Theorem prdsbnd 35070

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 2821	. . . 4 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
2		eqid 2821	. . . 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 2821	. . . 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 6684	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
9		prdsbnd.m	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Bnd‘𝑉))
10		bndmet 35058	. . . . 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 22979	. . 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 6723	. . . . . . . 8 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
17	15, 16	sylib 220	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
18	17	oveq2d 7171	. . . . . 6 ⊢ (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
19	14, 18	syl5eq 2868	. . . . 5 ⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
20	19	fveq2d 6673	. . . 4 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
21	13, 20	syl5eq 2868	. . 3 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
22		prdsbnd.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑌)
23	19	fveq2d 6673	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
24	22, 23	syl5eq 2868	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
25	24	fveq2d 6673	. . 3 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
26	12, 21, 25	3eltr4d 2928	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵))
27		isbnd3 35061	. . . . . . 7 ⊢ (𝐸 ∈ (Bnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤)))
28	27	simprbi 499	. . . . . 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 3182	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤))
31		oveq2 7163	. . . . . 6 ⊢ (𝑤 = (𝑘‘𝑥) → (0[,]𝑤) = (0[,](𝑘‘𝑥)))
32	31	feq3d 6500	. . . . 5 ⊢ (𝑤 = (𝑘‘𝑥) → (𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤) ↔ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥))))
33	32	ac6sfi 8761	. . . 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 586	. . 3 ⊢ (𝜑 → ∃𝑘(𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥))))
35		frn 6519	. . . . . . . 8 ⊢ (𝑘:𝐼⟶ℝ → ran 𝑘 ⊆ ℝ)
36	35	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ran 𝑘 ⊆ ℝ)
37		0red 10643	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
38	37	snssd 4741	. . . . . . . 8 ⊢ (𝜑 → {0} ⊆ ℝ)
39	38	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → {0} ⊆ ℝ)
40	36, 39	unssd 4161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
41		ffn 6513	. . . . . . . . . 10 ⊢ (𝑘:𝐼⟶ℝ → 𝑘 Fn 𝐼)
42		dffn4 6595	. . . . . . . . . 10 ⊢ (𝑘 Fn 𝐼 ↔ 𝑘:𝐼–onto→ran 𝑘)
43	41, 42	sylib 220	. . . . . . . . 9 ⊢ (𝑘:𝐼⟶ℝ → 𝑘:𝐼–onto→ran 𝑘)
44		fofi 8809	. . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑘:𝐼–onto→ran 𝑘) → ran 𝑘 ∈ Fin)
45	7, 43, 44	syl2an 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ran 𝑘 ∈ Fin)
46		snfi 8593	. . . . . . . 8 ⊢ {0} ∈ Fin
47		unfi 8784	. . . . . . . 8 ⊢ ((ran 𝑘 ∈ Fin ∧ {0} ∈ Fin) → (ran 𝑘 ∪ {0}) ∈ Fin)
48	45, 46, 47	sylancl 588	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ∈ Fin)
49		ssun2 4148	. . . . . . . . 9 ⊢ {0} ⊆ (ran 𝑘 ∪ {0})
50		c0ex 10634	. . . . . . . . . 10 ⊢ 0 ∈ V
51	50	snid 4600	. . . . . . . . 9 ⊢ 0 ∈ {0}
52	49, 51	sselii 3963	. . . . . . . 8 ⊢ 0 ∈ (ran 𝑘 ∪ {0})
53		ne0i 4299	. . . . . . . 8 ⊢ (0 ∈ (ran 𝑘 ∪ {0}) → (ran 𝑘 ∪ {0}) ≠ ∅)
54	52, 53	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ≠ ∅)
55		ltso 10720	. . . . . . . 8 ⊢ < Or ℝ
56		fisupcl 8932	. . . . . . . 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 688	. . . . . . 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 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ (ran 𝑘 ∪ {0}))
59	40, 58	sseldd 3967	. . . . 5 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
60	59	adantrr 715	. . . 4 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
61		metf 22939	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝐵) → 𝐷:(𝐵 × 𝐵)⟶ℝ)
62		ffn 6513	. . . . . . 7 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ → 𝐷 Fn (𝐵 × 𝐵))
63	26, 61, 62	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
64	63	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷 Fn (𝐵 × 𝐵))
65	26	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷 ∈ (Met‘𝐵))
66		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ 𝐵)
67	66	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑓 ∈ 𝐵)
68		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
69	68	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑔 ∈ 𝐵)
70		metcl 22941	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝐵) ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (𝑓𝐷𝑔) ∈ ℝ)
71	65, 67, 69, 70	syl3anc 1367	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓𝐷𝑔) ∈ ℝ)
72		metge0 22954	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝐵) ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 0 ≤ (𝑓𝐷𝑔))
73	65, 67, 69, 72	syl3anc 1367	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 0 ≤ (𝑓𝐷𝑔))
74	21	oveqdr 7183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) = (𝑓(dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))𝑔))
75	6	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑆 ∈ 𝑊)
76	7	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐼 ∈ Fin)
77		fvexd 6684	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
78	77	ralrimiva 3182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑅‘𝑥) ∈ V)
79	24	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐵 = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
80	66, 79	eleqtrd 2915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
81	68, 79	eleqtrd 2915	. . . . . . . . . . . 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 16757	. . . . . . . . . . 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 2856	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))
84	83	adantr 483	. . . . . . . . 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 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
86	1, 2, 75, 76, 78, 3, 80	prdsbascl 16755	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
87	86	r19.21bi 3208	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉)
88	1, 2, 75, 76, 78, 3, 81	prdsbascl 16755	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
89	88	r19.21bi 3208	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ 𝑉)
90		metcl 22941	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ (Met‘𝑉) ∧ (𝑓‘𝑥) ∈ 𝑉 ∧ (𝑔‘𝑥) ∈ 𝑉) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
91	85, 87, 89, 90	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
92	91	ad2ant2r 745	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ ℝ)
93		ffvelrn 6848	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘:𝐼⟶ℝ ∧ 𝑥 ∈ 𝐼) → (𝑘‘𝑥) ∈ ℝ)
94	93	ad2ant2lr 746	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ ℝ)
95	59	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
96	95	adantr 483	. . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . 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 745	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓‘𝑥) ∈ 𝑉)
99	89	ad2ant2r 745	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑔‘𝑥) ∈ 𝑉)
100	97, 98, 99	fovrnd 7319	. . . . . . . . . . . . . . . . . 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 10642	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
102		elicc2 12800	. . . . . . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . 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 1140	. . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
107	106	adantr 483	. . . . . . . . . . . . . . . . 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 11585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ran 𝑘 ∪ {0}) ⊆ ℝ ∧ (ran 𝑘 ∪ {0}) ∈ Fin) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
110	40, 48, 109	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
111	110	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
112	111	adantr 483	. . . . . . . . . . . . . . . . 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 4147	. . . . . . . . . . . . . . . . . 18 ⊢ ran 𝑘 ⊆ (ran 𝑘 ∪ {0})
114	41	ad2antlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑘 Fn 𝐼)
115		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑥 ∈ 𝐼)
116		fnfvelrn 6847	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → (𝑘‘𝑥) ∈ ran 𝑘)
117	114, 115, 116	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ ran 𝑘)
118	113, 117	sseldi 3964	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ (ran 𝑘 ∪ {0}))
119		suprub 11601	. . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . 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 10796	. . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . 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 457	. . . . . . . . . . . 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 7188	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
126	125	rgenw 3150	. . . . . . . . . . . . 13 ⊢ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
127		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))
128		breq1 5068	. . . . . . . . . . . . . 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 6859	. . . . . . . . . . . . 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 236	. . . . . . . . . . 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 745	. . . . . . . . . . . . . 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 745	. . . . . . . . . . . . . 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 11601	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 4583	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {0} → 𝑤 = 0)
139	138	breq1d 5075	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {0} → (𝑤 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < ) ↔ 0 ≤ sup((ran 𝑘 ∪ {0}), ℝ, < )))
140	137, 139	syl5ibrcom 249	. . . . . . . . . . . 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 3181	. . . . . . . . . . 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 4166	. . . . . . . . . . 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 585	. . . . . . . . . 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 6878	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))):𝐼⟶ℝ)
145	144	frnd 6520	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ⊆ ℝ)
146		0red 10643	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 0 ∈ ℝ)
147	146	snssd 4741	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → {0} ⊆ ℝ)
148	145, 147	unssd 4161	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ)
149		ressxr 10684	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ*
150	148, 149	sstrdi 3978	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ*)
151	150	adantr 483	. . . . . . . . . . 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 713	. . . . . . . . . . . 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 10690	. . . . . . . . . . 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 12718	. . . . . . . . . . 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 586	. . . . . . . . . 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 259	. . . . . . . . 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 5087	. . . . . . . 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 12800	. . . . . . . . 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 589	. . . . . . . 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 1338	. . . . . . 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 650	. . . . . 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 3191	. . . . 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 7277	. . . . 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 585	. . . 4 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷:(𝐵 × 𝐵)⟶(0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
165		oveq2 7163	. . . . . 6 ⊢ (𝑚 = sup((ran 𝑘 ∪ {0}), ℝ, < ) → (0[,]𝑚) = (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
166	165	feq3d 6500	. . . . 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 586	. . 3 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚))
169	34, 168	exlimddv 1932	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚))
170		isbnd3 35061	. 2 ⊢ (𝐷 ∈ (Bnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚)))
171	26, 169, 170	sylanbrc 585	1 ⊢ (𝜑 → 𝐷 ∈ (Bnd‘𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∪ cun 3933   ⊆ wss 3935  ∅c0 4290  {csn 4566   class class class wbr 5065   ↦ cmpt 5145   Or wor 5472   × cxp 5552  ran crn 5555   ↾ cres 5556   Fn wfn 6349  ⟶wf 6350  –onto→wfo 6352  ‘cfv 6354  (class class class)co 7155  Fincfn 8508  supcsup 8903  ℝcr 10535  0cc0 10536  ℝ^*cxr 10673   < clt 10674   ≤ cle 10675  [,]cicc 12740  Basecbs 16482  distcds 16573  X_scprds 16718  Metcmet 20530  Bndcbnd 35044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-ec 8290  df-map 8407  df-ixp 8461  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-rp 12389  df-xneg 12506  df-xadd 12507  df-xmul 12508  df-icc 12744  df-fz 12892  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-plusg 16577  df-mulr 16578  df-sca 16580  df-vsca 16581  df-ip 16582  df-tset 16583  df-ple 16584  df-ds 16586  df-hom 16588  df-cco 16589  df-prds 16720  df-psmet 20536  df-xmet 20537  df-met 20538  df-bl 20539  df-bnd 35056

This theorem is referenced by: (None)