Theorem prdsdsf 21982

Description: The product metric is a function into the nonnegative extended reals. In general this means that it is not a metric but rather an *extended* metric (even when all the factors are metrics), but it will be a metric when restricted to regions where it does not take infinite values. (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
prdsdsf.y	⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅))
prdsdsf.b	⊢ 𝐵 = (Base‘𝑌)
prdsdsf.v	⊢ 𝑉 = (Base‘𝑅)
prdsdsf.e	⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
prdsdsf.d	⊢ 𝐷 = (dist‘𝑌)
prdsdsf.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
prdsdsf.i	⊢ (𝜑 → 𝐼 ∈ 𝑋)
prdsdsf.r	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)
prdsdsf.m	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))

Assertion

Ref	Expression
prdsdsf	⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))

Distinct variable groups:   𝑥,𝐼   𝜑,𝑥

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

Proof of Theorem prdsdsf

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

Step	Hyp	Ref	Expression
1		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
2		prdsdsf.r	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)
3		elex 3185	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ 𝑍 → 𝑅 ∈ V)
4	2, 3	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ V)
5	4	ralrimiva 2949	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ V)
6	5	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 𝑅 ∈ V)
7		nfcsb1v 3515	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅
8	7	nfel1 2765	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 ∈ V
9		csbeq1a 3508	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅)
10	9	eleq1d 2672	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝑅 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝑅 ∈ V))
11	8, 10	rspc 3276	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 𝑅 ∈ V → ⦋𝑦 / 𝑥⦌𝑅 ∈ V))
12	6, 11	mpan9 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → ⦋𝑦 / 𝑥⦌𝑅 ∈ V)
13		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑅) = (𝑥 ∈ 𝐼 ↦ 𝑅)
14	13	fvmpts 6194	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐼 ∧ ⦋𝑦 / 𝑥⦌𝑅 ∈ V) → ((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦) = ⦋𝑦 / 𝑥⦌𝑅)
15	1, 12, 14	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦) = ⦋𝑦 / 𝑥⦌𝑅)
16	15	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → (dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦)) = (dist‘⦋𝑦 / 𝑥⦌𝑅))
17	16	oveqd 6566	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦)) = ((𝑓‘𝑦)(dist‘⦋𝑦 / 𝑥⦌𝑅)(𝑔‘𝑦)))
18		prdsdsf.y	. . . . . . . . . . . . . 14 ⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅))
19		prdsdsf.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝑌)
20		prdsdsf.s	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
21	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑆 ∈ 𝑊)
22		prdsdsf.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ 𝑋)
23	22	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐼 ∈ 𝑋)
24		prdsdsf.v	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (Base‘𝑅)
25		simprl 790	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ 𝐵)
26	18, 19, 21, 23, 6, 24, 25	prdsbascl 15966	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
27		nfcsb1v 3515	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑉
28	27	nfel2 2767	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉
29		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
30		csbeq1a 3508	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → 𝑉 = ⦋𝑦 / 𝑥⦌𝑉)
31	29, 30	eleq12d 2682	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ 𝑉 ↔ (𝑓‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉))
32	28, 31	rspc 3276	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉 → (𝑓‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉))
33	26, 32	mpan9 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → (𝑓‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉)
34		simprr 792	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
35	18, 19, 21, 23, 6, 24, 34	prdsbascl 15966	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
36	27	nfel2 2767	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑔‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉
37		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
38	37, 30	eleq12d 2682	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑥) ∈ 𝑉 ↔ (𝑔‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉))
39	36, 38	rspc 3276	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉 → (𝑔‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉))
40	35, 39	mpan9 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → (𝑔‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉)
41	33, 40	ovresd 6699	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → ((𝑓‘𝑦)((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉))(𝑔‘𝑦)) = ((𝑓‘𝑦)(dist‘⦋𝑦 / 𝑥⦌𝑅)(𝑔‘𝑦)))
42	17, 41	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦)) = ((𝑓‘𝑦)((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉))(𝑔‘𝑦)))
43		prdsdsf.m	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
44	43	ralrimiva 2949	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐸 ∈ (∞Met‘𝑉))
45	44	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 𝐸 ∈ (∞Met‘𝑉))
46		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥dist
47	46, 7	nffv 6110	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(dist‘⦋𝑦 / 𝑥⦌𝑅)
48	27, 27	nfxp 5066	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉)
49	47, 48	nfres 5319	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉))
50		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥∞Met
51	50, 27	nffv 6110	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(∞Met‘⦋𝑦 / 𝑥⦌𝑉)
52	49, 51	nfel 2763	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉)) ∈ (∞Met‘⦋𝑦 / 𝑥⦌𝑉)
53		prdsdsf.e	. . . . . . . . . . . . . . 15 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
54	9	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘⦋𝑦 / 𝑥⦌𝑅))
55	30	sqxpeqd 5065	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝑉 × 𝑉) = (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉))
56	54, 55	reseq12d 5318	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉)))
57	53, 56	syl5eq 2656	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → 𝐸 = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉)))
58	30	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘⦋𝑦 / 𝑥⦌𝑉))
59	57, 58	eleq12d 2682	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐸 ∈ (∞Met‘𝑉) ↔ ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉)) ∈ (∞Met‘⦋𝑦 / 𝑥⦌𝑉)))
60	52, 59	rspc 3276	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 𝐸 ∈ (∞Met‘𝑉) → ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉)) ∈ (∞Met‘⦋𝑦 / 𝑥⦌𝑉)))
61	45, 60	mpan9 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉)) ∈ (∞Met‘⦋𝑦 / 𝑥⦌𝑉))
62		xmetcl 21946	. . . . . . . . . . 11 ⊢ ((((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉)) ∈ (∞Met‘⦋𝑦 / 𝑥⦌𝑉) ∧ (𝑓‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉 ∧ (𝑔‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝑉) → ((𝑓‘𝑦)((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉))(𝑔‘𝑦)) ∈ ℝ*)
63	61, 33, 40, 62	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → ((𝑓‘𝑦)((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ (⦋𝑦 / 𝑥⦌𝑉 × ⦋𝑦 / 𝑥⦌𝑉))(𝑔‘𝑦)) ∈ ℝ*)
64	42, 63	eqeltrd 2688	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑦 ∈ 𝐼) → ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦)) ∈ ℝ*)
65		eqid 2610	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) = (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦)))
66	64, 65	fmptd 6292	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))):𝐼⟶ℝ*)
67		frn 5966	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))):𝐼⟶ℝ* → ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ⊆ ℝ*)
68	66, 67	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ⊆ ℝ*)
69		0xr 9965	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
70	69	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 0 ∈ ℝ*)
71	70	snssd 4281	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → {0} ⊆ ℝ*)
72	68, 71	unssd 3751	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}) ⊆ ℝ*)
73		supxrcl 12017	. . . . . 6 ⊢ ((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}) ⊆ ℝ* → sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ) ∈ ℝ*)
74	72, 73	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ) ∈ ℝ*)
75		ssun2 3739	. . . . . . 7 ⊢ {0} ⊆ (ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0})
76		c0ex 9913	. . . . . . . 8 ⊢ 0 ∈ V
77	76	snss 4259	. . . . . . 7 ⊢ (0 ∈ (ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}) ↔ {0} ⊆ (ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}))
78	75, 77	mpbir 220	. . . . . 6 ⊢ 0 ∈ (ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0})
79		supxrub 12026	. . . . . 6 ⊢ (((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}) ⊆ ℝ* ∧ 0 ∈ (ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0})) → 0 ≤ sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ))
80	72, 78, 79	sylancl 693	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 0 ≤ sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ))
81		elxrge0 12152	. . . . 5 ⊢ (sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞) ↔ (sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < )))
82	74, 80, 81	sylanbrc 695	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
83	82	ralrimivva 2954	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞))
84		eqid 2610	. . . 4 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < )) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ))
85	84	fmpt2 7126	. . 3 ⊢ (∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < ) ∈ (0[,]+∞) ↔ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞))
86	83, 85	sylib 207	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞))
87		mptexg 6389	. . . . 5 ⊢ (𝐼 ∈ 𝑋 → (𝑥 ∈ 𝐼 ↦ 𝑅) ∈ V)
88	22, 87	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑅) ∈ V)
89	2	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍)
90		dmmptg 5549	. . . . 5 ⊢ (∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍 → dom (𝑥 ∈ 𝐼 ↦ 𝑅) = 𝐼)
91	89, 90	syl 17	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐼 ↦ 𝑅) = 𝐼)
92		prdsdsf.d	. . . 4 ⊢ 𝐷 = (dist‘𝑌)
93	18, 20, 88, 19, 91, 92	prdsds 15947	. . 3 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < )))
94	93	feq1d 5943	. 2 ⊢ (𝜑 → (𝐷:(𝐵 × 𝐵)⟶(0[,]+∞) ↔ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑦 ∈ 𝐼 ↦ ((𝑓‘𝑦)(dist‘((𝑥 ∈ 𝐼 ↦ 𝑅)‘𝑦))(𝑔‘𝑦))) ∪ {0}), ℝ*, < )):(𝐵 × 𝐵)⟶(0[,]+∞)))
95	86, 94	mpbird 246	1 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⦋csb 3499   ∪ cun 3538   ⊆ wss 3540  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  supcsup 8229  0cc0 9815  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  [,]cicc 12049  Basecbs 15695  distcds 15777  X_scprds 15929  ∞Metcxmt 19552

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-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-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-xmet 19560

This theorem is referenced by:  prdsxmetlem  21983  prdsmet  21985