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Theorem ubthlem2 30112

Description: Lemma for ubth 30114. 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 13011	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
3	2, 2	rpaddcld 13028	. . . 4 ⊢ (𝜑 → (𝐾 + 𝐾) ∈ ℝ⁺)
4		ubthlem.12	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
5	3, 4	rpdivcld 13030	. . 3 ⊢ (𝜑 → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ⁺)
6	5	rpred 13013	. 2 ⊢ (𝜑 → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ)
7		oveq2 7414	. . . . . . . . . 10 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑃𝐷𝑧) = (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))))
8	7	breq1d 5158	. . . . . . . . 9 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → ((𝑃𝐷𝑧) ≤ 𝑅 ↔ (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅))
9		eleq1 2822	. . . . . . . . 9 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑧 ∈ (𝐴‘𝐾) ↔ (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ (𝐴‘𝐾)))
10	8, 9	imbi12d 345	. . . . . . . 8 ⊢ (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))))
11		ubthlem.13	. . . . . . . . . 10 ⊢ (𝜑 → {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴‘𝐾))
12		rabss 4069	. . . . . . . . . 10 ⊢ ({𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴‘𝐾) ↔ ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)))
13	11, 12	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)))
14	13	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝐴‘𝐾)))
15		ubthlem.5	. . . . . . . . . . 11 ⊢ 𝑈 ∈ CBan
16		bnnv 30107	. . . . . . . . . . 11 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec
18	17	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ NrmCVec)
19		ubthlem.11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
20	19	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑃 ∈ 𝑋)
21	4	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ⁺)
22	21	rpcnd 13015	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℂ)
23		simpr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
24		ubth.1	. . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈)
25		eqid 2733	. . . . . . . . . . 11 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
26	24, 25	nvscl 29867	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑅 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋)
27	18, 22, 23, 26	syl3anc 1372	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋)
28		eqid 2733	. . . . . . . . . 10 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
29	24, 28	nvgcl 29861	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑃 ∈ 𝑋 ∧ (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋) → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋)
30	18, 20, 27, 29	syl3anc 1372	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ 𝑋)
31	10, 14, 30	rspcdva 3614	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅 → (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) ∈ (𝐴‘𝐾)))
32		ubthlem.3	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 = (IndMet‘𝑈)
33	24, 32	cbncms 30106	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
34	15, 33	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ (CMet‘𝑋)
35		cmetmet 24795	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
36		metxmet 23832	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
37	34, 35, 36	mp2b 10	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ (∞Met‘𝑋)
38	37	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
39		xmetsym 23845	. . . . . . . . . . . 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))
40	38, 20, 30, 39	syl3anc 1372	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))𝐷𝑃))
41		eqid 2733	. . . . . . . . . . . . 13 ⊢ ( −_𝑣 ‘𝑈) = ( −_𝑣 ‘𝑈)
42		eqid 2733	. . . . . . . . . . . . 13 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
43	24, 41, 42, 32	imsdval 29927	. . . . . . . . . . . 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)))
44	18, 30, 20, 43	syl3anc 1372	. . . . . . . . . . 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)))
45	24, 28, 41	nvpncan2 29894	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑃 ∈ 𝑋 ∧ (𝑅( ·_𝑠OLD ‘𝑈)𝑥) ∈ 𝑋) → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃) = (𝑅( ·_𝑠OLD ‘𝑈)𝑥))
46	18, 20, 27, 45	syl3anc 1372	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃) = (𝑅( ·_𝑠OLD ‘𝑈)𝑥))
47	46	fveq2d 6893	. . . . . . . . . . 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)))
48	40, 44, 47	3eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = ((norm_CV‘𝑈)‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))
49	21	rprege0d 13020	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
50	24, 25, 42	nvsge0 29905	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅 · ((norm_CV‘𝑈)‘𝑥)))
51	18, 49, 23, 50	syl3anc 1372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅 · ((norm_CV‘𝑈)‘𝑥)))
52	48, 51	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) = (𝑅 · ((norm_CV‘𝑈)‘𝑥)))
53	22	mulridd 11228	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 · 1) = 𝑅)
54	53	eqcomd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 = (𝑅 · 1))
55	52, 54	breq12d 5161	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅 ↔ (𝑅 · ((norm_CV‘𝑈)‘𝑥)) ≤ (𝑅 · 1)))
56	24, 42	nvcl 29902	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘𝑥) ∈ ℝ)
57	17, 56	mpan 689	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → ((norm_CV‘𝑈)‘𝑥) ∈ ℝ)
58	57	adantl 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((norm_CV‘𝑈)‘𝑥) ∈ ℝ)
59		1red 11212	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ)
60	58, 59, 21	lemul2d 13057	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((norm_CV‘𝑈)‘𝑥) ≤ 1 ↔ (𝑅 · ((norm_CV‘𝑈)‘𝑥)) ≤ (𝑅 · 1)))
61	55, 60	bitr4d 282	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑃𝐷(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ≤ 𝑅 ↔ ((norm_CV‘𝑈)‘𝑥) ≤ 1))
62		breq2 5152	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → ((𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ (𝑁‘(𝑡‘𝑧)) ≤ 𝐾))
63	62	ralbidv 3178	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾))
64	63	rabbidv 3441	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾})
65		ubthlem.9	. . . . . . . . . . . 12 ⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
66	24	fvexi 6903	. . . . . . . . . . . . 13 ⊢ 𝑋 ∈ V
67	66	rabex 5332	. . . . . . . . . . . 12 ⊢ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾} ∈ V
68	64, 65, 67	fvmpt 6996	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → (𝐴‘𝐾) = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾})
69	1, 68	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝐾) = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾})
70	69	eleq2d 2820	. . . . . . . . 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}))
71		2fveq3 6894	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → (𝑁‘(𝑡‘𝑧)) = (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))))
72	71	breq1d 5158	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) → ((𝑁‘(𝑡‘𝑧)) ≤ 𝐾 ↔ (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾))
73	72	ralbidv 3178	. . . . . . . . . 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))
74	73	elrab 3683	. . . . . . . . 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))
75	70, 74	bitrdi 287	. . . . . . . 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)))
76	75	ad2antrr 725	. . . . . . 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)))
77	31, 61, 76	3imtr3d 293	. . . . . 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)))
78		rsp 3245	. . . . . . . . . 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))
79	78	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))
80	79	ad2antlr 726	. . . . . . . 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))
81		xmet0 23840	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃𝐷𝑃) = 0)
82	37, 19, 81	sylancr 588	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑃𝐷𝑃) = 0)
83	4	rpge0d 13017	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ≤ 𝑅)
84	82, 83	eqbrtrd 5170	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃𝐷𝑃) ≤ 𝑅)
85		oveq2 7414	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑃 → (𝑃𝐷𝑧) = (𝑃𝐷𝑃))
86	85	breq1d 5158	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑃 → ((𝑃𝐷𝑧) ≤ 𝑅 ↔ (𝑃𝐷𝑃) ≤ 𝑅))
87	86	elrab 3683	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ↔ (𝑃 ∈ 𝑋 ∧ (𝑃𝐷𝑃) ≤ 𝑅))
88	19, 84, 87	sylanbrc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅})
89	11, 88	sseldd 3983	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ (𝐴‘𝐾))
90	89, 69	eleqtrd 2836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾})
91		2fveq3 6894	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑃 → (𝑁‘(𝑡‘𝑧)) = (𝑁‘(𝑡‘𝑃)))
92	91	breq1d 5158	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑃 → ((𝑁‘(𝑡‘𝑧)) ≤ 𝐾 ↔ (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
93	92	ralbidv 3178	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑃 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
94	93	elrab 3683	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝐾} ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
95	90, 94	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾))
96	95	simprd 497	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑃)) ≤ 𝐾)
97	96	r19.21bi 3249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑃)) ≤ 𝐾)
98	97	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘𝑃)) ≤ 𝐾)
99		ubthlem.6	. . . . . . . . . . . . 13 ⊢ 𝑊 ∈ NrmCVec
100		ubthlem.7	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊))
101	100	sselda 3982	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑈 BLnOp 𝑊))
102		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊)
103		ubthlem.4	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 = (MetOpen‘𝐷)
104		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘(IndMet‘𝑊)) = (MetOpen‘(IndMet‘𝑊))
105		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑈 BLnOp 𝑊) = (𝑈 BLnOp 𝑊)
106	32, 102, 103, 104, 105, 17, 99	blocn2 30049	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))))
107	103	mopntopon 23937	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
108	37, 107	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ∈ (TopOn‘𝑋)
109		eqid 2733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
110	109, 102	imsxmet 29933	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 ∈ NrmCVec → (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)))
111	104	mopntopon 23937	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) → (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊)))
112	99, 110, 111	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊))
113		iscncl 22765	. . . . . . . . . . . . . . . . . . 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))))
114	108, 112, 113	mp2an 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(^◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
115	106, 114	sylib 217	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(^◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
116	101, 115	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(^◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
117	116	simpld 496	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
118	117	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑡:𝑋⟶(BaseSet‘𝑊))
119	118, 30	ffvelcdmd 7085	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ∈ (BaseSet‘𝑊))
120		ubth.2	. . . . . . . . . . . . . 14 ⊢ 𝑁 = (norm_CV‘𝑊)
121	109, 120	nvcl 29902	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ∈ ℝ)
122	99, 119, 121	sylancr 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ∈ ℝ)
123	118, 20	ffvelcdmd 7085	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑃) ∈ (BaseSet‘𝑊))
124	109, 120	nvcl 29902	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘𝑃) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘𝑃)) ∈ ℝ)
125	99, 123, 124	sylancr 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘𝑃)) ∈ ℝ)
126	1	nnred 12224	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℝ)
127	126	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ ℝ)
128		le2add 11693	. . . . . . . . . . . 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)))
129	122, 125, 127, 127, 128	syl22anc 838	. . . . . . . . . . 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)))
130	98, 129	mpan2d 693	. . . . . . . . . 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)))
131	46	fveq2d 6893	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘((𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥))( −_𝑣 ‘𝑈)𝑃)) = (𝑡‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))
132	99	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ NrmCVec)
133		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊)
134	133, 105	bloln 30025	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑡 ∈ (𝑈 BLnOp 𝑊)) → 𝑡 ∈ (𝑈 LnOp 𝑊))
135	17, 99, 134	mp3an12 1452	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝑈 LnOp 𝑊))
136	101, 135	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑈 LnOp 𝑊))
137	136	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑡 ∈ (𝑈 LnOp 𝑊))
138		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ ( −_𝑣 ‘𝑊) = ( −_𝑣 ‘𝑊)
139	24, 41, 138, 133	lnosub 30000	. . . . . . . . . . . . . . . 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)))
140	18, 132, 137, 30, 20, 139	syl32anc 1379	. . . . . . . . . . . . . . 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)))
141		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
142	24, 25, 141, 133	lnomul 30001	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑡 ∈ (𝑈 LnOp 𝑊)) ∧ (𝑅 ∈ ℂ ∧ 𝑥 ∈ 𝑋)) → (𝑡‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥)))
143	18, 132, 137, 22, 23, 142	syl32anc 1379	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘(𝑅( ·_𝑠OLD ‘𝑈)𝑥)) = (𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥)))
144	131, 140, 143	3eqtr3d 2781	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))( −_𝑣 ‘𝑊)(𝑡‘𝑃)) = (𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥)))
145	144	fveq2d 6893	. . . . . . . . . . . . 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))))
146	117	ffvelcdmda 7084	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
147	109, 141, 120	nvsge0 29905	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ (𝑡‘𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥))) = (𝑅 · (𝑁‘(𝑡‘𝑥))))
148	132, 49, 146, 147	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑅( ·_𝑠OLD ‘𝑊)(𝑡‘𝑥))) = (𝑅 · (𝑁‘(𝑡‘𝑥))))
149	145, 148	eqtrd 2773	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘((𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))( −_𝑣 ‘𝑊)(𝑡‘𝑃))) = (𝑅 · (𝑁‘(𝑡‘𝑥))))
150	109, 138, 120	nvmtri 29912	. . . . . . . . . . . . 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))))
151	132, 119, 123, 150	syl3anc 1372	. . . . . . . . . . . 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))))
152	149, 151	eqbrtrrd 5172	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))))
153	21	rpred 13013	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ)
154	109, 120	nvcl 29902	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
155	99, 146, 154	sylancr 588	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
156	153, 155	remulcld 11241	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑅 · (𝑁‘(𝑡‘𝑥))) ∈ ℝ)
157	122, 125	readdcld 11240	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) + (𝑁‘(𝑡‘𝑃))) ∈ ℝ)
158	3	rpred 13013	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 + 𝐾) ∈ ℝ)
159	158	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝐾 + 𝐾) ∈ ℝ)
160		letr 11305	. . . . . . . . . . . 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)))
161	156, 157, 159, 160	syl3anc 1372	. . . . . . . . . . 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)))
162	152, 161	mpand 694	. . . . . . . . . 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)))
163	130, 162	syld 47	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 → (𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ (𝐾 + 𝐾)))
164	155, 159, 21	lemuldiv2d 13063	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑅 · (𝑁‘(𝑡‘𝑥))) ≤ (𝐾 + 𝐾) ↔ (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
165	163, 164	sylibd 238	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘(𝑃( +_𝑣 ‘𝑈)(𝑅( ·_𝑠OLD ‘𝑈)𝑥)))) ≤ 𝐾 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
166	80, 165	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)))
167	166	adantld 492	. . . . . 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)))
168	77, 167	syld 47	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (((norm_CV‘𝑈)‘𝑥) ≤ 1 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
169	168	ralrimiva 3147	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∀𝑥 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑥) ≤ 1 → (𝑁‘(𝑡‘𝑥)) ≤ ((𝐾 + 𝐾) / 𝑅)))
170	5	rpxrd 13014	. . . . . 6 ⊢ (𝜑 → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ^*)
171	170	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐾 + 𝐾) / 𝑅) ∈ ℝ^*)
172		eqid 2733	. . . . . 6 ⊢ (𝑈 normOp_OLD 𝑊) = (𝑈 normOp_OLD 𝑊)
173	24, 109, 42, 120, 172, 17, 99	nmoubi 30013	. . . . 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))))
174	117, 171, 173	syl2anc 585	. . . 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))))
175	169, 174	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅))
176	175	ralrimiva 3147	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ ((𝐾 + 𝐾) / 𝑅))
177		brralrspcev 5208	. 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)
178	6, 176, 177	syl2anc 585	1 ⊢ (𝜑 → ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 ⊆ wss 3948 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5675 “ cima 5679 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 + caddc 11110 · cmul 11112 ℝ^*cxr 11244 ≤ cle 11246 / cdiv 11868 ℕcn 12209 ℝ⁺crp 12971 ∞Metcxmet 20922 Metcmet 20923 MetOpencmopn 20927 TopOnctopon 22404 Clsdccld 22512 Cn ccn 22720 CMetccmet 24763 NrmCVeccnv 29825 +_𝑣 cpv 29826 BaseSetcba 29827 ·_𝑠OLD cns 29828 −_𝑣 cnsb 29830 norm_CVcnmcv 29831 IndMetcims 29832 LnOp clno 29981 normOp_OLD cnmoo 29982 BLnOp cblo 29983 CBanccbn 30103

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-topgen 17386 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-top 22388 df-topon 22405 df-bases 22441 df-cld 22515 df-cn 22723 df-cnp 22724 df-cmet 24766 df-grpo 29734 df-gid 29735 df-ginv 29736 df-gdiv 29737 df-ablo 29786 df-vc 29800 df-nv 29833 df-va 29836 df-ba 29837 df-sm 29838 df-0v 29839 df-vs 29840 df-nmcv 29841 df-ims 29842 df-lno 29985 df-nmoo 29986 df-blo 29987 df-0o 29988 df-cbn 30104

This theorem is referenced by: ubthlem3 30113

W3C validator