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Theorem smcnlem 30726

Description: Lemma for smcn 30727. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
smcn.c	⊢ 𝐶 = (IndMet‘𝑈)
smcn.j	⊢ 𝐽 = (MetOpen‘𝐶)
smcn.s	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
smcn.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
smcn.x	⊢ 𝑋 = (BaseSet‘𝑈)
smcn.n	⊢ 𝑁 = (norm_CV‘𝑈)
smcn.u	⊢ 𝑈 ∈ NrmCVec
smcn.t	⊢ 𝑇 = (1 / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)))

**Assertion**
Ref	Expression
smcnlem	⊢ 𝑆 ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Distinct variable groups: 𝑥,𝑟,𝑦,𝐶 𝐽,𝑟,𝑥,𝑦 𝑈,𝑟,𝑥,𝑦 𝐾,𝑟,𝑥,𝑦 𝑆,𝑟,𝑥,𝑦 𝑋,𝑟,𝑥,𝑦 Allowed substitution hints: 𝑇(𝑥,𝑦,𝑟) 𝑁(𝑥,𝑦,𝑟)

Proof of Theorem smcnlem Dummy variables 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smcn.u	. . 3 ⊢ 𝑈 ∈ NrmCVec
2		smcn.x	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
3		smcn.s	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
4	2, 3	nvsf 30648	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋)
5	1, 4	ax-mp 5	. 2 ⊢ 𝑆:(ℂ × 𝑋)⟶𝑋
6		smcn.t	. . . . . 6 ⊢ 𝑇 = (1 / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)))
7		1rp 13036	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
8		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
9		smcn.n	. . . . . . . . . . . . 13 ⊢ 𝑁 = (norm_CV‘𝑈)
10	2, 9	nvcl 30690	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (𝑁‘𝑦) ∈ ℝ)
11	1, 8, 10	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (𝑁‘𝑦) ∈ ℝ)
12		abscl 15314	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ)
13	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (abs‘𝑥) ∈ ℝ)
14	11, 13	readdcld 11288	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → ((𝑁‘𝑦) + (abs‘𝑥)) ∈ ℝ)
15	2, 9	nvge0 30702	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑁‘𝑦))
16	1, 8, 15	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑁‘𝑦))
17		absge0 15323	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → 0 ≤ (abs‘𝑥))
18	17	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → 0 ≤ (abs‘𝑥))
19	11, 13, 16, 18	addge0d 11837	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → 0 ≤ ((𝑁‘𝑦) + (abs‘𝑥)))
20	14, 19	ge0p1rpd 13105	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (((𝑁‘𝑦) + (abs‘𝑥)) + 1) ∈ ℝ⁺)
21		rpdivcl 13058	. . . . . . . . 9 ⊢ (((((𝑁‘𝑦) + (abs‘𝑥)) + 1) ∈ ℝ⁺ ∧ 𝑟 ∈ ℝ⁺) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) ∈ ℝ⁺)
22	20, 21	sylan 580	. . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) ∈ ℝ⁺)
23		rpaddcl 13055	. . . . . . . 8 ⊢ ((1 ∈ ℝ⁺ ∧ ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) ∈ ℝ⁺) → (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)) ∈ ℝ⁺)
24	7, 22, 23	sylancr 587	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)) ∈ ℝ⁺)
25	24	rpreccld 13085	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (1 / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))) ∈ ℝ⁺)
26	6, 25	eqeltrid 2843	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → 𝑇 ∈ ℝ⁺)
27		smcn.c	. . . . . . . . . . . 12 ⊢ 𝐶 = (IndMet‘𝑈)
28	2, 27	imsmet 30720	. . . . . . . . . . 11 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (Met‘𝑋))
29	1, 28	ax-mp 5	. . . . . . . . . 10 ⊢ 𝐶 ∈ (Met‘𝑋)
30	29	a1i 11	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝐶 ∈ (Met‘𝑋))
31	1	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑈 ∈ NrmCVec)
32		simplll 775	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑥 ∈ ℂ)
33		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑦 ∈ 𝑋)
34	2, 3	nvscl 30655	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (𝑥𝑆𝑦) ∈ 𝑋)
35	31, 32, 33, 34	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑥𝑆𝑦) ∈ 𝑋)
36		simprll 779	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑧 ∈ ℂ)
37		simprlr 780	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑤 ∈ 𝑋)
38	2, 3	nvscl 30655	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) → (𝑧𝑆𝑤) ∈ 𝑋)
39	31, 36, 37, 38	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑧𝑆𝑤) ∈ 𝑋)
40		metcl 24358	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Met‘𝑋) ∧ (𝑥𝑆𝑦) ∈ 𝑋 ∧ (𝑧𝑆𝑤) ∈ 𝑋) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) ∈ ℝ)
41	30, 35, 39, 40	syl3anc 1370	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) ∈ ℝ)
42	2, 3	nvscl 30655	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (𝑧𝑆𝑦) ∈ 𝑋)
43	31, 36, 33, 42	syl3anc 1370	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑧𝑆𝑦) ∈ 𝑋)
44		metcl 24358	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Met‘𝑋) ∧ (𝑥𝑆𝑦) ∈ 𝑋 ∧ (𝑧𝑆𝑦) ∈ 𝑋) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) ∈ ℝ)
45	30, 35, 43, 44	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) ∈ ℝ)
46		metcl 24358	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Met‘𝑋) ∧ (𝑧𝑆𝑦) ∈ 𝑋 ∧ (𝑧𝑆𝑤) ∈ 𝑋) → ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤)) ∈ ℝ)
47	30, 43, 39, 46	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤)) ∈ ℝ)
48	45, 47	readdcld 11288	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) + ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤))) ∈ ℝ)
49		rpre 13041	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
50	49	ad2antlr 727	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑟 ∈ ℝ)
51		mettri 24378	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Met‘𝑋) ∧ ((𝑥𝑆𝑦) ∈ 𝑋 ∧ (𝑧𝑆𝑤) ∈ 𝑋 ∧ (𝑧𝑆𝑦) ∈ 𝑋)) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) ≤ (((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) + ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤))))
52	30, 35, 39, 43, 51	syl13anc 1371	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) ≤ (((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) + ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤))))
53	1, 33, 10	sylancr 587	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑁‘𝑦) ∈ ℝ)
54	32	abscld 15472	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘𝑥) ∈ ℝ)
55	53, 54	readdcld 11288	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑁‘𝑦) + (abs‘𝑥)) ∈ ℝ)
56		peano2re 11432	. . . . . . . . . . 11 ⊢ (((𝑁‘𝑦) + (abs‘𝑥)) ∈ ℝ → (((𝑁‘𝑦) + (abs‘𝑥)) + 1) ∈ ℝ)
57	55, 56	syl 17	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((𝑁‘𝑦) + (abs‘𝑥)) + 1) ∈ ℝ)
58	26	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑇 ∈ ℝ⁺)
59	58	rpred 13075	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑇 ∈ ℝ)
60	57, 59	remulcld 11289	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) · 𝑇) ∈ ℝ)
61	32, 36	subcld 11618	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑥 − 𝑧) ∈ ℂ)
62	61	abscld 15472	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘(𝑥 − 𝑧)) ∈ ℝ)
63	62, 53	remulcld 11289	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((abs‘(𝑥 − 𝑧)) · (𝑁‘𝑦)) ∈ ℝ)
64	36	abscld 15472	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘𝑧) ∈ ℝ)
65		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ ( −_𝑣 ‘𝑈) = ( −_𝑣 ‘𝑈)
66	2, 65	nvmcl 30675	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑦( −_𝑣 ‘𝑈)𝑤) ∈ 𝑋)
67	31, 33, 37, 66	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑦( −_𝑣 ‘𝑈)𝑤) ∈ 𝑋)
68	2, 9	nvcl 30690	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑦( −_𝑣 ‘𝑈)𝑤) ∈ 𝑋) → (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)) ∈ ℝ)
69	1, 67, 68	sylancr 587	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)) ∈ ℝ)
70	64, 69	remulcld 11289	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((abs‘𝑧) · (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤))) ∈ ℝ)
71	53, 59	remulcld 11289	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑁‘𝑦) · 𝑇) ∈ ℝ)
72		peano2re 11432	. . . . . . . . . . . . 13 ⊢ ((abs‘𝑥) ∈ ℝ → ((abs‘𝑥) + 1) ∈ ℝ)
73	54, 72	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((abs‘𝑥) + 1) ∈ ℝ)
74	73, 59	remulcld 11289	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((abs‘𝑥) + 1) · 𝑇) ∈ ℝ)
75	1, 33, 15	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 0 ≤ (𝑁‘𝑦))
76	32, 36	abssubd 15489	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘(𝑥 − 𝑧)) = (abs‘(𝑧 − 𝑥)))
77	36, 32	subcld 11618	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑧 − 𝑥) ∈ ℂ)
78	77	abscld 15472	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘(𝑧 − 𝑥)) ∈ ℝ)
79		eqid 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs ∘ − ) = (abs ∘ − )
80	79	cnmetdval 24807	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(abs ∘ − )𝑧) = (abs‘(𝑥 − 𝑧)))
81	32, 36, 80	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑥(abs ∘ − )𝑧) = (abs‘(𝑥 − 𝑧)))
82	81, 76	eqtrd 2775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑥(abs ∘ − )𝑧) = (abs‘(𝑧 − 𝑥)))
83		simprrl 781	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑥(abs ∘ − )𝑧) < 𝑇)
84	82, 83	eqbrtrrd 5172	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘(𝑧 − 𝑥)) < 𝑇)
85	78, 59, 84	ltled 11407	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘(𝑧 − 𝑥)) ≤ 𝑇)
86	76, 85	eqbrtrd 5170	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘(𝑥 − 𝑧)) ≤ 𝑇)
87	62, 59, 53, 75, 86	lemul1ad 12205	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((abs‘(𝑥 − 𝑧)) · (𝑁‘𝑦)) ≤ (𝑇 · (𝑁‘𝑦)))
88	58	rpcnd 13077	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑇 ∈ ℂ)
89	53	recnd 11287	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑁‘𝑦) ∈ ℂ)
90	88, 89	mulcomd 11280	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑇 · (𝑁‘𝑦)) = ((𝑁‘𝑦) · 𝑇))
91	87, 90	breqtrd 5174	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((abs‘(𝑥 − 𝑧)) · (𝑁‘𝑦)) ≤ ((𝑁‘𝑦) · 𝑇))
92	36	absge0d 15480	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 0 ≤ (abs‘𝑧))
93	2, 9	nvge0 30702	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑦( −_𝑣 ‘𝑈)𝑤) ∈ 𝑋) → 0 ≤ (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)))
94	1, 67, 93	sylancr 587	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 0 ≤ (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)))
95	54, 78	readdcld 11288	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((abs‘𝑥) + (abs‘(𝑧 − 𝑥))) ∈ ℝ)
96	32, 36	pncan3d 11621	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑥 + (𝑧 − 𝑥)) = 𝑧)
97	96	fveq2d 6911	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘(𝑥 + (𝑧 − 𝑥))) = (abs‘𝑧))
98	32, 77	abstrid 15492	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘(𝑥 + (𝑧 − 𝑥))) ≤ ((abs‘𝑥) + (abs‘(𝑧 − 𝑥))))
99	97, 98	eqbrtrrd 5172	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘𝑧) ≤ ((abs‘𝑥) + (abs‘(𝑧 − 𝑥))))
100		1red 11260	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 1 ∈ ℝ)
101		1re 11259	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ
102	22	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) ∈ ℝ⁺)
103		ltaddrp 13070	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ ∧ ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) ∈ ℝ⁺) → 1 < (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)))
104	101, 102, 103	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 1 < (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)))
105	24	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)) ∈ ℝ⁺)
106	105	recgt1d 13089	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (1 < (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)) ↔ (1 / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))) < 1))
107	104, 106	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (1 / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))) < 1)
108	6, 107	eqbrtrid 5183	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑇 < 1)
109	59, 100, 108	ltled 11407	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑇 ≤ 1)
110	78, 59, 100, 85, 109	letrd 11416	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘(𝑧 − 𝑥)) ≤ 1)
111	78, 100, 54, 110	leadd2dd 11876	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((abs‘𝑥) + (abs‘(𝑧 − 𝑥))) ≤ ((abs‘𝑥) + 1))
112	64, 95, 73, 99, 111	letrd 11416	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘𝑧) ≤ ((abs‘𝑥) + 1))
113	2, 65, 9, 27	imsdval 30715	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑦𝐶𝑤) = (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)))
114	31, 33, 37, 113	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑦𝐶𝑤) = (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)))
115		simprrr 782	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑦𝐶𝑤) < 𝑇)
116	114, 115	eqbrtrrd 5172	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)) < 𝑇)
117	69, 59, 116	ltled 11407	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)) ≤ 𝑇)
118	64, 73, 69, 59, 92, 94, 112, 117	lemul12ad 12208	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((abs‘𝑧) · (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤))) ≤ (((abs‘𝑥) + 1) · 𝑇))
119	63, 70, 71, 74, 91, 118	le2addd 11880	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((abs‘(𝑥 − 𝑧)) · (𝑁‘𝑦)) + ((abs‘𝑧) · (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)))) ≤ (((𝑁‘𝑦) · 𝑇) + (((abs‘𝑥) + 1) · 𝑇)))
120		eqid 2735	. . . . . . . . . . . . . 14 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
121	2, 120, 3, 9, 27	imsdval2 30716	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝑆𝑦) ∈ 𝑋 ∧ (𝑧𝑆𝑦) ∈ 𝑋) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) = (𝑁‘((𝑥𝑆𝑦)( +_𝑣 ‘𝑈)(-1𝑆(𝑧𝑆𝑦)))))
122	31, 35, 43, 121	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) = (𝑁‘((𝑥𝑆𝑦)( +_𝑣 ‘𝑈)(-1𝑆(𝑧𝑆𝑦)))))
123		neg1cn 12378	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℂ
124		mulcl 11237	. . . . . . . . . . . . . . . 16 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (-1 · 𝑧) ∈ ℂ)
125	123, 36, 124	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (-1 · 𝑧) ∈ ℂ)
126	2, 120, 3	nvdir 30660	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ ℂ ∧ (-1 · 𝑧) ∈ ℂ ∧ 𝑦 ∈ 𝑋)) → ((𝑥 + (-1 · 𝑧))𝑆𝑦) = ((𝑥𝑆𝑦)( +_𝑣 ‘𝑈)((-1 · 𝑧)𝑆𝑦)))
127	31, 32, 125, 33, 126	syl13anc 1371	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥 + (-1 · 𝑧))𝑆𝑦) = ((𝑥𝑆𝑦)( +_𝑣 ‘𝑈)((-1 · 𝑧)𝑆𝑦)))
128	36	mulm1d 11713	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (-1 · 𝑧) = -𝑧)
129	128	oveq2d 7447	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑥 + (-1 · 𝑧)) = (𝑥 + -𝑧))
130	32, 36	negsubd 11624	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑥 + -𝑧) = (𝑥 − 𝑧))
131	129, 130	eqtrd 2775	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑥 + (-1 · 𝑧)) = (𝑥 − 𝑧))
132	131	oveq1d 7446	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥 + (-1 · 𝑧))𝑆𝑦) = ((𝑥 − 𝑧)𝑆𝑦))
133	123	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → -1 ∈ ℂ)
134	2, 3	nvsass 30657	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑦 ∈ 𝑋)) → ((-1 · 𝑧)𝑆𝑦) = (-1𝑆(𝑧𝑆𝑦)))
135	31, 133, 36, 33, 134	syl13anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((-1 · 𝑧)𝑆𝑦) = (-1𝑆(𝑧𝑆𝑦)))
136	135	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥𝑆𝑦)( +_𝑣 ‘𝑈)((-1 · 𝑧)𝑆𝑦)) = ((𝑥𝑆𝑦)( +_𝑣 ‘𝑈)(-1𝑆(𝑧𝑆𝑦))))
137	127, 132, 136	3eqtr3d 2783	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥 − 𝑧)𝑆𝑦) = ((𝑥𝑆𝑦)( +_𝑣 ‘𝑈)(-1𝑆(𝑧𝑆𝑦))))
138	137	fveq2d 6911	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑁‘((𝑥 − 𝑧)𝑆𝑦)) = (𝑁‘((𝑥𝑆𝑦)( +_𝑣 ‘𝑈)(-1𝑆(𝑧𝑆𝑦)))))
139	2, 3, 9	nvs 30692	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 − 𝑧) ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (𝑁‘((𝑥 − 𝑧)𝑆𝑦)) = ((abs‘(𝑥 − 𝑧)) · (𝑁‘𝑦)))
140	31, 61, 33, 139	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑁‘((𝑥 − 𝑧)𝑆𝑦)) = ((abs‘(𝑥 − 𝑧)) · (𝑁‘𝑦)))
141	122, 138, 140	3eqtr2d 2781	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) = ((abs‘(𝑥 − 𝑧)) · (𝑁‘𝑦)))
142	2, 65, 9, 27	imsdval 30715	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑧𝑆𝑦) ∈ 𝑋 ∧ (𝑧𝑆𝑤) ∈ 𝑋) → ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤)) = (𝑁‘((𝑧𝑆𝑦)( −_𝑣 ‘𝑈)(𝑧𝑆𝑤))))
143	31, 43, 39, 142	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤)) = (𝑁‘((𝑧𝑆𝑦)( −_𝑣 ‘𝑈)(𝑧𝑆𝑤))))
144	2, 65, 3	nvmdi 30677	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑧 ∈ ℂ ∧ 𝑦 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧𝑆(𝑦( −_𝑣 ‘𝑈)𝑤)) = ((𝑧𝑆𝑦)( −_𝑣 ‘𝑈)(𝑧𝑆𝑤)))
145	31, 36, 33, 37, 144	syl13anc 1371	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑧𝑆(𝑦( −_𝑣 ‘𝑈)𝑤)) = ((𝑧𝑆𝑦)( −_𝑣 ‘𝑈)(𝑧𝑆𝑤)))
146	145	fveq2d 6911	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑁‘(𝑧𝑆(𝑦( −_𝑣 ‘𝑈)𝑤))) = (𝑁‘((𝑧𝑆𝑦)( −_𝑣 ‘𝑈)(𝑧𝑆𝑤))))
147	2, 3, 9	nvs 30692	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑧 ∈ ℂ ∧ (𝑦( −_𝑣 ‘𝑈)𝑤) ∈ 𝑋) → (𝑁‘(𝑧𝑆(𝑦( −_𝑣 ‘𝑈)𝑤))) = ((abs‘𝑧) · (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤))))
148	31, 36, 67, 147	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (𝑁‘(𝑧𝑆(𝑦( −_𝑣 ‘𝑈)𝑤))) = ((abs‘𝑧) · (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤))))
149	143, 146, 148	3eqtr2d 2781	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤)) = ((abs‘𝑧) · (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤))))
150	141, 149	oveq12d 7449	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) + ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤))) = (((abs‘(𝑥 − 𝑧)) · (𝑁‘𝑦)) + ((abs‘𝑧) · (𝑁‘(𝑦( −_𝑣 ‘𝑈)𝑤)))))
151	54	recnd 11287	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (abs‘𝑥) ∈ ℂ)
152		1cnd 11254	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 1 ∈ ℂ)
153	89, 151, 152	addassd 11281	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((𝑁‘𝑦) + (abs‘𝑥)) + 1) = ((𝑁‘𝑦) + ((abs‘𝑥) + 1)))
154	153	oveq1d 7446	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) · 𝑇) = (((𝑁‘𝑦) + ((abs‘𝑥) + 1)) · 𝑇))
155	73	recnd 11287	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((abs‘𝑥) + 1) ∈ ℂ)
156	89, 155, 88	adddird 11284	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((𝑁‘𝑦) + ((abs‘𝑥) + 1)) · 𝑇) = (((𝑁‘𝑦) · 𝑇) + (((abs‘𝑥) + 1) · 𝑇)))
157	154, 156	eqtrd 2775	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) · 𝑇) = (((𝑁‘𝑦) · 𝑇) + (((abs‘𝑥) + 1) · 𝑇)))
158	119, 150, 157	3brtr4d 5180	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) + ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤))) ≤ ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) · 𝑇))
159	6	oveq2i 7442	. . . . . . . . . . 11 ⊢ ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) · 𝑇) = ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) · (1 / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))))
160	57	recnd 11287	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((𝑁‘𝑦) + (abs‘𝑥)) + 1) ∈ ℂ)
161	105	rpcnd 13077	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)) ∈ ℂ)
162	105	rpne0d 13080	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)) ≠ 0)
163	160, 161, 162	divrecd 12044	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))) = ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) · (1 / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)))))
164	159, 163	eqtr4id 2794	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) · 𝑇) = ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))))
165		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → 𝑟 ∈ ℝ⁺)
166	102	rpred 13075	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) ∈ ℝ)
167	166	ltp1d 12196	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) < (((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) + 1))
168	102	rpcnd 13077	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) ∈ ℂ)
169	168, 152	addcomd 11461	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) + 1) = (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)))
170	167, 169	breqtrd 5174	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟) < (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟)))
171	57, 165, 105, 170	ltdiv23d 13142	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))) < 𝑟)
172	164, 171	eqbrtrd 5170	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) · 𝑇) < 𝑟)
173	48, 60, 50, 158, 172	lelttrd 11417	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → (((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑦)) + ((𝑧𝑆𝑦)𝐶(𝑧𝑆𝑤))) < 𝑟)
174	41, 48, 50, 52, 173	lelttrd 11417	. . . . . . 7 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋) ∧ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇))) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟)
175	174	expr 456	. . . . . 6 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ 𝑋)) → (((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟))
176	175	ralrimivva 3200	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟))
177		breq2 5152	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((𝑥(abs ∘ − )𝑧) < 𝑠 ↔ (𝑥(abs ∘ − )𝑧) < 𝑇))
178		breq2 5152	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((𝑦𝐶𝑤) < 𝑠 ↔ (𝑦𝐶𝑤) < 𝑇))
179	177, 178	anbi12d 632	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → (((𝑥(abs ∘ − )𝑧) < 𝑠 ∧ (𝑦𝐶𝑤) < 𝑠) ↔ ((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇)))
180	179	imbi1d 341	. . . . . . 7 ⊢ (𝑠 = 𝑇 → ((((𝑥(abs ∘ − )𝑧) < 𝑠 ∧ (𝑦𝐶𝑤) < 𝑠) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟) ↔ (((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟)))
181	180	2ralbidv 3219	. . . . . 6 ⊢ (𝑠 = 𝑇 → (∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑠 ∧ (𝑦𝐶𝑤) < 𝑠) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟) ↔ ∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟)))
182	181	rspcev 3622	. . . . 5 ⊢ ((𝑇 ∈ ℝ⁺ ∧ ∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑇 ∧ (𝑦𝐶𝑤) < 𝑇) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟)) → ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑠 ∧ (𝑦𝐶𝑤) < 𝑠) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟))
183	26, 176, 182	syl2anc 584	. . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑠 ∧ (𝑦𝐶𝑤) < 𝑠) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟))
184	183	ralrimiva 3144	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑠 ∧ (𝑦𝐶𝑤) < 𝑠) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟))
185	184	rgen2 3197	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑠 ∧ (𝑦𝐶𝑤) < 𝑠) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟)
186		cnxmet 24809	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
187	2, 27	imsxmet 30721	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (∞Met‘𝑋))
188	1, 187	ax-mp 5	. . 3 ⊢ 𝐶 ∈ (∞Met‘𝑋)
189		smcn.k	. . . . 5 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
190	189	cnfldtopn 24818	. . . 4 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
191		smcn.j	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐶)
192	190, 191, 191	txmetcn 24577	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ (∞Met‘𝑋)) → (𝑆 ∈ ((𝐾 ×_t 𝐽) Cn 𝐽) ↔ (𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑠 ∧ (𝑦𝐶𝑤) < 𝑠) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟))))
193	186, 188, 188, 192	mp3an 1460	. 2 ⊢ (𝑆 ∈ ((𝐾 ×_t 𝐽) Cn 𝐽) ↔ (𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ ℂ ∀𝑤 ∈ 𝑋 (((𝑥(abs ∘ − )𝑧) < 𝑠 ∧ (𝑦𝐶𝑤) < 𝑠) → ((𝑥𝑆𝑦)𝐶(𝑧𝑆𝑤)) < 𝑟)))
194	5, 185, 193	mpbir2an 711	1 ⊢ 𝑆 ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 class class class wbr 5148 × cxp 5687 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 -cneg 11491 / cdiv 11918 ℝ⁺crp 13032 abscabs 15270 TopOpenctopn 17468 ∞Metcxmet 21367 Metcmet 21368 MetOpencmopn 21372 ℂ_fldccnfld 21382 Cn ccn 23248 ×_t ctx 23584 NrmCVeccnv 30613 +_𝑣 cpv 30614 BaseSetcba 30615 ·_𝑠OLD cns 30616 −_𝑣 cnsb 30618 norm_CVcnmcv 30619 IndMetcims 30620

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cn 23251 df-cnp 23252 df-tx 23586 df-hmeo 23779 df-xms 24346 df-tms 24348 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-ims 30630

This theorem is referenced by: smcn 30727

W3C validator