prdsbnd - Mathbox for Jeff Madsen

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Theorem prdsbnd 38707

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 = (𝑆X_s𝑅)
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 2761	. . . 4 ⊢ (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) = (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
2		eqid 2761	. . . 4 ⊢ (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
3		prdsbnd.v	. . . 4 ⊢ 𝑉 = (Base‘(𝑅‘𝑥))
4		prdsbnd.e	. . . 4 ⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉))
5		eqid 2761	. . . 4 ⊢ (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
6		prdsbnd.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
7		prdsbnd.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
8		fvexd 6898	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
9		prdsbnd.m	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Bnd‘𝑉))
10		bndmet 38695	. . . . 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 24682	. . 3 ⊢ (𝜑 → (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) ∈ (Met‘(Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
13		prdsbnd.d	. . . 4 ⊢ 𝐷 = (dist‘𝑌)
14		prdsbnd.y	. . . . . 6 ⊢ 𝑌 = (𝑆X_s𝑅)
15		prdsbnd.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
16		dffn5 6941	. . . . . . . 8 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
17	15, 16	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
18	17	oveq2d 7434	. . . . . 6 ⊢ (𝜑 → (𝑆X_s𝑅) = (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
19	14, 18	eqtrid 2808	. . . . 5 ⊢ (𝜑 → 𝑌 = (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
20	19	fveq2d 6887	. . . 4 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
21	13, 20	eqtrid 2808	. . 3 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
22		prdsbnd.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑌)
23	19	fveq2d 6887	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
24	22, 23	eqtrid 2808	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
25	24	fveq2d 6887	. . 3 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
26	12, 21, 25	3eltr4d 2876	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵))
27		isbnd3 38698	. . . . . . 7 ⊢ (𝐸 ∈ (Bnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤)))
28	27	simprbi 503	. . . . . 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 3155	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ ℝ 𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤))
31		oveq2 7426	. . . . . 6 ⊢ (𝑤 = (𝑘‘𝑥) → (0[,]𝑤) = (0[,](𝑘‘𝑥)))
32	31	feq3d 6692	. . . . 5 ⊢ (𝑤 = (𝑘‘𝑥) → (𝐸:(𝑉 × 𝑉)⟶(0[,]𝑤) ↔ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥))))
33	32	ac6sfi 9268	. . . 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 596	. . 3 ⊢ (𝜑 → ∃𝑘(𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥))))
35		frn 6715	. . . . . . . 8 ⊢ (𝑘:𝐼⟶ℝ → ran 𝑘 ⊆ ℝ)
36	35	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ran 𝑘 ⊆ ℝ)
37		0red 11304	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
38	37	snssd 4747	. . . . . . . 8 ⊢ (𝜑 → {0} ⊆ ℝ)
39	38	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → {0} ⊆ ℝ)
40	36, 39	unssd 4138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
41		ffn 6707	. . . . . . . . . 10 ⊢ (𝑘:𝐼⟶ℝ → 𝑘 Fn 𝐼)
42		dffn4 6800	. . . . . . . . . 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 9298	. . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑘:𝐼–onto→ran 𝑘) → ran 𝑘 ∈ Fin)
45	7, 43, 44	syl2an 608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ran 𝑘 ∈ Fin)
46		snfi 9064	. . . . . . . 8 ⊢ {0} ∈ Fin
47		unfi 9179	. . . . . . . 8 ⊢ ((ran 𝑘 ∈ Fin ∧ {0} ∈ Fin) → (ran 𝑘 ∪ {0}) ∈ Fin)
48	45, 46, 47	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ∈ Fin)
49		ssun2 4125	. . . . . . . . 9 ⊢ {0} ⊆ (ran 𝑘 ∪ {0})
50		c0ex 11293	. . . . . . . . . 10 ⊢ 0 ∈ V
51	50	snid 4623	. . . . . . . . 9 ⊢ 0 ∈ {0}
52	49, 51	sselii 3928	. . . . . . . 8 ⊢ 0 ∈ (ran 𝑘 ∪ {0})
53		ne0i 4287	. . . . . . . 8 ⊢ (0 ∈ (ran 𝑘 ∪ {0}) → (ran 𝑘 ∪ {0}) ≠ ∅)
54	52, 53	mp1i 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ≠ ∅)
55		ltso 11383	. . . . . . . 8 ⊢ < Or ℝ
56		fisupcl 9455	. . . . . . . 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 703	. . . . . . 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 3932	. . . . 5 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
60	59	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
61		metf 24642	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝐵) → 𝐷:(𝐵 × 𝐵)⟶ℝ)
62		ffn 6707	. . . . . . 7 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ → 𝐷 Fn (𝐵 × 𝐵))
63	26, 61, 62	3syl 19	. . . . . 6 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
64	63	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷 Fn (𝐵 × 𝐵))
65	26	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷 ∈ (Met‘𝐵))
66		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ 𝐵)
67	66	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑓 ∈ 𝐵)
68		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ 𝐵)
69	68	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑔 ∈ 𝐵)
70		metcl 24644	. . . . . . . . 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 24657	. . . . . . . . 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 7446	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) = (𝑓(dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))𝑔))
75	6	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑆 ∈ 𝑊)
76	7	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐼 ∈ Fin)
77		fvexd 6898	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
78	77	ralrimiva 3155	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑅‘𝑥) ∈ V)
79	24	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝐵 = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
80	66, 79	eleqtrd 2863	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑓 ∈ (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
81	68, 79	eleqtrd 2863	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 𝑔 ∈ (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
82	1, 2, 75, 76, 78, 80, 81, 3, 4, 5	prdsdsval3 17649	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓(dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))
83	74, 82	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑓𝐷𝑔) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))
84	83	adantr 486	. . . . . . . . 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 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
86	1, 2, 75, 76, 78, 3, 80	prdsbascl 17647	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ 𝑉)
87	86	r19.21bi 3255	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ 𝑉)
88	1, 2, 75, 76, 78, 3, 81	prdsbascl 17647	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
89	88	r19.21bi 3255	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑔‘𝑥) ∈ 𝑉)
90		metcl 24644	. . . . . . . . . . . . . . . . . 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 760	. . . . . . . . . . . . . . . 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 7079	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘:𝐼⟶ℝ ∧ 𝑥 ∈ 𝐼) → (𝑘‘𝑥) ∈ ℝ)
94	93	ad2ant2lr 761	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ ℝ)
95	59	adantlr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → sup((ran 𝑘 ∪ {0}), ℝ, < ) ∈ ℝ)
96	95	adantr 486	. . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . 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 760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑓‘𝑥) ∈ 𝑉)
99	89	ad2ant2r 760	. . . . . . . . . . . . . . . . . . 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 7591	. . . . . . . . . . . . . . . . . 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 11303	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
102		elicc2 13535	. . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → (ran 𝑘 ∪ {0}) ⊆ ℝ)
107	106	adantr 486	. . . . . . . . . . . . . . . . 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 12255	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ran 𝑘 ∪ {0}) ⊆ ℝ ∧ (ran 𝑘 ∪ {0}) ∈ Fin) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
110	40, 48, 109	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘:𝐼⟶ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
111	110	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (ran 𝑘 ∪ {0})𝑤 ≤ 𝑧)
112	111	adantr 486	. . . . . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . . . . . . 18 ⊢ ran 𝑘 ⊆ (ran 𝑘 ∪ {0})
114	41	ad2antlr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑘 Fn 𝐼)
115		simprl 783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝑥 ∈ 𝐼)
116		fnfvelrn 7078	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → (𝑘‘𝑥) ∈ ran 𝑘)
117	114, 115, 116	syl2anc 596	. . . . . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) ∧ 𝑘:𝐼⟶ℝ) ∧ (𝑥 ∈ 𝐼 ∧ 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → (𝑘‘𝑥) ∈ (ran 𝑘 ∪ {0}))
119		suprub 12271	. . . . . . . . . . . . . . . . 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 11460	. . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . 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 460	. . . . . . . . . . . 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 7451	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
126	125	rgenw 3081	. . . . . . . . . . . . 13 ⊢ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥)𝐸(𝑔‘𝑥)) ∈ V
127		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥)))
128		breq1 5106	. . . . . . . . . . . . . 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 7092	. . . . . . . . . . . . 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 760	. . . . . . . . . . . . . 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 760	. . . . . . . . . . . . . 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 12271	. . . . . . . . . . . . . 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 4601	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {0} → 𝑤 = 0)
139	138	breq1d 5113	. . . . . . . . . . . . 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 3154	. . . . . . . . . . 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 4143	. . . . . . . . . . 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 595	. . . . . . . . . 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 7113	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))):𝐼⟶ℝ)
145	144	frnd 6716	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ⊆ ℝ)
146		0red 11304	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → 0 ∈ ℝ)
147	146	snssd 4747	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → {0} ⊆ ℝ)
148	145, 147	unssd 4138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ)
149		ressxr 11346	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
150	148, 149	sstrdi 3943	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)𝐸(𝑔‘𝑥))) ∪ {0}) ⊆ ℝ^*)
151	150	adantr 486	. . . . . . . . . . 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 728	. . . . . . . . . . . 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 11352	. . . . . . . . . . 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 13449	. . . . . . . . . . 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 596	. . . . . . . . . 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 5127	. . . . . . . 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 13535	. . . . . . . . 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 599	. . . . . . . 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 665	. . . . . 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 3206	. . . . 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 7544	. . . . 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 595	. . . 4 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → 𝐷:(𝐵 × 𝐵)⟶(0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
165		oveq2 7426	. . . . . 6 ⊢ (𝑚 = sup((ran 𝑘 ∪ {0}), ℝ, < ) → (0[,]𝑚) = (0[,]sup((ran 𝑘 ∪ {0}), ℝ, < )))
166	165	feq3d 6692	. . . . 5 ⊢ (𝑚 = sup((ran 𝑘 ∪ {0}), ℝ, < ) → (𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚) ↔ 𝐷:(𝐵 × 𝐵)⟶(0[,]sup((ran 𝑘 ∪ {0}), ℝ, < ))))
167	166	rspcev 3577	. . . 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 596	. . 3 ⊢ ((𝜑 ∧ (𝑘:𝐼⟶ℝ ∧ ∀𝑥 ∈ 𝐼 𝐸:(𝑉 × 𝑉)⟶(0[,](𝑘‘𝑥)))) → ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚))
169	34, 168	exlimddv 1968	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚))
170		isbnd3 38698	. 2 ⊢ (𝐷 ∈ (Bnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∃𝑚 ∈ ℝ 𝐷:(𝐵 × 𝐵)⟶(0[,]𝑚)))
171	26, 169, 170	sylanbrc 595	1 ⊢ (𝜑 → 𝐷 ∈ (Bnd‘𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 Vcvv 3451 ∪ cun 3897 ⊆ wss 3899 ∅c0 4279 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 Or wor 5558 × cxp 5649 ran crn 5652 ↾ cres 5653 Fn wfn 6532 ⟶wf 6533 –onto→wfo 6535 ‘cfv 6537 (class class class)co 7418 Fincfn 8966 supcsup 9425 ℝcr 11192 0cc0 11193 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 [,]cicc 13472 Basecbs 17380 distcds 17430 X_scprds 17609 Metcmet 21657 Bndcbnd 38681

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-ec 8712 df-map 8842 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9427 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-icc 13476 df-fz 13633 df-struct 17318 df-slot 17353 df-ndx 17365 df-base 17381 df-plusg 17434 df-mulr 17435 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-hom 17445 df-cco 17446 df-prds 17611 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-bnd 38693

This theorem is used by: (None)

W3C validator