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Theorem ubthlem2 27727

Description: Lemma for ubth 27729. Given that there is a closed ball 𝐵(𝑃, 𝑅) in 𝐴‘𝐾, for any 𝑥 ∈ 𝐵(0, 1), we have 𝑃 + 𝑅 · 𝑥 ∈ 𝐵(𝑃, 𝑅) and 𝑃 ∈ 𝐵(𝑃, 𝑅), so both of these have norm(𝑡(𝑧)) ≤ 𝐾 and so norm(𝑡(𝑥 )) ≤ (norm(𝑡(𝑃)) + norm(𝑡(𝑃 + 𝑅 · 𝑥))) / 𝑅 ≤ ( 𝐾 + 𝐾) / 𝑅, which is our desired uniform bound. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ubth.1	⊢ 𝑋 = (BaseSet‘𝑈)
ubth.2	⊢ 𝑁 = (norm_CV‘𝑊)
ubthlem.3	⊢ 𝐷 = (IndMet‘𝑈)
ubthlem.4	⊢ 𝐽 = (MetOpen‘𝐷)
ubthlem.5	⊢ 𝑈 ∈ CBan
ubthlem.6	⊢ 𝑊 ∈ NrmCVec
ubthlem.7	⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊))
ubthlem.8	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐)
ubthlem.9	⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
ubthlem.10	⊢ (𝜑 → 𝐾 ∈ ℕ)
ubthlem.11	⊢ (𝜑 → 𝑃 ∈ 𝑋)
ubthlem.12	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
ubthlem.13	⊢ (𝜑 → {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴‘𝐾))

**Assertion**
Ref	Expression
ubthlem2	⊢ (𝜑 → ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑)

Distinct variable groups: 𝑘,𝑐,𝑥,𝑧,𝐴 𝑡,𝑐,𝐷,𝑘,𝑥,𝑧 𝑘,𝐽,𝑡,𝑥 𝑘,𝑑,𝑡,𝑥,𝑧,𝐾 𝑐,𝑑,𝑁,𝑘,𝑡,𝑥,𝑧 𝑡,𝑃,𝑧 𝜑,𝑐,𝑘,𝑡,𝑥 𝑅,𝑑,𝑡,𝑥,𝑧 𝑇,𝑐,𝑑,𝑘,𝑡,𝑥,𝑧 𝑈,𝑐,𝑑,𝑡,𝑥,𝑧 𝑊,𝑐,𝑑,𝑡,𝑥 𝑋,𝑐,𝑑,𝑘,𝑡,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑧,𝑑) 𝐴(𝑡,𝑑) 𝐷(𝑑) 𝑃(𝑥,𝑘,𝑐,𝑑) 𝑅(𝑘,𝑐) 𝑈(𝑘) 𝐽(𝑧,𝑐,𝑑) 𝐾(𝑐) 𝑊(𝑧,𝑘)

**Proof of Theorem ubthlem2**
Step	Hyp	Ref	Expression
1		ubthlem.10	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ)
2	1	nnrpd 11870	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
3	2, 2	rpaddcld 11887	. . . 4 ⊢ (𝜑 → (𝐾 + 𝐾) ∈ ℝ⁺)
4		ubthlem.12	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
5	3, 4	rpdivcld 11889	. . 3 ⊢ (𝜑 → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ⁺)
6	5	rpred 11872	. 2 ⊢ (𝜑 → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ)
7		ubthlem.13	. . . . . . . . . 10 ⊢ (𝜑 → {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴‘𝐾))
8		rabss 3679	. . . . . . . . . 10 ⊢ ({𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴‘𝐾) ↔ ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)))
9	7, 8	sylib 208	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)))
10	9	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)))
11		ubthlem.5	. . . . . . . . . . 11 ⊢ 𝑈 ∈ CBan
12		bnnv 27722	. . . . . . . . . . 11 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
13	11, 12	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec
14	13	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ NrmCVec)
15		ubthlem.11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
16	15	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑃 ∈ 𝑋)
17	4	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ⁺)
18	17	rpcnd 11874	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℂ)
19		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
20		ubth.1	. . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈)
21		eqid 2622	. . . . . . . . . . 11 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
22	20, 21	nvscl 27481	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑅 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋)
23	14, 18, 19, 22	syl3anc 1326	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋)
24		eqid 2622	. . . . . . . . . 10 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
25	20, 24	nvgcl 27475	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑃 ∈ 𝑋 ∧ (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋) → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋)
26	14, 16, 23, 25	syl3anc 1326	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋)
27		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑃𝐷𝑧) = (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))))
28	27	breq1d 4663	. . . . . . . . . 10 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → ((𝑃𝐷𝑧) ≤ 𝑅 ↔ (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅))
29		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑧 ∈ (𝐴‘𝐾) ↔ (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ (𝐴‘𝐾)))
30	28, 29	imbi12d 334	. . . . . . . . 9 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)) ↔ ((𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅 → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ (𝐴‘𝐾))))
31	30	rspccv 3306	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)) → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋 → ((𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅 → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ (𝐴‘𝐾))))
32	10, 26, 31	sylc 65	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅 → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ (𝐴‘𝐾)))
33		ubthlem.3	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 = (IndMet‘𝑈)
34	20, 33	cbncms 27721	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
35	11, 34	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ (CMet‘𝑋)
36		cmetmet 23084	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
37		metxmet 22139	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
38	35, 36, 37	mp2b 10	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ (∞Met‘𝑋)
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
40		xmetsym 22152	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))𝐷𝑃))
41	39, 16, 26, 40	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))𝐷𝑃))
42		eqid 2622	. . . . . . . . . . . . 13 ⊢ ( −_𝑣 ‘𝑈) = ( −_𝑣 ‘𝑈)
43		eqid 2622	. . . . . . . . . . . . 13 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
44	20, 42, 43, 33	imsdval 27541	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))𝐷𝑃) = ((norm_CV‘𝑈)‘((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃)))
45	14, 26, 16, 44	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))𝐷𝑃) = ((norm_CV‘𝑈)‘((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃)))
46	20, 24, 42	nvpncan2 27508	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑃 ∈ 𝑋 ∧ (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋) → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃) = (𝑅( ·_𝑠OLD ‘𝑈)𝑥))
47	14, 16, 23, 46	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃) = (𝑅( ·_𝑠OLD ‘𝑈)𝑥))
48	47	fveq2d 6195	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃)) = ((norm_CV‘𝑈)‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))
49	41, 45, 48	3eqtrd 2660	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = ((norm_CV‘𝑈)‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))
50	17	rprege0d 11879	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
51	20, 21, 43	nvsge0 27519	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅 · ((norm_CV‘𝑈)‘𝑥)))
52	14, 50, 19, 51	syl3anc 1326	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅 · ((norm_CV‘𝑈)‘𝑥)))
53	49, 52	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = (𝑅 · ((norm_CV‘𝑈)‘𝑥)))
54	18	mulid1d 10057	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 · 1) = 𝑅)
55	54	eqcomd 2628	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 = (𝑅 · 1))
56	53, 55	breq12d 4666	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅 ↔ (𝑅 · ((norm_CV‘𝑈)‘𝑥)) ≤ (𝑅 · 1)))
57	20, 43	nvcl 27516	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘𝑥) ∈ ℝ)
58	13, 57	mpan 706	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → ((norm_CV‘𝑈)‘𝑥) ∈ ℝ)
59	58	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘𝑥) ∈ ℝ)
60		1red 10055	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
61	59, 60, 17	lemul2d 11916	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((norm_CV‘𝑈)‘𝑥) ≤ 1 ↔ (𝑅 · ((norm_CV‘𝑈)‘𝑥)) ≤ (𝑅 · 1)))
62	56, 61	bitr4d 271	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅 ↔ ((norm_CV‘𝑈)‘𝑥) ≤ 1))
63		breq2 4657	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → ((𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ (𝑁‘(𝑡‘𝑧)) ≤ 𝐾))
64	63	ralbidv 2986	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾))
65	64	rabbidv 3189	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾})
66		ubthlem.9	. . . . . . . . . . . 12 ⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
67		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (BaseSet‘𝑈) ∈ V
68	20, 67	eqeltri 2697	. . . . . . . . . . . . 13 ⊢ 𝑋 ∈ V
69	68	rabex 4813	. . . . . . . . . . . 12 ⊢ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾} ∈ V
70	65, 66, 69	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → (𝐴‘𝐾) = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾})
71	1, 70	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝐾) = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾})
72	71	eleq2d 2687	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ (𝐴‘𝐾) ↔ (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾}))
73		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑡‘𝑧) = (𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))))
74	73	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑁‘(𝑡‘𝑧)) = (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))))
75	74	breq1d 4663	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → ((𝑁‘(𝑡‘𝑧)) ≤ 𝐾 ↔ (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾))
76	75	ralbidv 2986	. . . . . . . . . 10 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾))
77	76	elrab 3363	. . . . . . . . 9 ⊢ ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾} ↔ ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾))
78	72, 77	syl6bb 276	. . . . . . . 8 ⊢ (𝜑 → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ (𝐴‘𝐾) ↔ ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾)))
79	78	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ (𝐴‘𝐾) ↔ ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾)))
80	32, 62, 79	3imtr3d 282	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((norm_CV‘𝑈)‘𝑥) ≤ 1 → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾)))
81		rsp 2929	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 → (𝑡 ∈ 𝑇 → (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾))
82	81	com12 32	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝑇 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 → (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾))
83	82	ad2antlr 763	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 → (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾))
84		xmet0 22147	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃𝐷𝑃) = 0)
85	38, 15, 84	sylancr 695	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑃𝐷𝑃) = 0)
86	4	rpge0d 11876	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ≤ 𝑅)
87	85, 86	eqbrtrd 4675	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃𝐷𝑃) ≤ 𝑅)
88		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑃 → (𝑃𝐷𝑧) = (𝑃𝐷𝑃))
89	88	breq1d 4663	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑃 → ((𝑃𝐷𝑧) ≤ 𝑅 ↔ (𝑃𝐷𝑃) ≤ 𝑅))
90	89	elrab 3363	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ↔ (𝑃 ∈ 𝑋 ∧ (𝑃𝐷𝑃) ≤ 𝑅))
91	15, 87, 90	sylanbrc 698	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅})
92	7, 91	sseldd 3604	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ (𝐴‘𝐾))
93	92, 71	eleqtrd 2703	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾})
94		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑃 → (𝑡‘𝑧) = (𝑡‘𝑃))
95	94	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑃 → (𝑁‘(𝑡‘𝑧)) = (𝑁‘(𝑡‘𝑃)))
96	95	breq1d 4663	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑃 → ((𝑁‘(𝑡‘𝑧)) ≤ 𝐾 ↔ (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
97	96	ralbidv 2986	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑃 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
98	97	elrab 3363	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾} ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
99	93, 98	sylib 208	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
100	99	simprd 479	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾)
101	100	r19.21bi 2932	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑃)) ≤ 𝐾)
102	101	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘𝑃)) ≤ 𝐾)
103		ubthlem.6	. . . . . . . . . . . . 13 ⊢ 𝑊 ∈ NrmCVec
104		ubthlem.7	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊))
105	104	sselda 3603	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑈 BLnOp 𝑊))
106		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊)
107		ubthlem.4	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 = (MetOpen‘𝐷)
108		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘(IndMet‘𝑊)) = (MetOpen‘(IndMet‘𝑊))
109		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 BLnOp 𝑊) = (𝑈 BLnOp 𝑊)
110	33, 106, 107, 108, 109, 13, 103	blocn2 27663	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))))
111	107	mopntopon 22244	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
112	38, 111	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ∈ (TopOn‘𝑋)
113		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
114	113, 106	imsxmet 27547	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 ∈ NrmCVec → (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)))
115	108	mopntopon 22244	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) → (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊)))
116	103, 114, 115	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊))
117		iscncl 21073	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊))) → (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(^◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽))))
118	112, 116, 117	mp2an 708	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(^◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
119	110, 118	sylib 208	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(^◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
120	105, 119	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(^◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
121	120	simpld 475	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
122	121	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑡:𝑋⟶(BaseSet‘𝑊))
123	122, 26	ffvelrnd 6360	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ∈ (BaseSet‘𝑊))
124		ubth.2	. . . . . . . . . . . . . 14 ⊢ 𝑁 = (norm_CV‘𝑊)
125	113, 124	nvcl 27516	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ∈ ℝ)
126	103, 123, 125	sylancr 695	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ∈ ℝ)
127	122, 16	ffvelrnd 6360	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑃) ∈ (BaseSet‘𝑊))
128	113, 124	nvcl 27516	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘𝑃) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘𝑃)) ∈ ℝ)
129	103, 127, 128	sylancr 695	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘𝑃)) ∈ ℝ)
130	1	nnred 11035	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℝ)
131	130	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ ℝ)
132		le2add 10510	. . . . . . . . . . . 12 ⊢ ((((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ∈ ℝ ∧ (𝑁‘(𝑡‘𝑃)) ∈ ℝ) ∧ (𝐾 ∈ ℝ ∧ 𝐾 ∈ ℝ)) → (((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 ∧ (𝑁‘(𝑡‘𝑃)) ≤ 𝐾) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ≤ (𝐾 + 𝐾)))
133	126, 129, 131, 131, 132	syl22anc 1327	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 ∧ (𝑁‘(𝑡‘𝑃)) ≤ 𝐾) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ≤ (𝐾 + 𝐾)))
134	102, 133	mpan2d 710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ≤ (𝐾 + 𝐾)))
135	47	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃)) = (𝑡‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))
136	103	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ NrmCVec)
137		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊)
138	137, 109	bloln 27639	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑡 ∈ (𝑈 BLnOp 𝑊)) → 𝑡 ∈ (𝑈 LnOp 𝑊))
139	13, 103, 138	mp3an12 1414	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝑈 LnOp 𝑊))
140	105, 139	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑈 LnOp 𝑊))
141	140	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑡 ∈ (𝑈 LnOp 𝑊))
142		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ ( −_𝑣 ‘𝑊) = ( −_𝑣 ‘𝑊)
143	20, 42, 142, 137	lnosub 27614	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑡 ∈ (𝑈 LnOp 𝑊)) ∧ ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋 ∧ 𝑃 ∈ 𝑋)) → (𝑡‘((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃)) = ((𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))( −_𝑣 ‘𝑊)(𝑡‘𝑃)))
144	14, 136, 141, 26, 16, 143	syl32anc 1334	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃)) = ((𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))( −_𝑣 ‘𝑊)(𝑡‘𝑃)))
145		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
146	20, 21, 145, 137	lnomul 27615	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑡 ∈ (𝑈 LnOp 𝑊)) ∧ (𝑅 ∈ ℂ ∧ 𝑥 ∈ 𝑋)) → (𝑡‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥)))
147	14, 136, 141, 18, 19, 146	syl32anc 1334	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥)))
148	135, 144, 147	3eqtr3d 2664	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))( −_𝑣 ‘𝑊)(𝑡‘𝑃)) = (𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥)))
149	148	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘((𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))( −_𝑣 ‘𝑊)(𝑡‘𝑃))) = (𝑁‘(𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥))))
150	121	ffvelrnda 6359	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
151	113, 145, 124	nvsge0 27519	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ (𝑡‘𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥))) = (𝑅 · (𝑁‘(𝑡‘𝑥))))
152	136, 50, 150, 151	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥))) = (𝑅 · (𝑁‘(𝑡‘𝑥))))
153	149, 152	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘((𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))( −_𝑣 ‘𝑊)(𝑡‘𝑃))) = (𝑅 · (𝑁‘(𝑡‘𝑥))))
154	113, 142, 124	nvmtri 27526	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ∈ (BaseSet‘𝑊) ∧ (𝑡‘𝑃) ∈ (BaseSet‘𝑊)) → (𝑁‘((𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))( −_𝑣 ‘𝑊)(𝑡‘𝑃))) ≤ ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))))
155	136, 123, 127, 154	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘((𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))( −_𝑣 ‘𝑊)(𝑡‘𝑃))) ≤ ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))))
156	153, 155	eqbrtrrd 4677	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))))
157	17	rpred 11872	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ)
158	113, 124	nvcl 27516	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
159	103, 150, 158	sylancr 695	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
160	157, 159	remulcld 10070	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 · (𝑁‘(𝑡‘𝑥))) ∈ ℝ)
161	126, 129	readdcld 10069	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ∈ ℝ)
162	3	rpred 11872	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 + 𝐾) ∈ ℝ)
163	162	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝐾 + 𝐾) ∈ ℝ)
164		letr 10131	. . . . . . . . . . . 12 ⊢ (((𝑅 · (𝑁‘(𝑡‘𝑥))) ∈ ℝ ∧ ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ∈ ℝ ∧ (𝐾 + 𝐾) ∈ ℝ) → (((𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ∧ ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ≤ (𝐾 + 𝐾)) → (𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ (𝐾 + 𝐾)))
165	160, 161, 163, 164	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ∧ ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ≤ (𝐾 + 𝐾)) → (𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ (𝐾 + 𝐾)))
166	156, 165	mpand 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ≤ (𝐾 + 𝐾) → (𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ (𝐾 + 𝐾)))
167	134, 166	syld 47	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 → (𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ (𝐾 + 𝐾)))
168	159, 163, 17	lemuldiv2d 11922	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ (𝐾 + 𝐾) ↔ (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
169	167, 168	sylibd 229	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
170	83, 169	syld 47	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
171	170	adantld 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾) → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
172	80, 171	syld 47	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((norm_CV‘𝑈)‘𝑥) ≤ 1 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
173	172	ralrimiva 2966	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∀𝑥 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑥) ≤ 1 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
174	5	rpxrd 11873	. . . . . 6 ⊢ (𝜑 → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ^*)
175	174	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ^*)
176		eqid 2622	. . . . . 6 ⊢ (𝑈 normOp_OLD 𝑊) = (𝑈 normOp_OLD 𝑊)
177	20, 113, 43, 124, 176, 13, 103	nmoubi 27627	. . . . 5 ⊢ ((𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ((𝐾 + 𝐾) / 𝑅) ∈ ℝ^*) → (((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅) ↔ ∀𝑥 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑥) ≤ 1 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅))))
178	121, 175, 177	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅) ↔ ∀𝑥 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑥) ≤ 1 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅))))
179	173, 178	mpbird 247	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅))
180	179	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅))
181		breq2 4657	. . . 4 ⊢ (𝑑 = ((𝐾 + 𝐾) / 𝑅) → (((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑 ↔ ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅)))
182	181	ralbidv 2986	. . 3 ⊢ (𝑑 = ((𝐾 + 𝐾) / 𝑅) → (∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑 ↔ ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅)))
183	182	rspcev 3309	. 2 ⊢ ((((𝐾 + 𝐾) / 𝑅) ∈ ℝ ∧ ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅)) → ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑)
184	6, 180, 183	syl2anc 693	1 ⊢ (𝜑 → ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ^◡ccnv 5113 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℝ^*cxr 10073 ≤ cle 10075 / cdiv 10684 ℕcn 11020 ℝ⁺crp 11832 ∞Metcxmt 19731 Metcme 19732 MetOpencmopn 19736 TopOnctopon 20715 Clsdccld 20820 Cn ccn 21028 CMetcms 23052 NrmCVeccnv 27439 +_𝑣 cpv 27440 BaseSetcba 27441 ·_𝑠OLD cns 27442 −_𝑣 cnsb 27444 norm_CVcnmcv 27445 IndMetcims 27446 LnOp clno 27595 normOp_OLD cnmoo 27596 BLnOp cblo 27597 CBanccbn 27718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-cn 21031 df-cnp 21032 df-cmet 23055 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-lno 27599 df-nmoo 27600 df-blo 27601 df-0o 27602 df-cbn 27719

This theorem is referenced by: ubthlem3 27728

W3C validator