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Theorem ubthlem2 27111

Description: Lemma for ubth 27113. 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 11746	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
3	2, 2	rpaddcld 11763	. . . 4 ⊢ (𝜑 → (𝐾 + 𝐾) ∈ ℝ⁺)
4		ubthlem.12	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
5	3, 4	rpdivcld 11765	. . 3 ⊢ (𝜑 → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ⁺)
6	5	rpred 11748	. 2 ⊢ (𝜑 → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ)
7		ubthlem.13	. . . . . . . . . 10 ⊢ (𝜑 → {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴‘𝐾))
8		rabss 3642	. . . . . . . . . 10 ⊢ ({𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴‘𝐾) ↔ ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)))
9	7, 8	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)))
10	9	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)))
11		ubthlem.5	. . . . . . . . . . 11 ⊢ 𝑈 ∈ CBan
12		bnnv 27106	. . . . . . . . . . 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 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑃 ∈ 𝑋)
17	4	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ⁺)
18	17	rpcnd 11750	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℂ)
19		simpr 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
20		ubth.1	. . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈)
21		eqid 2610	. . . . . . . . . . 11 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
22	20, 21	nvscl 26865	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑅 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋)
23	14, 18, 19, 22	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋)
24		eqid 2610	. . . . . . . . . 10 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
25	20, 24	nvgcl 26859	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑃 ∈ 𝑋 ∧ (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋) → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋)
26	14, 16, 23, 25	syl3anc 1318	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋)
27		oveq2 6557	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑃𝐷𝑧) = (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))))
28	27	breq1d 4593	. . . . . . . . . 10 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → ((𝑃𝐷𝑧) ≤ 𝑅 ↔ (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅))
29		eleq1 2676	. . . . . . . . . 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 333	. . . . . . . . 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 3279	. . . . . . . 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 63	. . . . . . 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 27105	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
35	11, 34	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ (CMet‘𝑋)
36		cmetmet 22892	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
37		metxmet 21949	. . . . . . . . . . . . . 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 21962	. . . . . . . . . . . 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 1318	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))𝐷𝑃))
42		eqid 2610	. . . . . . . . . . . . 13 ⊢ ( −_𝑣 ‘𝑈) = ( −_𝑣 ‘𝑈)
43		eqid 2610	. . . . . . . . . . . . 13 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
44	20, 42, 43, 33	imsdval 26925	. . . . . . . . . . . 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 1318	. . . . . . . . . . 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 26892	. . . . . . . . . . . . 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 1318	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃) = (𝑅( ·_𝑠OLD ‘𝑈)𝑥))
48	47	fveq2d 6107	. . . . . . . . . . 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 2648	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = ((norm_CV‘𝑈)‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))
50	17	rprege0d 11755	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
51	20, 21, 43	nvsge0 26903	. . . . . . . . . . 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 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅 · ((norm_CV‘𝑈)‘𝑥)))
53	49, 52	eqtrd 2644	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = (𝑅 · ((norm_CV‘𝑈)‘𝑥)))
54	18	mulid1d 9936	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 · 1) = 𝑅)
55	54	eqcomd 2616	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 = (𝑅 · 1))
56	53, 55	breq12d 4596	. . . . . . . 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 26900	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘𝑥) ∈ ℝ)
58	13, 57	mpan 702	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → ((norm_CV‘𝑈)‘𝑥) ∈ ℝ)
59	58	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘𝑥) ∈ ℝ)
60		1red 9934	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
61	59, 60, 17	lemul2d 11792	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((norm_CV‘𝑈)‘𝑥) ≤ 1 ↔ (𝑅 · ((norm_CV‘𝑈)‘𝑥)) ≤ (𝑅 · 1)))
62	56, 61	bitr4d 270	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅 ↔ ((norm_CV‘𝑈)‘𝑥) ≤ 1))
63		breq2 4587	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → ((𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ (𝑁‘(𝑡‘𝑧)) ≤ 𝐾))
64	63	ralbidv 2969	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾))
65	64	rabbidv 3164	. . . . . . . . . . . 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 6113	. . . . . . . . . . . . . 14 ⊢ (BaseSet‘𝑈) ∈ V
68	20, 67	eqeltri 2684	. . . . . . . . . . . . 13 ⊢ 𝑋 ∈ V
69	68	rabex 4740	. . . . . . . . . . . 12 ⊢ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾} ∈ V
70	65, 66, 69	fvmpt 6191	. . . . . . . . . . 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 2673	. . . . . . . . 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 6103	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑡‘𝑧) = (𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))))
74	73	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑁‘(𝑡‘𝑧)) = (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))))
75	74	breq1d 4593	. . . . . . . . . . 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 2969	. . . . . . . . . 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 3331	. . . . . . . . 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 275	. . . . . . . 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 758	. . . . . . 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 281	. . . . . 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 2913	. . . . . . . . . 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 759	. . . . . . . 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 21957	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃𝐷𝑃) = 0)
85	38, 15, 84	sylancr 694	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑃𝐷𝑃) = 0)
86	4	rpge0d 11752	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ≤ 𝑅)
87	85, 86	eqbrtrd 4605	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃𝐷𝑃) ≤ 𝑅)
88		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑃 → (𝑃𝐷𝑧) = (𝑃𝐷𝑃))
89	88	breq1d 4593	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑃 → ((𝑃𝐷𝑧) ≤ 𝑅 ↔ (𝑃𝐷𝑃) ≤ 𝑅))
90	89	elrab 3331	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ↔ (𝑃 ∈ 𝑋 ∧ (𝑃𝐷𝑃) ≤ 𝑅))
91	15, 87, 90	sylanbrc 695	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅})
92	7, 91	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ (𝐴‘𝐾))
93	92, 71	eleqtrd 2690	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾})
94		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑃 → (𝑡‘𝑧) = (𝑡‘𝑃))
95	94	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑃 → (𝑁‘(𝑡‘𝑧)) = (𝑁‘(𝑡‘𝑃)))
96	95	breq1d 4593	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑃 → ((𝑁‘(𝑡‘𝑧)) ≤ 𝐾 ↔ (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
97	96	ralbidv 2969	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑃 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
98	97	elrab 3331	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾} ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
99	93, 98	sylib 207	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
100	99	simprd 478	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾)
101	100	r19.21bi 2916	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑃)) ≤ 𝐾)
102	101	adantr 480	. . . . . . . . . . 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 3568	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑈 BLnOp 𝑊))
106		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊)
107		ubthlem.4	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 = (MetOpen‘𝐷)
108		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘(IndMet‘𝑊)) = (MetOpen‘(IndMet‘𝑊))
109		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 BLnOp 𝑊) = (𝑈 BLnOp 𝑊)
110	33, 106, 107, 108, 109, 13, 103	blocn2 27047	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))))
111	107	mopntopon 22054	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
112	38, 111	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ∈ (TopOn‘𝑋)
113		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
114	113, 106	imsxmet 26931	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 ∈ NrmCVec → (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)))
115	108	mopntopon 22054	. . . . . . . . . . . . . . . . . . . 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 20883	. . . . . . . . . . . . . . . . . . 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 704	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(^◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
119	110, 118	sylib 207	. . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
122	121	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑡:𝑋⟶(BaseSet‘𝑊))
123	122, 26	ffvelrnd 6268	. . . . . . . . . . . . 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 26900	. . . . . . . . . . . . 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 694	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ∈ ℝ)
127	122, 16	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑃) ∈ (BaseSet‘𝑊))
128	113, 124	nvcl 26900	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘𝑃) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘𝑃)) ∈ ℝ)
129	103, 127, 128	sylancr 694	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘𝑃)) ∈ ℝ)
130	1	nnred 10912	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℝ)
131	130	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ ℝ)
132		le2add 10389	. . . . . . . . . . . 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 1319	. . . . . . . . . . 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 706	. . . . . . . . . 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 6107	. . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊)
138	137, 109	bloln 27023	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑡 ∈ (𝑈 BLnOp 𝑊)) → 𝑡 ∈ (𝑈 LnOp 𝑊))
139	13, 103, 138	mp3an12 1406	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑡 ∈ (𝑈 LnOp 𝑊))
142		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ ( −_𝑣 ‘𝑊) = ( −_𝑣 ‘𝑊)
143	20, 42, 142, 137	lnosub 26998	. . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . 17 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
146	20, 21, 145, 137	lnomul 26999	. . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥)))
148	135, 144, 147	3eqtr3d 2652	. . . . . . . . . . . . . 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 6107	. . . . . . . . . . . . 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 6267	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
151	113, 145, 124	nvsge0 26903	. . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥))) = (𝑅 · (𝑁‘(𝑡‘𝑥))))
153	149, 152	eqtrd 2644	. . . . . . . . . . . 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 26910	. . . . . . . . . . . . 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 1318	. . . . . . . . . . . 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 4607	. . . . . . . . . . 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 11748	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ)
158	113, 124	nvcl 26900	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
159	103, 150, 158	sylancr 694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
160	157, 159	remulcld 9949	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 · (𝑁‘(𝑡‘𝑥))) ∈ ℝ)
161	126, 129	readdcld 9948	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ∈ ℝ)
162	3	rpred 11748	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 + 𝐾) ∈ ℝ)
163	162	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝐾 + 𝐾) ∈ ℝ)
164		letr 10010	. . . . . . . . . . . 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 1318	. . . . . . . . . . 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 707	. . . . . . . . . 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 46	. . . . . . . . 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 11798	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ (𝐾 + 𝐾) ↔ (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
169	167, 168	sylibd 228	. . . . . . . 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 46	. . . . . . 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 482	. . . . . 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 46	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((norm_CV‘𝑈)‘𝑥) ≤ 1 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
173	172	ralrimiva 2949	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∀𝑥 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑥) ≤ 1 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
174	5	rpxrd 11749	. . . . . 6 ⊢ (𝜑 → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ^*)
175	174	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ^*)
176		eqid 2610	. . . . . 6 ⊢ (𝑈 normOp_OLD 𝑊) = (𝑈 normOp_OLD 𝑊)
177	20, 113, 43, 124, 176, 13, 103	nmoubi 27011	. . . . 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 691	. . . 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 246	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅))
180	179	ralrimiva 2949	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅))
181		breq2 4587	. . . 4 ⊢ (𝑑 = ((𝐾 + 𝐾) / 𝑅) → (((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑 ↔ ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅)))
182	181	ralbidv 2969	. . 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 3282	. 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 691	1 ⊢ (𝜑 → ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ^◡ccnv 5037 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℝ^*cxr 9952 ≤ cle 9954 / cdiv 10563 ℕcn 10897 ℝ⁺crp 11708 ∞Metcxmt 19552 Metcme 19553 MetOpencmopn 19557 TopOnctopon 20518 Clsdccld 20630 Cn ccn 20838 CMetcms 22860 NrmCVeccnv 26823 +_𝑣 cpv 26824 BaseSetcba 26825 ·_𝑠OLD cns 26826 −_𝑣 cnsb 26828 norm_CVcnmcv 26829 IndMetcims 26830 LnOp clno 26979 normOp_OLD cnmoo 26980 BLnOp cblo 26981 CBanccbn 27102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-cn 20841 df-cnp 20842 df-cmet 22863 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-lno 26983 df-nmoo 26984 df-blo 26985 df-0o 26986 df-cbn 27103

This theorem is referenced by: ubthlem3 27112

W3C validator