Theorem prdsbnd 38444

Description: The product metric over finite index set is bounded if all the factors are bounded. (Contributed by Mario Carneiro, 13-Sep-2015.)

Hypotheses

Ref	Expression
prdsbnd.y	⊢ 𝑌 = (𝑆Xs𝑅)
prdsbnd.b	⊢ 𝐵 = (Base‘𝑌)
prdsbnd.v	⊢ 𝑉 = (Base‘(𝑅‘𝑥))
prdsbnd.e	⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉))
prdsbnd.d	⊢ 𝐷 = (dist‘𝑌)
prdsbnd.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
prdsbnd.i	⊢ (𝜑 → 𝐼 ∈ Fin)
prdsbnd.r	⊢ (𝜑 → 𝑅 Fn 𝐼)
prdsbnd.m	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Bnd‘𝑉))

Assertion

Ref	Expression
prdsbnd	⊢ (𝜑 → 𝐷 ∈ (Bnd‘𝐵))

Distinct variable groups:   𝑥,𝑅   𝑥,𝐵   𝜑,𝑥   𝑥,𝐼   𝑥,𝑆   𝑥,𝑌

Allowed substitution hints:   𝐷(𝑥)   𝐸(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem prdsbnd

Dummy variables 𝑧 𝑓 𝑔 𝑘 𝑚 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
2		eqid 2763	. . . 4 ⊢ (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
3		prdsbnd.v	. . . 4 ⊢ 𝑉 = (Base‘(𝑅‘𝑥))
4		prdsbnd.e	. . . 4 ⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉))
5		eqid 2763	. . . 4 ⊢ (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
6		prdsbnd.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
7		prdsbnd.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
8		fvexd 6896	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
9		prdsbnd.m	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Bnd‘𝑉))
10		bndmet 38432	. . . . 5 ⊢ (𝐸 ∈ (Bnd‘𝑉) → 𝐸 ∈ (Met‘𝑉))
11	9, 10	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
12	1, 2, 3, 4, 5, 6, 7, 8, 11	prdsmet 24527	. . 3 ⊢ (𝜑 → (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) ∈ (Met‘(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
13		prdsbnd.d	. . . 4 ⊢ 𝐷 = (dist‘𝑌)
14		prdsbnd.y	. . . . . 6 ⊢ 𝑌 = (𝑆Xs𝑅)
15		prdsbnd.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
16		dffn5 6939	. . . . . . . 8 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
17	15, 16	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
18	17	oveq2d 7426	. . . . . 6 ⊢ (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
19	14, 18	eqtrid 2810	. . . . 5 ⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
20	19	fveq2d 6885	. . . 4 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
21	13, 20	eqtrid 2810	. . 3 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
22		prdsbnd.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑌)
23	19	fveq2d 6885	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
24	22, 23	eqtrid 2810	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
25	24	fveq2d 6885	. . 3 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
26	12, 21, 25	3eltr4d 2878	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵))
27		isbnd3 38435	. . . . . . 7 ⊢ (𝐸 ∈ (Bnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤)))
28	27	simprbi 502	. . . . . 6 ⊢ (𝐸 ∈ (Bnd‘𝑉) → ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤))
29	9, 28	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤))
30	29	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤))
31		oveq2 7418	. . . . . 6 ⊢ (𝑤 = (𝑘‘𝑥) → (0[,]𝑤) = (0[,](𝑘‘𝑥)))
32	31	feq3d 6690	. . . . 5 ⊢ (𝑤 = (𝑘‘𝑥) → (𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤) ↔ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥))))
33	32	ac6sfi 9240	. . . 4 ⊢ ((𝐼 ∈ Fin ∧ ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤)) → ∃𝑘(𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥))))
34	7, 30, 33	syl2anc 595	. . 3 ⊢ (𝜑 → ∃𝑘(𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥))))
35		frn 6713	. . . . . . . 8 ⊢ (𝑘:𝐼⟶ℝ → ran 𝑘 ⊆ ℝ)
36	35	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ran 𝑘 ⊆ ℝ)
37		0red 11206	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
38	37	snssd 4752	. . . . . . . 8 ⊢ (𝜑 → {0} ⊆ ℝ)
39	38	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → {0} ⊆ ℝ)
40	36, 39	unssd 4145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
41		ffn 6705	. . . . . . . . . 10 ⊢ (𝑘:𝐼⟶ℝ → 𝑘 Fn 𝐼)
42		dffn4 6798	. . . . . . . . . 10 ⊢ (𝑘 Fn 𝐼 ↔ 𝑘:𝐼–onto→ran 𝑘)
43	41, 42	sylib 221	. . . . . . . . 9 ⊢ (𝑘:𝐼⟶ℝ → 𝑘:𝐼–onto→ran 𝑘)
44		fofi 9269	. . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑘:𝐼–onto→ran 𝑘) → ran 𝑘 ∈ Fin)
45	7, 43, 44	syl2an 607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ran 𝑘 ∈ Fin)
46		snfi 9036	. . . . . . . 8 ⊢ {0} ∈ Fin
47		unfi 9151	. . . . . . . 8 ⊢ ((ran 𝑘 ∈ Fin ∧ {0} ∈ Fin) → (ran 𝑘 ∪ {0}) ∈ Fin)
48	45, 46, 47	sylancl 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ∈ Fin)
49		ssun2 4132	. . . . . . . . 9 ⊢ {0} ⊆ (ran 𝑘 ∪ {0})
50		c0ex 11195	. . . . . . . . . 10 ⊢ 0 ∈ V
51	50	snid 4628	. . . . . . . . 9 ⊢ 0 ∈ {0}
52	49, 51	sselii 3934	. . . . . . . 8 ⊢ 0 ∈ (ran 𝑘 ∪ {0})
53		ne0i 4294	. . . . . . . 8 ⊢ (0 ∈ (ran 𝑘 ∪ {0}) → (ran 𝑘 ∪ {0}) ≠ ∅)
54	52, 53	mp1i 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ≠ ∅)
55		ltso 11285	. . . . . . . 8 ⊢ < Or ℝ
56		fisupcl 9426	. . . . . . . 8 ⊢ (( < Or ℝ ∧ ((ran 𝑘 ∪ {0}) ∈ Fin ∧ (ran 𝑘 ∪ {0}) ≠ ∅ ∧ (ran 𝑘 ∪ {0}) ⊆ ℝ)) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ (ran 𝑘 ∪ {0}))
57	55, 56	mpan 702	. . . . . . 7 ⊢ (((ran 𝑘 ∪ {0}) ∈ Fin ∧ (ran 𝑘 ∪ {0}) ≠ ∅ ∧ (ran 𝑘 ∪ {0}) ⊆ ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ (ran 𝑘 ∪ {0}))
58	48, 54, 40, 57	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ (ran 𝑘 ∪ {0}))
59	40, 58	sseldd 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
60	59	adantrr 729	. . . 4 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
61		metf 24487	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝐵) → 𝐷:(𝐵 × 𝐵)⟶ℝ)
62		ffn 6705	. . . . . . 7 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ → 𝐷 Fn (𝐵 × 𝐵))
63	26, 61, 62	3syl 19	. . . . . 6 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
64	63	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷 Fn (𝐵 × 𝐵))
65	26	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷 ∈ (Met‘𝐵))
66		simprl 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ 𝐵)
67	66	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑓 ∈ 𝐵)
68		simprr 784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
69	68	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑔 ∈ 𝐵)
70		metcl 24489	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝐵) ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓𝐷𝑔) ∈ ℝ)
71	65, 67, 69, 70	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓𝐷𝑔) ∈ ℝ)
72		metge0 24502	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝐵) ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 0 ≤ (𝑓𝐷𝑔))
73	65, 67, 69, 72	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 0 ≤ (𝑓𝐷𝑔))
74	21	oveqdr 7438	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) = (𝑓(dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))𝑔))
75	6	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑆 ∈ 𝑊)
76	7	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐼 ∈ Fin)
77		fvexd 6896	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
78	77	ralrimiva 3157	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑅‘𝑥) ∈ V)
79	24	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐵 = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
80	66, 79	eleqtrd 2865	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
81	68, 79	eleqtrd 2865	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
82	1, 2, 75, 76, 78, 80, 81, 3, 4, 5	prdsdsval3 17533	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))
83	74, 82	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))
84	83	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))
85	11	adantlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
86	1, 2, 75, 76, 78, 3, 80	prdsbascl 17531	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
87	86	r19.21bi 3257	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉)
88	1, 2, 75, 76, 78, 3, 81	prdsbascl 17531	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
89	88	r19.21bi 3257	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ 𝑉)
90		metcl 24489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ (Met‘𝑉) ∧ (𝑓‘𝑥) ∈ 𝑉 ∧ (𝑔‘𝑥) ∈ 𝑉) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
91	85, 87, 89, 90	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
92	91	ad2ant2r 759	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
93		ffvelcdm 7076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘:𝐼⟶ℝ ∧ 𝑥 ∈ 𝐼) → (𝑘‘𝑥) ∈ ℝ)
94	93	ad2ant2lr 760	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ ℝ)
95	59	adantlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
96	95	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
97		simprr 784	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))
98	87	ad2ant2r 759	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓‘𝑥) ∈ 𝑉)
99	89	ad2ant2r 759	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑔‘𝑥) ∈ 𝑉)
100	97, 98, 99	fovcdmd 7582	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ (0[,](𝑘‘𝑥)))
101		0re 11205	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
102		elicc2 13433	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ (𝑘‘𝑥) ∈ ℝ) → (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ (0[,](𝑘‘𝑥)) ↔ (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ ∧ 0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∧ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ (𝑘‘𝑥))))
103	101, 94, 102	sylancr 598	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ (0[,](𝑘‘𝑥)) ↔ (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ ∧ 0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∧ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ (𝑘‘𝑥))))
104	100, 103	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ ∧ 0 ≤ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∧ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ (𝑘‘𝑥)))
105	104	simp3d 1162	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ (𝑘‘𝑥))
106	40	adantlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
107	106	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
108	52, 53	mp1i 14	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (ran 𝑘 ∪ {0}) ≠ ∅)
109		fimaxre2 12155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ran 𝑘 ∪ {0}) ⊆ ℝ ∧ (ran 𝑘 ∪ {0}) ∈ Fin) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
110	40, 48, 109	syl2anc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
111	110	adantlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
112	111	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
113		ssun1 4131	. . . . . . . . . . . . . . . . . 18 ⊢ ran 𝑘 ⊆ (ran 𝑘 ∪ {0})
114	41	ad2antlr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑘 Fn 𝐼)
115		simprl 782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑥 ∈ 𝐼)
116		fnfvelrn 7075	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → (𝑘‘𝑥) ∈ ran 𝑘)
117	114, 115, 116	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ ran 𝑘)
118	113, 117	sselid 3935	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ (ran 𝑘 ∪ {0}))
119		suprub 12171	. . . . . . . . . . . . . . . . 17 ⊢ ((((ran 𝑘 ∪ {0}) ⊆ ℝ ∧ (ran 𝑘 ∪ {0}) ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧) ∧ (𝑘‘𝑥) ∈ (ran 𝑘 ∪ {0})) → (𝑘‘𝑥) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
120	107, 108, 112, 118, 119	syl31anc 1400	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
121	92, 94, 96, 105, 120	letrd 11362	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
122	121	expr 461	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ 𝑥 ∈ 𝐼) → (𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
123	122	ralimdva 3177	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → (∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)) → ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
124	123	impr 459	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
125		ovex 7443	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
126	125	rgenw 3083	. . . . . . . . . . . . 13 ⊢ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
127		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))
128		breq1 5112	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) → (𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ↔ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
129	127, 128	ralrnmptw 7089	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V → (∀𝑤 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ↔ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
130	126, 129	ax-mp 5	. . . . . . . . . . . 12 ⊢ (∀𝑤 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ↔ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
131	124, 130	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∀𝑤 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
132	40	ad2ant2r 759	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
133	52, 53	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (ran 𝑘 ∪ {0}) ≠ ∅)
134	110	ad2ant2r 759	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
135	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 0 ∈ (ran 𝑘 ∪ {0}))
136		suprub 12171	. . . . . . . . . . . . . 14 ⊢ ((((ran 𝑘 ∪ {0}) ⊆ ℝ ∧ (ran 𝑘 ∪ {0}) ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧) ∧ 0 ∈ (ran 𝑘 ∪ {0})) → 0 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
137	132, 133, 134, 135, 136	syl31anc 1400	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 0 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
138		elsni 4606	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {0} → 𝑤 = 0)
139	138	breq1d 5119	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {0} → (𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ↔ 0 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
140	137, 139	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑤 ∈ {0} → 𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
141	140	ralrimiv 3156	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∀𝑤 ∈ {0}𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
142		ralunb 4150	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ↔ (∀𝑤 ∈ ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ∧ ∀𝑤 ∈ {0}𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
143	131, 141, 142	sylanbrc 594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∀𝑤 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
144	91	fmpttd 7110	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))):𝐼⟶ℝ)
145	144	frnd 6714	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ⊆ ℝ)
146		0red 11206	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 0 ∈ ℝ)
147	146	snssd 4752	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → {0} ⊆ ℝ)
148	145, 147	unssd 4145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ)
149		ressxr 11248	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ*
150	148, 149	sstrdi 3949	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ*)
151	150	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ*)
152	60	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
153	152	rexrd 11254	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ*)
154		supxrleub 13347	. . . . . . . . . . 11 ⊢ (((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ* ∧ sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ*) → (sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ*, < ) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ↔ ∀𝑤 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
155	151, 153, 154	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ*, < ) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ↔ ∀𝑤 ∈ (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0})𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
156	143, 155	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ*, < ) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
157	84, 156	eqbrtrd 5133	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓𝐷𝑔) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))
158		elicc2 13433	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ) → ((𝑓𝐷𝑔) ∈ (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )) ↔ ((𝑓𝐷𝑔) ∈ ℝ ∧ 0 ≤ (𝑓𝐷𝑔) ∧ (𝑓𝐷𝑔) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))))
159	101, 152, 158	sylancr 598	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ((𝑓𝐷𝑔) ∈ (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )) ↔ ((𝑓𝐷𝑔) ∈ ℝ ∧ 0 ≤ (𝑓𝐷𝑔) ∧ (𝑓𝐷𝑔) ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ))))
160	71, 73, 157, 159	mpbir3and 1361	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓𝐷𝑔) ∈ (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
161	160	an32s 664	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) ∈ (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
162	161	ralrimivva 3208	. . . . 5 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑓𝐷𝑔) ∈ (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
163		ffnov 7536	. . . . 5 ⊢ (𝐷:(𝐵 × 𝐵)⟶(0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )) ↔ (𝐷 Fn (𝐵 × 𝐵) ∧ ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑓𝐷𝑔) ∈ (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < ))))
164	64, 162, 163	sylanbrc 594	. . . 4 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷:(𝐵 × 𝐵)⟶(0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
165		oveq2 7418	. . . . . 6 ⊢ (𝑚 = sup((ran 𝑘 ∪ {0}), ℝ, < ) → (0[,]𝑚) = (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
166	165	feq3d 6690	. . . . 5 ⊢ (𝑚 = sup((ran 𝑘 ∪ {0}), ℝ, < ) → (𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚) ↔ 𝐷:(𝐵 × 𝐵)⟶(0[,]sup((ran 𝑘 ∪ {0}), ℝ, < ))))
167	166	rspcev 3581	. . . 4 ⊢ ((sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ ∧ 𝐷:(𝐵 × 𝐵)⟶(0[,]sup((ran 𝑘 ∪ {0}), ℝ, < ))) → ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚))
168	60, 164, 167	syl2anc 595	. . 3 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚))
169	34, 168	exlimddv 1965	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚))
170		isbnd3 38435	. 2 ⊢ (𝐷 ∈ (Bnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚)))
171	26, 169, 170	sylanbrc 594	1 ⊢ (𝜑 → 𝐷 ∈ (Bnd‘𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  ∅c0 4286  {csn 4589   class class class wbr 5109   ↦ cmpt 5192   Or wor 5568   × cxp 5659  ran crn 5662   ↾ cres 5663   Fn wfn 6531  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7410  Fincfn 8939  supcsup 9396  ℝcr 11094  0cc0 11095  ℝ^*cxr 11237   < clt 11238   ≤ cle 11239  [,]cicc 13370  Basecbs 17264  distcds 17314  X_scprds 17493  Metcmet 21508  Bndcbnd 38418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-ec 8692  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-rp 13012  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-icc 13374  df-fz 13531  df-struct 17202  df-slot 17237  df-ndx 17249  df-base 17265  df-plusg 17318  df-mulr 17319  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-hom 17329  df-cco 17330  df-prds 17495  df-psmet 21514  df-xmet 21515  df-met 21516  df-bl 21517  df-bnd 38430

This theorem is referenced by: (None)