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Theorem prdsxmetlem 23266

Description: The product metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
prdsdsf.y	⊢ 𝑌 = (𝑆X_s(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
prdsxmetlem	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Distinct variable groups: 𝑥,𝐼 𝜑,𝑥 𝑥,𝐵 𝑥,𝐷 Allowed substitution hints: 𝑅(𝑥) 𝑆(𝑥) 𝐸(𝑥) 𝑉(𝑥) 𝑊(𝑥) 𝑋(𝑥) 𝑌(𝑥) 𝑍(𝑥)

Proof of Theorem prdsxmetlem Dummy variables 𝑓 𝑔 ℎ 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsdsf.b	. . . 4 ⊢ 𝐵 = (Base‘𝑌)
2	1	fvexi 6731	. . 3 ⊢ 𝐵 ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
4		prdsdsf.y	. . . 4 ⊢ 𝑌 = (𝑆X_s(𝑥 ∈ 𝐼 ↦ 𝑅))
5		prdsdsf.v	. . . 4 ⊢ 𝑉 = (Base‘𝑅)
6		prdsdsf.e	. . . 4 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
7		prdsdsf.d	. . . 4 ⊢ 𝐷 = (dist‘𝑌)
8		prdsdsf.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
9		prdsdsf.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑋)
10		prdsdsf.r	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)
11		prdsdsf.m	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
12	4, 1, 5, 6, 7, 8, 9, 10, 11	prdsdsf 23265	. . 3 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))
13		iccssxr 13018	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
14		fss 6562	. . 3 ⊢ ((𝐷:(𝐵 × 𝐵)⟶(0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ^) → 𝐷:(𝐵 × 𝐵)⟶ℝ^)
15	12, 13, 14	sylancl 589	. 2 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶ℝ^*)
16	12	fovrnda 7379	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) ∈ (0[,]+∞))
17		elxrge0 13045	. . . 4 ⊢ ((𝑓𝐷𝑔) ∈ (0[,]+∞) ↔ ((𝑓𝐷𝑔) ∈ ℝ^* ∧ 0 ≤ (𝑓𝐷𝑔)))
18	17	simprbi 500	. . 3 ⊢ ((𝑓𝐷𝑔) ∈ (0[,]+∞) → 0 ≤ (𝑓𝐷𝑔))
19	16, 18	syl 17	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 0 ≤ (𝑓𝐷𝑔))
20	8	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑆 ∈ 𝑊)
21	9	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐼 ∈ 𝑋)
22	10	ralrimiva 3105	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍)
23	22	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍)
24		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ 𝐵)
25		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
26	4, 1, 20, 21, 23, 24, 25, 5, 6, 7	prdsdsval3 16990	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))
27	26	breq1d 5063	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ((𝑓𝐷𝑔) ≤ 0 ↔ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ) ≤ 0))
28	11	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
29	4, 1, 20, 21, 23, 5, 24	prdsbascl 16988	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
30	29	r19.21bi 3130	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉)
31	4, 1, 20, 21, 23, 5, 25	prdsbascl 16988	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
32	31	r19.21bi 3130	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ 𝑉)
33		xmetcl 23229	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑓‘𝑥) ∈ 𝑉 ∧ (𝑔‘𝑥) ∈ 𝑉) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^*)
34	28, 30, 32, 33	syl3anc 1373	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^*)
35	34	fmpttd 6932	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))):𝐼⟶ℝ^*)
36	35	frnd 6553	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ⊆ ℝ^*)
37		0xr 10880	. . . . . . 7 ⊢ 0 ∈ ℝ^*
38	37	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 0 ∈ ℝ^*)
39	38	snssd 4722	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → {0} ⊆ ℝ^*)
40	36, 39	unssd 4100	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ^*)
41		supxrleub 12916	. . . 4 ⊢ (((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ^* ∧ 0 ∈ ℝ^) → (sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^, < ) ≤ 0 ↔ ∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ 0))
42	40, 37, 41	sylancl 589	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ) ≤ 0 ↔ ∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ 0))
43		0le0 11931	. . . . . . 7 ⊢ 0 ≤ 0
44		c0ex 10827	. . . . . . . 8 ⊢ 0 ∈ V
45		breq1 5056	. . . . . . . 8 ⊢ (𝑧 = 0 → (𝑧 ≤ 0 ↔ 0 ≤ 0))
46	44, 45	ralsn 4597	. . . . . . 7 ⊢ (∀𝑧 ∈ {0}𝑧 ≤ 0 ↔ 0 ≤ 0)
47	43, 46	mpbir 234	. . . . . 6 ⊢ ∀𝑧 ∈ {0}𝑧 ≤ 0
48		ralunb 4105	. . . . . 6 ⊢ (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ 0 ↔ (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑧 ≤ 0 ∧ ∀𝑧 ∈ {0}𝑧 ≤ 0))
49	47, 48	mpbiran2 710	. . . . 5 ⊢ (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ 0 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑧 ≤ 0)
50		ovex 7246	. . . . . . 7 ⊢ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
51	50	rgenw 3073	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
52		eqid 2737	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))
53		breq1 5056	. . . . . . 7 ⊢ (𝑧 = ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) → (𝑧 ≤ 0 ↔ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0))
54	52, 53	ralrnmptw 6913	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑧 ≤ 0 ↔ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0))
55	51, 54	ax-mp 5	. . . . 5 ⊢ (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑧 ≤ 0 ↔ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0)
56	49, 55	bitri 278	. . . 4 ⊢ (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ 0 ↔ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0)
57		xmetge0 23242	. . . . . . . . 9 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑓‘𝑥) ∈ 𝑉 ∧ (𝑔‘𝑥) ∈ 𝑉) → 0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))
58	28, 30, 32, 57	syl3anc 1373	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))
59	58	biantrud 535	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0 ↔ (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0 ∧ 0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))))
60		xrletri3 12744	. . . . . . . 8 ⊢ ((((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) = 0 ↔ (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0 ∧ 0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))))
61	34, 37, 60	sylancl 589	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) = 0 ↔ (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0 ∧ 0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))))
62		xmeteq0 23236	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑓‘𝑥) ∈ 𝑉 ∧ (𝑔‘𝑥) ∈ 𝑉) → (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) = 0 ↔ (𝑓‘𝑥) = (𝑔‘𝑥)))
63	28, 30, 32, 62	syl3anc 1373	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) = 0 ↔ (𝑓‘𝑥) = (𝑔‘𝑥)))
64	59, 61, 63	3bitr2d 310	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0 ↔ (𝑓‘𝑥) = (𝑔‘𝑥)))
65	64	ralbidva 3117	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0 ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = (𝑔‘𝑥)))
66		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑅) = (𝑥 ∈ 𝐼 ↦ 𝑅)
67	66	fnmpt 6518	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍 → (𝑥 ∈ 𝐼 ↦ 𝑅) Fn 𝐼)
68	22, 67	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑅) Fn 𝐼)
69	68	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ 𝑅) Fn 𝐼)
70	4, 1, 20, 21, 69, 24	prdsbasfn 16976	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 Fn 𝐼)
71	4, 1, 20, 21, 69, 25	prdsbasfn 16976	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 Fn 𝐼)
72		eqfnfv 6852	. . . . . 6 ⊢ ((𝑓 Fn 𝐼 ∧ 𝑔 Fn 𝐼) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = (𝑔‘𝑥)))
73	70, 71, 72	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) = (𝑔‘𝑥)))
74	65, 73	bitr4d 285	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ 0 ↔ 𝑓 = 𝑔))
75	56, 74	syl5bb 286	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ 0 ↔ 𝑓 = 𝑔))
76	27, 42, 75	3bitrd 308	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ((𝑓𝐷𝑔) ≤ 0 ↔ 𝑓 = 𝑔))
77	26	3adantr3 1173	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → (𝑓𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))
78	77	3adant3 1134	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (𝑓𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))
79	11	3ad2antl1 1187	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
80	29	3adantr3 1173	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
81	80	3adant3 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
82	81	r19.21bi 3130	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉)
83	31	3adantr3 1173	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
84	83	3adant3 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
85	84	r19.21bi 3130	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ 𝑉)
86	79, 82, 85, 33	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^*)
87	8	3ad2ant1 1135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → 𝑆 ∈ 𝑊)
88	9	3ad2ant1 1135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → 𝐼 ∈ 𝑋)
89	22	3ad2ant1 1135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑍)
90		simp23 1210	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ℎ ∈ 𝐵)
91	4, 1, 87, 88, 89, 5, 90	prdsbascl 16988	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ∀𝑥 ∈ 𝐼 (ℎ‘𝑥) ∈ 𝑉)
92	91	r19.21bi 3130	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (ℎ‘𝑥) ∈ 𝑉)
93		xmetcl 23229	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (ℎ‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ^*)
94	79, 92, 82, 93	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ^*)
95		simp3l 1203	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (ℎ𝐷𝑓) ∈ ℝ)
96	95	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (ℎ𝐷𝑓) ∈ ℝ)
97		xmetge0 23242	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (ℎ‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉) → 0 ≤ ((ℎ‘𝑥)𝐸(𝑓‘𝑥)))
98	79, 92, 82, 97	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → 0 ≤ ((ℎ‘𝑥)𝐸(𝑓‘𝑥)))
99	94	fmpttd 6932	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))):𝐼⟶ℝ^*)
100	99	frnd 6553	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ⊆ ℝ^*)
101	37	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → 0 ∈ ℝ^*)
102	101	snssd 4722	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → {0} ⊆ ℝ^*)
103	100, 102	unssd 4100	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ^*)
104		ssun1 4086	. . . . . . . . . . . . . 14 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})
105		ovex 7246	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ V
106	105	elabrex 7056	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐼 → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐼 𝑧 = ((ℎ‘𝑥)𝐸(𝑓‘𝑥))})
107	106	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐼 𝑧 = ((ℎ‘𝑥)𝐸(𝑓‘𝑥))})
108		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥)))
109	108	rnmpt 5824	. . . . . . . . . . . . . . 15 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) = {𝑧 ∣ ∃𝑥 ∈ 𝐼 𝑧 = ((ℎ‘𝑥)𝐸(𝑓‘𝑥))}
110	107, 109	eleqtrrdi 2849	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))))
111	104, 110	sseldi 3899	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}))
112		supxrub 12914	. . . . . . . . . . . . 13 ⊢ (((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}) ⊆ ℝ^* ∧ ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0})) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ≤ sup((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ^*, < ))
113	103, 111, 112	syl2an2r 685	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ≤ sup((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ^*, < ))
114		simp21 1208	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → 𝑓 ∈ 𝐵)
115	4, 1, 87, 88, 89, 90, 114, 5, 6, 7	prdsdsval3 16990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (ℎ𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ^*, < ))
116	115	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (ℎ𝐷𝑓) = sup((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑓‘𝑥))) ∪ {0}), ℝ^*, < ))
117	113, 116	breqtrrd 5081	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ≤ (ℎ𝐷𝑓))
118		xrrege0 12764	. . . . . . . . . . 11 ⊢ (((((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ^* ∧ (ℎ𝐷𝑓) ∈ ℝ) ∧ (0 ≤ ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∧ ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ≤ (ℎ𝐷𝑓))) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ)
119	94, 96, 98, 117, 118	syl22anc 839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑓‘𝑥)) ∈ ℝ)
120		xmetcl 23229	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (ℎ‘𝑥) ∈ 𝑉 ∧ (𝑔‘𝑥) ∈ 𝑉) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^*)
121	79, 92, 85, 120	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^*)
122		simp3r 1204	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (ℎ𝐷𝑔) ∈ ℝ)
123	122	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (ℎ𝐷𝑔) ∈ ℝ)
124		xmetge0 23242	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (ℎ‘𝑥) ∈ 𝑉 ∧ (𝑔‘𝑥) ∈ 𝑉) → 0 ≤ ((ℎ‘𝑥)𝐸(𝑔‘𝑥)))
125	79, 92, 85, 124	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → 0 ≤ ((ℎ‘𝑥)𝐸(𝑔‘𝑥)))
126	121	fmpttd 6932	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))):𝐼⟶ℝ^*)
127	126	frnd 6553	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ⊆ ℝ^*)
128	127, 102	unssd 4100	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ^*)
129		ssun1 4086	. . . . . . . . . . . . . 14 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ⊆ (ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})
130		ovex 7246	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
131	130	elabrex 7056	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐼 → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐼 𝑧 = ((ℎ‘𝑥)𝐸(𝑔‘𝑥))})
132	131	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐼 𝑧 = ((ℎ‘𝑥)𝐸(𝑔‘𝑥))})
133		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥)))
134	133	rnmpt 5824	. . . . . . . . . . . . . . 15 ⊢ ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) = {𝑧 ∣ ∃𝑥 ∈ 𝐼 𝑧 = ((ℎ‘𝑥)𝐸(𝑔‘𝑥))}
135	132, 134	eleqtrrdi 2849	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))))
136	129, 135	sseldi 3899	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}))
137		supxrub 12914	. . . . . . . . . . . . 13 ⊢ (((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ^* ∧ ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ (ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ≤ sup((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))
138	128, 136, 137	syl2an2r 685	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ≤ sup((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))
139		simp22 1209	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → 𝑔 ∈ 𝐵)
140	4, 1, 87, 88, 89, 90, 139, 5, 6, 7	prdsdsval3 16990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (ℎ𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))
141	140	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (ℎ𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))
142	138, 141	breqtrrd 5081	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ≤ (ℎ𝐷𝑔))
143		xrrege0 12764	. . . . . . . . . . 11 ⊢ (((((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^* ∧ (ℎ𝐷𝑔) ∈ ℝ) ∧ (0 ≤ ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∧ ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ≤ (ℎ𝐷𝑔))) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
144	121, 123, 125, 142, 143	syl22anc 839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
145	119, 144	readdcld 10862	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (((ℎ‘𝑥)𝐸(𝑓‘𝑥)) + ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∈ ℝ)
146	79, 82, 85, 57	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → 0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))
147		xmettri2 23238	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ ((ℎ‘𝑥) ∈ 𝑉 ∧ (𝑓‘𝑥) ∈ 𝑉 ∧ (𝑔‘𝑥) ∈ 𝑉)) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ (((ℎ‘𝑥)𝐸(𝑓‘𝑥)) +_𝑒 ((ℎ‘𝑥)𝐸(𝑔‘𝑥))))
148	79, 92, 82, 85, 147	syl13anc 1374	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ (((ℎ‘𝑥)𝐸(𝑓‘𝑥)) +_𝑒 ((ℎ‘𝑥)𝐸(𝑔‘𝑥))))
149	119, 144	rexaddd 12824	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (((ℎ‘𝑥)𝐸(𝑓‘𝑥)) +_𝑒 ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) = (((ℎ‘𝑥)𝐸(𝑓‘𝑥)) + ((ℎ‘𝑥)𝐸(𝑔‘𝑥))))
150	148, 149	breqtrd 5079	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ (((ℎ‘𝑥)𝐸(𝑓‘𝑥)) + ((ℎ‘𝑥)𝐸(𝑔‘𝑥))))
151		xrrege0 12764	. . . . . . . . 9 ⊢ (((((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^* ∧ (((ℎ‘𝑥)𝐸(𝑓‘𝑥)) + ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ∈ ℝ) ∧ (0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∧ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ (((ℎ‘𝑥)𝐸(𝑓‘𝑥)) + ((ℎ‘𝑥)𝐸(𝑔‘𝑥))))) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
152	86, 145, 146, 150, 151	syl22anc 839	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
153		readdcl 10812	. . . . . . . . . 10 ⊢ (((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ) → ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ∈ ℝ)
154	153	3ad2ant3 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ∈ ℝ)
155	154	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ∈ ℝ)
156	119, 144, 96, 123, 117, 142	le2addd 11451	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → (((ℎ‘𝑥)𝐸(𝑓‘𝑥)) + ((ℎ‘𝑥)𝐸(𝑔‘𝑥))) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
157	152, 145, 155, 150, 156	letrd 10989	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
158	157	ralrimiva 3105	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
159	86	ralrimiva 3105	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^*)
160		breq1 5056	. . . . . . . 8 ⊢ (𝑧 = ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) → (𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ↔ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔))))
161	52, 160	ralrnmptw 6913	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ^* → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ↔ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔))))
162	159, 161	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ↔ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔))))
163	158, 162	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
164	12	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → 𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))
165	164, 90, 114	fovrnd 7380	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (ℎ𝐷𝑓) ∈ (0[,]+∞))
166		elxrge0 13045	. . . . . . . . 9 ⊢ ((ℎ𝐷𝑓) ∈ (0[,]+∞) ↔ ((ℎ𝐷𝑓) ∈ ℝ^* ∧ 0 ≤ (ℎ𝐷𝑓)))
167	166	simprbi 500	. . . . . . . 8 ⊢ ((ℎ𝐷𝑓) ∈ (0[,]+∞) → 0 ≤ (ℎ𝐷𝑓))
168	165, 167	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → 0 ≤ (ℎ𝐷𝑓))
169	164, 90, 139	fovrnd 7380	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (ℎ𝐷𝑔) ∈ (0[,]+∞))
170		elxrge0 13045	. . . . . . . . 9 ⊢ ((ℎ𝐷𝑔) ∈ (0[,]+∞) ↔ ((ℎ𝐷𝑔) ∈ ℝ^* ∧ 0 ≤ (ℎ𝐷𝑔)))
171	170	simprbi 500	. . . . . . . 8 ⊢ ((ℎ𝐷𝑔) ∈ (0[,]+∞) → 0 ≤ (ℎ𝐷𝑔))
172	169, 171	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → 0 ≤ (ℎ𝐷𝑔))
173	95, 122, 168, 172	addge0d 11408	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → 0 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
174		breq1 5056	. . . . . . 7 ⊢ (𝑧 = 0 → (𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ↔ 0 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔))))
175	44, 174	ralsn 4597	. . . . . 6 ⊢ (∀𝑧 ∈ {0}𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ↔ 0 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
176	173, 175	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ∀𝑧 ∈ {0}𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
177		ralunb 4105	. . . . 5 ⊢ (∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ↔ (∀𝑧 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ∧ ∀𝑧 ∈ {0}𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔))))
178	163, 176, 177	sylanbrc 586	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
179	40	3adantr3 1173	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ^*)
180	179	3adant3 1134	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ^*)
181	154	rexrd 10883	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ∈ ℝ^*)
182		supxrleub 12916	. . . . 5 ⊢ (((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ^* ∧ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ∈ ℝ^) → (sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^, < ) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ↔ ∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔))))
183	180, 181, 182	syl2anc 587	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)) ↔ ∀𝑧 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑧 ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔))))
184	178, 183	mpbird 260	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
185	78, 184	eqbrtrd 5075	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) ∧ ((ℎ𝐷𝑓) ∈ ℝ ∧ (ℎ𝐷𝑔) ∈ ℝ)) → (𝑓𝐷𝑔) ≤ ((ℎ𝐷𝑓) + (ℎ𝐷𝑔)))
186	3, 15, 19, 76, 185	isxmet2d 23225	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 {cab 2714 ∀wral 3061 ∃wrex 3062 Vcvv 3408 ∪ cun 3864 ⊆ wss 3866 {csn 4541 class class class wbr 5053 ↦ cmpt 5135 × cxp 5549 ran crn 5552 ↾ cres 5553 Fn wfn 6375 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 supcsup 9056 ℝcr 10728 0cc0 10729 + caddc 10732 +∞cpnf 10864 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 +_𝑒 cxad 12702 [,]cicc 12938 Basecbs 16760 distcds 16811 X_scprds 16950 ∞Metcxmet 20348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-fz 13096 df-struct 16700 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-hom 16826 df-cco 16827 df-prds 16952 df-xmet 20356

This theorem is referenced by: prdsxmet 23267

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