Theorem prdsbnd 37800

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 2737	. . . 4 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
2		eqid 2737	. . . 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 2737	. . . 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 6921	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
9		prdsbnd.m	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Bnd‘𝑉))
10		bndmet 37788	. . . . 5 ⊢ (𝐸 ∈ (Bnd‘𝑉) → 𝐸 ∈ (Met‘𝑉))
11	9, 10	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
12	1, 2, 3, 4, 5, 6, 7, 8, 11	prdsmet 24380	. . 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 6967	. . . . . . . 8 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
17	15, 16	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
18	17	oveq2d 7447	. . . . . 6 ⊢ (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
19	14, 18	eqtrid 2789	. . . . 5 ⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
20	19	fveq2d 6910	. . . 4 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
21	13, 20	eqtrid 2789	. . 3 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
22		prdsbnd.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑌)
23	19	fveq2d 6910	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
24	22, 23	eqtrid 2789	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
25	24	fveq2d 6910	. . 3 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
26	12, 21, 25	3eltr4d 2856	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵))
27		isbnd3 37791	. . . . . . 7 ⊢ (𝐸 ∈ (Bnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤)))
28	27	simprbi 496	. . . . . 6 ⊢ (𝐸 ∈ (Bnd‘𝑉) → ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤))
29	9, 28	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤))
30	29	ralrimiva 3146	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤))
31		oveq2 7439	. . . . . 6 ⊢ (𝑤 = (𝑘‘𝑥) → (0[,]𝑤) = (0[,](𝑘‘𝑥)))
32	31	feq3d 6723	. . . . 5 ⊢ (𝑤 = (𝑘‘𝑥) → (𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤) ↔ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥))))
33	32	ac6sfi 9320	. . . 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 584	. . 3 ⊢ (𝜑 → ∃𝑘(𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥))))
35		frn 6743	. . . . . . . 8 ⊢ (𝑘:𝐼⟶ℝ → ran 𝑘 ⊆ ℝ)
36	35	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ran 𝑘 ⊆ ℝ)
37		0red 11264	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
38	37	snssd 4809	. . . . . . . 8 ⊢ (𝜑 → {0} ⊆ ℝ)
39	38	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → {0} ⊆ ℝ)
40	36, 39	unssd 4192	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
41		ffn 6736	. . . . . . . . . 10 ⊢ (𝑘:𝐼⟶ℝ → 𝑘 Fn 𝐼)
42		dffn4 6826	. . . . . . . . . 10 ⊢ (𝑘 Fn 𝐼 ↔ 𝑘:𝐼–onto→ran 𝑘)
43	41, 42	sylib 218	. . . . . . . . 9 ⊢ (𝑘:𝐼⟶ℝ → 𝑘:𝐼–onto→ran 𝑘)
44		fofi 9351	. . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑘:𝐼–onto→ran 𝑘) → ran 𝑘 ∈ Fin)
45	7, 43, 44	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ran 𝑘 ∈ Fin)
46		snfi 9083	. . . . . . . 8 ⊢ {0} ∈ Fin
47		unfi 9211	. . . . . . . 8 ⊢ ((ran 𝑘 ∈ Fin ∧ {0} ∈ Fin) → (ran 𝑘 ∪ {0}) ∈ Fin)
48	45, 46, 47	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ∈ Fin)
49		ssun2 4179	. . . . . . . . 9 ⊢ {0} ⊆ (ran 𝑘 ∪ {0})
50		c0ex 11255	. . . . . . . . . 10 ⊢ 0 ∈ V
51	50	snid 4662	. . . . . . . . 9 ⊢ 0 ∈ {0}
52	49, 51	sselii 3980	. . . . . . . 8 ⊢ 0 ∈ (ran 𝑘 ∪ {0})
53		ne0i 4341	. . . . . . . 8 ⊢ (0 ∈ (ran 𝑘 ∪ {0}) → (ran 𝑘 ∪ {0}) ≠ ∅)
54	52, 53	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ≠ ∅)
55		ltso 11341	. . . . . . . 8 ⊢ < Or ℝ
56		fisupcl 9509	. . . . . . . 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 690	. . . . . . 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 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ (ran 𝑘 ∪ {0}))
59	40, 58	sseldd 3984	. . . . 5 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
60	59	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
61		metf 24340	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝐵) → 𝐷:(𝐵 × 𝐵)⟶ℝ)
62		ffn 6736	. . . . . . 7 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ → 𝐷 Fn (𝐵 × 𝐵))
63	26, 61, 62	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
64	63	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷 Fn (𝐵 × 𝐵))
65	26	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷 ∈ (Met‘𝐵))
66		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ 𝐵)
67	66	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑓 ∈ 𝐵)
68		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
69	68	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑔 ∈ 𝐵)
70		metcl 24342	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝐵) ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓𝐷𝑔) ∈ ℝ)
71	65, 67, 69, 70	syl3anc 1373	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓𝐷𝑔) ∈ ℝ)
72		metge0 24355	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝐵) ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 0 ≤ (𝑓𝐷𝑔))
73	65, 67, 69, 72	syl3anc 1373	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 0 ≤ (𝑓𝐷𝑔))
74	21	oveqdr 7459	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) = (𝑓(dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))𝑔))
75	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑆 ∈ 𝑊)
76	7	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐼 ∈ Fin)
77		fvexd 6921	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
78	77	ralrimiva 3146	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑅‘𝑥) ∈ V)
79	24	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐵 = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
80	66, 79	eleqtrd 2843	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
81	68, 79	eleqtrd 2843	. . . . . . . . . . . 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 17530	. . . . . . . . . . 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 2777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))
84	83	adantr 480	. . . . . . . . 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 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
86	1, 2, 75, 76, 78, 3, 80	prdsbascl 17528	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
87	86	r19.21bi 3251	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉)
88	1, 2, 75, 76, 78, 3, 81	prdsbascl 17528	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
89	88	r19.21bi 3251	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ 𝑉)
90		metcl 24342	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ (Met‘𝑉) ∧ (𝑓‘𝑥) ∈ 𝑉 ∧ (𝑔‘𝑥) ∈ 𝑉) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
91	85, 87, 89, 90	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
92	91	ad2ant2r 747	. . . . . . . . . . . . . . . 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 7101	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘:𝐼⟶ℝ ∧ 𝑥 ∈ 𝐼) → (𝑘‘𝑥) ∈ ℝ)
94	93	ad2ant2lr 748	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ ℝ)
95	59	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
96	95	adantr 480	. . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓‘𝑥) ∈ 𝑉)
99	89	ad2ant2r 747	. . . . . . . . . . . . . . . . . . 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 7605	. . . . . . . . . . . . . . . . . 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 11263	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
102		elicc2 13452	. . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . 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 1145	. . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
107	106	adantr 480	. . . . . . . . . . . . . . . . 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 13	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (ran 𝑘 ∪ {0}) ≠ ∅)
109		fimaxre2 12213	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ran 𝑘 ∪ {0}) ⊆ ℝ ∧ (ran 𝑘 ∪ {0}) ∈ Fin) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
110	40, 48, 109	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
111	110	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
112	111	adantr 480	. . . . . . . . . . . . . . . . 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 4178	. . . . . . . . . . . . . . . . . 18 ⊢ ran 𝑘 ⊆ (ran 𝑘 ∪ {0})
114	41	ad2antlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑘 Fn 𝐼)
115		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑥 ∈ 𝐼)
116		fnfvelrn 7100	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → (𝑘‘𝑥) ∈ ran 𝑘)
117	114, 115, 116	syl2anc 584	. . . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ (ran 𝑘 ∪ {0}))
119		suprub 12229	. . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . 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 11418	. . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . 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 3167	. . . . . . . . . . . . 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 454	. . . . . . . . . . . 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 7464	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
126	125	rgenw 3065	. . . . . . . . . . . . 13 ⊢ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
127		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))
128		breq1 5146	. . . . . . . . . . . . . 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 7114	. . . . . . . . . . . . 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 234	. . . . . . . . . . 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 747	. . . . . . . . . . . . . 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 13	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (ran 𝑘 ∪ {0}) ≠ ∅)
134	110	ad2ant2r 747	. . . . . . . . . . . . . 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 12229	. . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . 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 4643	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {0} → 𝑤 = 0)
139	138	breq1d 5153	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {0} → (𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ↔ 0 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
140	137, 139	syl5ibrcom 247	. . . . . . . . . . . 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 3145	. . . . . . . . . . 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 4197	. . . . . . . . . . 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 583	. . . . . . . . . 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 7135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))):𝐼⟶ℝ)
145	144	frnd 6744	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ⊆ ℝ)
146		0red 11264	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 0 ∈ ℝ)
147	146	snssd 4809	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → {0} ⊆ ℝ)
148	145, 147	unssd 4192	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ)
149		ressxr 11305	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ*
150	148, 149	sstrdi 3996	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ*)
151	150	adantr 480	. . . . . . . . . . 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 715	. . . . . . . . . . . 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 11311	. . . . . . . . . . 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 13368	. . . . . . . . . . 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 584	. . . . . . . . . 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 257	. . . . . . . . 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 5165	. . . . . . . 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 13452	. . . . . . . . 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 587	. . . . . . . 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 1343	. . . . . . 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 652	. . . . . 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 3202	. . . . 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 7559	. . . . 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 583	. . . 4 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷:(𝐵 × 𝐵)⟶(0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
165		oveq2 7439	. . . . . 6 ⊢ (𝑚 = sup((ran 𝑘 ∪ {0}), ℝ, < ) → (0[,]𝑚) = (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
166	165	feq3d 6723	. . . . 5 ⊢ (𝑚 = sup((ran 𝑘 ∪ {0}), ℝ, < ) → (𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚) ↔ 𝐷:(𝐵 × 𝐵)⟶(0[,]sup((ran 𝑘 ∪ {0}), ℝ, < ))))
167	166	rspcev 3622	. . . 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 584	. . 3 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚))
169	34, 168	exlimddv 1935	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚))
170		isbnd3 37791	. 2 ⊢ (𝐷 ∈ (Bnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚)))
171	26, 169, 170	sylanbrc 583	1 ⊢ (𝜑 → 𝐷 ∈ (Bnd‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626   class class class wbr 5143   ↦ cmpt 5225   Or wor 5591   × cxp 5683  ran crn 5686   ↾ cres 5687   Fn wfn 6556  ⟶wf 6557  –onto→wfo 6559  ‘cfv 6561  (class class class)co 7431  Fincfn 8985  supcsup 9480  ℝcr 11154  0cc0 11155  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296  [,]cicc 13390  Basecbs 17247  distcds 17306  X_scprds 17490  Metcmet 21350  Bndcbnd 37774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-ec 8747  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-icc 13394  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-prds 17492  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-bnd 37786

This theorem is referenced by: (None)