	. . . . . . . . 9 ⊢ ((((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ* ∧ (𝑃𝐷𝑓) ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → ((((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓) ∧ (𝑃𝐷𝑓) < 𝐴) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
97	34, 95, 19, 96	syl3anc 1374	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓) ∧ (𝑃𝐷𝑓) < 𝐴) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
98	89, 97	mpand 696	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃𝐷𝑓) < 𝐴 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
99	98	ralrimdva 3138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝑃𝐷𝑓) < 𝐴 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
100	79, 99	impbid 212	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴 ↔ (𝑃𝐷𝑓) < 𝐴))
101	15, 32, 100	3bitrrd 306	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝑃𝐷𝑓) < 𝐴 ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
102	101	pm5.32da 579	. . 3 ⊢ (𝜑 → ((𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴) ↔ (𝑓 ∈ 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))))
103		elbl 24366	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝐴 ∈ ℝ*) → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴)))
104	90, 20, 18, 103	syl3anc 1374	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴)))
105	21	r19.21bi 3230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) ∈ 𝑉)
106	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ ℝ*)
107		blssm 24396	. . . . . . . . 9 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ 𝐴 ∈ ℝ*) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉)
108	16, 105, 106, 107	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉)
109	108	ralrimiva 3130	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉)
110		ss2ixp 8852	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉 → X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ X𝑥 ∈ 𝐼 𝑉)
111	109, 110	syl 17	. . . . . 6 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ X𝑥 ∈ 𝐼 𝑉)
112	111, 8	sseqtrrd 3960	. . . . 5 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝐵)
113	112	sseld 3921	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) → 𝑓 ∈ 𝐵))
114	113	pm4.71rd 562	. . 3 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))))
115	102, 104, 114	3bitr4d 311	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
116	115	eqrdv 2735	1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  {csn 4568   class class class wbr 5086   ↦ cmpt 5167   Or wor 5532   × cxp 5623  ran crn 5626   ↾ cres 5627   Fn wfn 6488  ‘cfv 6493  (class class class)co 7361  Xcixp 8839  Fincfn 8887  supcsup 9347  0cc0 11032  ℝ^*cxr 11172   < clt 11173   ≤ cle 11174  Basecbs 17173  distcds 17223  X_scprds 17402  ∞Metcxmet 21332  ballcbl 21334

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-rp 12937  df-xneg 13057  df-xadd 13058  df-xmul 13059  df-icc 13299  df-fz 13456  df-struct 17111  df-slot 17146  df-ndx 17158  df-base 17174  df-plusg 17227  df-mulr 17228  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-hom 17238  df-cco 17239  df-prds 17404  df-psmet 21339  df-xmet 21340  df-bl 21342

This theorem is referenced by:  prdsxmslem2  24507  prdstotbnd  38132  prdsbnd2  38133