Theorem prdsbl 23237

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 23251) - 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 3096	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍)
7		prdsbl.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑅)
8	1, 2, 3, 4, 6, 7	prdsbas3 16850	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝑉)
9	8	eleq2d 2818	. . . . . . 7 ⊢ (𝜑 → (𝑓 ∈ 𝐵 ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉))
10	9	biimpa 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉)
11		ixpfn 8506	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐼 𝑉 → 𝑓 Fn 𝐼)
12		vex 3401	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	elixp 8507	. . . . . . 7 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ (𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
14	13	baib 539	. . . . . 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 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
18		prdsbl.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
19	18	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ ℝ*)
20		prdsbl.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
21	1, 2, 3, 4, 6, 7, 20	prdsbascl 16852	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉)
22	21	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉)
23	22	r19.21bi 3120	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) ∈ 𝑉)
24	3	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑆 ∈ 𝑊)
25	4	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝐼 ∈ Fin)
26	6	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍)
27		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵)
28	1, 2, 24, 25, 26, 7, 27	prdsbascl 16852	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
29	28	r19.21bi 3120	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉)
30		elbl2 23136	. . . . . . 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 838	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
32	31	ralbidva 3108	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
33		xmetcl 23077	. . . . . . . . . 10 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ*)
34	17, 23, 29, 33	syl3anc 1372	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ*)
35	34	ralrimiva 3096	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ*)
36		eqid 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))
37		breq1 5030	. . . . . . . . 9 ⊢ (𝑧 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) → (𝑧 < 𝐴 ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
38	36, 37	ralrnmptw 6864	. . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 0 < 𝐴)
42		c0ex 10706	. . . . . . . . . 10 ⊢ 0 ∈ V
43		breq1 5030	. . . . . . . . . 10 ⊢ (𝑧 = 0 → (𝑧 < 𝐴 ↔ 0 < 𝐴))
44	42, 43	ralsn 4569	. . . . . . . . 9 ⊢ (∀𝑧 ∈ {0}𝑧 < 𝐴 ↔ 0 < 𝐴)
45	41, 44	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑧 ∈ {0}𝑧 < 𝐴)
46		ralunb 4079	. . . . . . . . 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 484	. . . . . . . . . . . 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 16854	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < ))
51		xrltso 12610	. . . . . . . . . . . . 13 ⊢ < Or ℝ*
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → < Or ℝ*)
53	36	rnmpt 5792	. . . . . . . . . . . . . . 15 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))}
54		abrexfi 8890	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ Fin → {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))} ∈ Fin)
55	53, 54	eqeltrid 2837	. . . . . . . . . . . . . 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 8635	. . . . . . . . . . . . 13 ⊢ {0} ∈ Fin
58		unfi 8764	. . . . . . . . . . . . 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 589	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ∈ Fin)
60		ssun2 4061	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
61	42	snss 4671	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 13 ⊢ 0 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
63		ne0i 4221	. . . . . . . . . . . . 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 6883	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))):𝐼⟶ℝ*)
66	65	frnd 6506	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆ ℝ*)
67		0xr 10759	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ*
68	67	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 0 ∈ ℝ*)
69	68	snssd 4694	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → {0} ⊆ ℝ*)
70	66, 69	unssd 4074	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ*)
71		fisupcl 8999	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 2833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
74		breq1 5030	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑃𝐷𝑓) → (𝑧 < 𝐴 ↔ (𝑃𝐷𝑓) < 𝐴))
75	74	rspcv 3519	. . . . . . . . . 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	syl5bir 246	. . . . . . . 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 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴))
79	39, 78	sylbird 263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴 → (𝑃𝐷𝑓) < 𝐴))
80		ssun1 4060	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
81		ovex 7197	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ V
82	81	elabrex 7007	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐼 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))})
83	82	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))})
84	83, 53	eleqtrrdi 2844	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))))
85	80, 84	sseldi 3873	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
86		supxrub 12793	. . . . . . . . . 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 685	. . . . . . . . 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 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ*, < ))
89	87, 88	breqtrrd 5055	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓))
90	1, 2, 7, 48, 49, 3, 4, 5, 16	prdsxmet 23115	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
91	90	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐷 ∈ (∞Met‘𝐵))
92	20	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ 𝐵)
93	27	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑓 ∈ 𝐵)
94		xmetcl 23077	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈ ℝ*)
95	91, 92, 93, 94	syl3anc 1372	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) ∈ ℝ*)
96		xrlelttr 12625	. . . . . . . . 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 1372	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓) ∧ (𝑃𝐷𝑓) < 𝐴) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
98	89, 97	mpand 695	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃𝐷𝑓) < 𝐴 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
99	98	ralrimdva 3101	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝑃𝐷𝑓) < 𝐴 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
100	79, 99	impbid 215	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴 ↔ (𝑃𝐷𝑓) < 𝐴))
101	15, 32, 100	3bitrrd 309	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝑃𝐷𝑓) < 𝐴 ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
102	101	pm5.32da 582	. . 3 ⊢ (𝜑 → ((𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴) ↔ (𝑓 ∈ 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))))
103		elbl 23134	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝐴 ∈ ℝ*) → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴)))
104	90, 20, 18, 103	syl3anc 1372	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴)))
105	21	r19.21bi 3120	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) ∈ 𝑉)
106	18	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ ℝ*)
107		blssm 23164	. . . . . . . . 9 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ 𝐴 ∈ ℝ*) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉)
108	16, 105, 106, 107	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉)
109	108	ralrimiva 3096	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉)
110		ss2ixp 8513	. . . . . . 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 3916	. . . . 5 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝐵)
113	112	sseld 3874	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) → 𝑓 ∈ 𝐵))
114	113	pm4.71rd 566	. . 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 314	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
116	115	eqrdv 2736	1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2113  {cab 2716   ≠ wne 2934  ∀wral 3053  ∃wrex 3054   ∪ cun 3839   ⊆ wss 3841  ∅c0 4209  {csn 4513   class class class wbr 5027   ↦ cmpt 5107   Or wor 5437   × cxp 5517  ran crn 5520   ↾ cres 5521   Fn wfn 6328  ‘cfv 6333  (class class class)co 7164  Xcixp 8500  Fincfn 8548  supcsup 8970  0cc0 10608  ℝ^*cxr 10745   < clt 10746   ≤ cle 10747  Basecbs 16579  distcds 16670  X_scprds 16815  ∞Metcxmet 20195  ballcbl 20197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685  ax-pre-sup 10686

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-er 8313  df-map 8432  df-ixp 8501  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-sup 8972  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-div 11369  df-nn 11710  df-2 11772  df-3 11773  df-4 11774  df-5 11775  df-6 11776  df-7 11777  df-8 11778  df-9 11779  df-n0 11970  df-z 12056  df-dec 12173  df-uz 12318  df-rp 12466  df-xneg 12583  df-xadd 12584  df-xmul 12585  df-icc 12821  df-fz 12975  df-struct 16581  df-ndx 16582  df-slot 16583  df-base 16585  df-plusg 16674  df-mulr 16675  df-sca 16677  df-vsca 16678  df-ip 16679  df-tset 16680  df-ple 16681  df-ds 16683  df-hom 16685  df-cco 16686  df-prds 16817  df-psmet 20202  df-xmet 20203  df-bl 20205

This theorem is referenced by:  prdsxmslem2  23275  prdstotbnd  35564  prdsbnd2  35565