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Theorem prdsbl 22106

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 22120) - 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 = (𝑆X_s(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 = (𝑆X_s(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 2949	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍)
7		prdsbl.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑅)
8	1, 2, 3, 4, 6, 7	prdsbas3 15964	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝑉)
9	8	eleq2d 2673	. . . . . . 7 ⊢ (𝜑 → (𝑓 ∈ 𝐵 ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉))
10	9	biimpa 500	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ X𝑥 ∈ 𝐼 𝑉)
11		ixpfn 7800	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐼 𝑉 → 𝑓 Fn 𝐼)
12		vex 3176	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	elixp 7801	. . . . . . 7 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ (𝑓 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
14	13	baib 942	. . . . . 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 747	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
18		prdsbl.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
19	18	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ ℝ^*)
20		prdsbl.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
21	1, 2, 3, 4, 6, 7, 20	prdsbascl 15966	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉)
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑃‘𝑥) ∈ 𝑉)
23	22	r19.21bi 2916	. . . . . . 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 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵)
28	1, 2, 24, 25, 26, 7, 27	prdsbascl 15966	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
29	28	r19.21bi 2916	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉)
30		elbl2 22005	. . . . . . 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 1319	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
32	31	ralbidva 2968	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
33		xmetcl 21946	. . . . . . . . . 10 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ^*)
34	17, 23, 29, 33	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ^*)
35	34	ralrimiva 2949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ^*)
36		eqid 2610	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))
37		breq1 4586	. . . . . . . . 9 ⊢ (𝑧 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) → (𝑧 < 𝐴 ↔ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
38	36, 37	ralrnmpt 6276	. . . . . . . 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 9913	. . . . . . . . . 10 ⊢ 0 ∈ V
43		breq1 4586	. . . . . . . . . 10 ⊢ (𝑧 = 0 → (𝑧 < 𝐴 ↔ 0 < 𝐴))
44	42, 43	ralsn 4169	. . . . . . . . 9 ⊢ (∀𝑧 ∈ {0}𝑧 < 𝐴 ↔ 0 < 𝐴)
45	41, 44	sylibr 223	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ∀𝑧 ∈ {0}𝑧 < 𝐴)
46		ralunb 3756	. . . . . . . . 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 15968	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ^*, < ))
51		xrltso 11850	. . . . . . . . . . . . 13 ⊢ < Or ℝ^*
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → < Or ℝ^*)
53	36	rnmpt 5292	. . . . . . . . . . . . . . 15 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) = {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))}
54		abrexfi 8149	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ Fin → {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))} ∈ Fin)
55	53, 54	syl5eqel 2692	. . . . . . . . . . . . . 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 7923	. . . . . . . . . . . . 13 ⊢ {0} ∈ Fin
58		unfi 8112	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ∈ Fin)
60		ssun2 3739	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
61	42	snss 4259	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . . 13 ⊢ 0 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
63		ne0i 3880	. . . . . . . . . . . . 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, 36	fmptd 6292	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))):𝐼⟶ℝ^*)
66		frn 5966	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))):𝐼⟶ℝ^* → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆ ℝ^*)
67	65, 66	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆ ℝ^*)
68		0xr 9965	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
69	68	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 0 ∈ ℝ^*)
70	69	snssd 4281	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → {0} ⊆ ℝ^*)
71	67, 70	unssd 3751	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ^*)
72		fisupcl 8258	. . . . . . . . . . . 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}))
73	52, 59, 64, 71, 72	syl13anc 1320	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ^*, < ) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
74	50, 73	eqeltrd 2688	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
75		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑃𝐷𝑓) → (𝑧 < 𝐴 ↔ (𝑃𝐷𝑓) < 𝐴))
76	75	rspcv 3278	. . . . . . . . . 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))
77	74, 76	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴))
78	46, 77	syl5bir 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 ∧ ∀𝑧 ∈ {0}𝑧 < 𝐴) → (𝑃𝐷𝑓) < 𝐴))
79	45, 78	mpan2d 706	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)))𝑧 < 𝐴 → (𝑃𝐷𝑓) < 𝐴))
80	39, 79	sylbird 249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴 → (𝑃𝐷𝑓) < 𝐴))
81	71	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ^*)
82		ssun1 3738	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
83		ovex 6577	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ V
84	83	elabrex 6405	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐼 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))})
85	84	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐼 𝑦 = ((𝑃‘𝑥)𝐸(𝑓‘𝑥))})
86	85, 53	syl6eleqr 2699	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))))
87	82, 86	sseldi 3566	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
88		supxrub 12026	. . . . . . . . . 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^*, < ))
89	81, 87, 88	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ^*, < ))
90	50	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ^*, < ))
91	89, 90	breqtrrd 4611	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓))
92	1, 2, 7, 48, 49, 3, 4, 5, 16	prdsxmet 21984	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
93	92	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝐷 ∈ (∞Met‘𝐵))
94	20	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ 𝐵)
95	27	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → 𝑓 ∈ 𝐵)
96		xmetcl 21946	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → (𝑃𝐷𝑓) ∈ ℝ^*)
97	93, 94, 95, 96	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑃𝐷𝑓) ∈ ℝ^*)
98		xrlelttr 11863	. . . . . . . . 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))
99	34, 97, 19, 98	syl3anc 1318	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((((𝑃‘𝑥)𝐸(𝑓‘𝑥)) ≤ (𝑃𝐷𝑓) ∧ (𝑃𝐷𝑓) < 𝐴) → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
100	91, 99	mpand 707	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → ((𝑃𝐷𝑓) < 𝐴 → ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
101	100	ralrimdva 2952	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝑃𝐷𝑓) < 𝐴 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴))
102	80, 101	impbid 201	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)𝐸(𝑓‘𝑥)) < 𝐴 ↔ (𝑃𝐷𝑓) < 𝐴))
103	15, 32, 102	3bitrrd 294	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → ((𝑃𝐷𝑓) < 𝐴 ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
104	103	pm5.32da 671	. . 3 ⊢ (𝜑 → ((𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴) ↔ (𝑓 ∈ 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))))
105		elbl 22003	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑃 ∈ 𝐵 ∧ 𝐴 ∈ ℝ^*) → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴)))
106	92, 20, 18, 105	syl3anc 1318	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ (𝑃𝐷𝑓) < 𝐴)))
107	21	r19.21bi 2916	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) ∈ 𝑉)
108	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ ℝ^*)
109		blssm 22033	. . . . . . . . 9 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑃‘𝑥) ∈ 𝑉 ∧ 𝐴 ∈ ℝ^*) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉)
110	16, 107, 108, 109	syl3anc 1318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉)
111	110	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉)
112		ss2ixp 7807	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝑉 → X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ X𝑥 ∈ 𝐼 𝑉)
113	111, 112	syl 17	. . . . . 6 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ X𝑥 ∈ 𝐼 𝑉)
114	113, 8	sseqtr4d 3605	. . . . 5 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ⊆ 𝐵)
115	114	sseld 3567	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) → 𝑓 ∈ 𝐵))
116	115	pm4.71rd 665	. . 3 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴) ↔ (𝑓 ∈ 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))))
117	104, 106, 116	3bitr4d 299	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑃(ball‘𝐷)𝐴) ↔ 𝑓 ∈ X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴)))
118	117	eqrdv 2608	1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 Or wor 4958 × cxp 5036 ran crn 5039 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Xcixp 7794 Fincfn 7841 supcsup 8229 0cc0 9815 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 Basecbs 15695 distcds 15777 X_scprds 15929 ∞Metcxmt 19552 ballcbl 19554

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-prds 15931 df-psmet 19559 df-xmet 19560 df-bl 19562

This theorem is referenced by: prdsxmslem2 22144 prdstotbnd 32763 prdsbnd2 32764

W3C validator