Theorem norm3lemt 28928

Description: Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
norm3lemt	⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℝ)) → (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷))

Proof of Theorem norm3lemt

Step	Hyp	Ref	Expression
1		fvoveq1 7178	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (normℎ‘(𝐴 −ℎ 𝐶)) = (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)))
2	1	breq1d 5075	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ↔ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2)))
3	2	anbi1d 631	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) ↔ ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2))))
4		fvoveq1 7178	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵)))
5	4	breq1d 5075	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷 ↔ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵)) < 𝐷))
6	3, 5	imbi12d 347	. 2 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷) ↔ (((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵)) < 𝐷)))
7		oveq2 7163	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (𝐶 −ℎ 𝐵) = (𝐶 −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)))
8	7	fveq2d 6673	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (normℎ‘(𝐶 −ℎ 𝐵)) = (normℎ‘(𝐶 −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))))
9	8	breq1d 5075	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2) ↔ (normℎ‘(𝐶 −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2)))
10	9	anbi2d 630	. . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) ↔ ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2))))
11		oveq2 7163	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)))
12	11	fveq2d 6673	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵)) = (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))))
13	12	breq1d 5075	. . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵)) < 𝐷 ↔ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < 𝐷))
14	10, 13	imbi12d 347	. 2 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵)) < 𝐷) ↔ (((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2)) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < 𝐷)))
15		oveq2 7163	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ)))
16	15	fveq2d 6673	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) = (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))))
17	16	breq1d 5075	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2) ↔ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (𝐷 / 2)))
18		fvoveq1 7178	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → (normℎ‘(𝐶 −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) = (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))))
19	18	breq1d 5075	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → ((normℎ‘(𝐶 −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2) ↔ (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2)))
20	17, 19	anbi12d 632	. . 3 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → (((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2)) ↔ ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (𝐷 / 2) ∧ (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2))))
21	20	imbi1d 344	. 2 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → ((((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2)) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < 𝐷) ↔ (((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (𝐷 / 2) ∧ (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2)) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < 𝐷)))
22		oveq1 7162	. . . . 5 ⊢ (𝐷 = if(𝐷 ∈ ℝ, 𝐷, 2) → (𝐷 / 2) = (if(𝐷 ∈ ℝ, 𝐷, 2) / 2))
23	22	breq2d 5077	. . . 4 ⊢ (𝐷 = if(𝐷 ∈ ℝ, 𝐷, 2) → ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (𝐷 / 2) ↔ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (if(𝐷 ∈ ℝ, 𝐷, 2) / 2)))
24	22	breq2d 5077	. . . 4 ⊢ (𝐷 = if(𝐷 ∈ ℝ, 𝐷, 2) → ((normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2) ↔ (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (if(𝐷 ∈ ℝ, 𝐷, 2) / 2)))
25	23, 24	anbi12d 632	. . 3 ⊢ (𝐷 = if(𝐷 ∈ ℝ, 𝐷, 2) → (((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (𝐷 / 2) ∧ (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2)) ↔ ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (if(𝐷 ∈ ℝ, 𝐷, 2) / 2) ∧ (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (if(𝐷 ∈ ℝ, 𝐷, 2) / 2))))
26		breq2 5069	. . 3 ⊢ (𝐷 = if(𝐷 ∈ ℝ, 𝐷, 2) → ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < 𝐷 ↔ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < if(𝐷 ∈ ℝ, 𝐷, 2)))
27	25, 26	imbi12d 347	. 2 ⊢ (𝐷 = if(𝐷 ∈ ℝ, 𝐷, 2) → ((((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (𝐷 / 2) ∧ (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (𝐷 / 2)) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < 𝐷) ↔ (((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (if(𝐷 ∈ ℝ, 𝐷, 2) / 2) ∧ (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (if(𝐷 ∈ ℝ, 𝐷, 2) / 2)) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < if(𝐷 ∈ ℝ, 𝐷, 2))))
28		ifhvhv0 28798	. . 3 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ
29		ifhvhv0 28798	. . 3 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ∈ ℋ
30		ifhvhv0 28798	. . 3 ⊢ if(𝐶 ∈ ℋ, 𝐶, 0ℎ) ∈ ℋ
31		2re 11710	. . . 4 ⊢ 2 ∈ ℝ
32	31	elimel 4533	. . 3 ⊢ if(𝐷 ∈ ℝ, 𝐷, 2) ∈ ℝ
33	28, 29, 30, 32	norm3lem 28925	. 2 ⊢ (((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) < (if(𝐷 ∈ ℝ, 𝐷, 2) / 2) ∧ (normℎ‘(if(𝐶 ∈ ℋ, 𝐶, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < (if(𝐷 ∈ ℝ, 𝐷, 2) / 2)) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) < if(𝐷 ∈ ℝ, 𝐷, 2))
34	6, 14, 21, 27, 33	dedth4h 4525	1 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℝ)) → (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ifcif 4466   class class class wbr 5065  ‘cfv 6354  (class class class)co 7155  ℝcr 10535   < clt 10674   / cdiv 11296  2c2 11691   ℋchba 28695  norm_ℎcno 28699  0_ℎc0v 28700   −_ℎ cmv 28701

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614  ax-hfvadd 28776  ax-hvcom 28777  ax-hvass 28778  ax-hv0cl 28779  ax-hvaddid 28780  ax-hfvmul 28781  ax-hvmulid 28782  ax-hvmulass 28783  ax-hvdistr2 28785  ax-hvmul0 28786  ax-hfi 28855  ax-his1 28858  ax-his2 28859  ax-his3 28860  ax-his4 28861

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-sup 8905  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-seq 13369  df-exp 13429  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-hnorm 28744  df-hvsub 28747

This theorem is referenced by: (None)