Theorem pjthlem10 5234

Description: Lemma for Theorem 3.7(i) of [Beran] p. 102.

Hypotheses

Ref	Expression
pjthlem9.1	⊢ A ∈ ℋ
pjthlem9.2	⊢ B ∈ ℋ
pjthlem9.3	⊢ D ∈ ℋ
pjthlem9.4	⊢ R = (1 / (D ·i D))
pjthlem9.5	⊢ S = (R · (C ·i D))
pjthlem9.6	⊢ C = (A −v B)

Assertion

Ref	Expression
pjthlem10	⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → (R · ((abs ‘(C ·i D))↑2)) = 0)

Proof of Theorem pjthlem10

Step	Hyp	Ref	Expression
1		mulge0t 4375	. . . . 5 ⊢ ((R ∈ ℝ ∧ ((abs ‘(C ·i D))↑2) ∈ ℝ) → ((0 ≤ R ∧ 0 ≤ ((abs ‘(C ·i D))↑2)) → 0 ≤ (R · ((abs ‘(C ·i D))↑2))))
2		pjthlem9.3	. . . . . . 7 ⊢ D ∈ ℋ
3		pjthlem9.4	. . . . . . 7 ⊢ R = (1 / (D ·i D))
4	2, 3	pjthlem2 5226	. . . . . 6 ⊢ (¬ D = 0v → R ∈ ℝ)
5		pjthlem9.6	. . . . . . . . . 10 ⊢ C = (A −v B)
6		pjthlem9.1	. . . . . . . . . . 11 ⊢ A ∈ ℋ
7		pjthlem9.2	. . . . . . . . . . 11 ⊢ B ∈ ℋ
8	6, 7	hvsubcl 5002	. . . . . . . . . 10 ⊢ (A −v B) ∈ ℋ
9	5, 8	eqeltr 1159	. . . . . . . . 9 ⊢ C ∈ ℋ
10	9, 2	hicl 5044	. . . . . . . 8 ⊢ (C ·i D) ∈ ℂ
11	10	abscl 4840	. . . . . . 7 ⊢ (abs ‘(C ·i D)) ∈ ℝ
12	11	sqrecl 4699	. . . . . 6 ⊢ ((abs ‘(C ·i D))↑2) ∈ ℝ
13	4, 12	jctir 241	. . . . 5 ⊢ (¬ D = 0v → (R ∈ ℝ ∧ ((abs ‘(C ·i D))↑2) ∈ ℝ))
14		ltlet 4286	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ R ∈ ℝ) → (0 < R → 0 ≤ R))
15		ax0re 4063	. . . . . . . 8 ⊢ 0 ∈ ℝ
16	4, 15	jctil 240	. . . . . . 7 ⊢ (¬ D = 0v → (0 ∈ ℝ ∧ R ∈ ℝ))
17	2, 3	pjthlem3 5227	. . . . . . 7 ⊢ (¬ D = 0v → 0 < R)
18	14, 16, 17	sylc 62	. . . . . 6 ⊢ (¬ D = 0v → 0 ≤ R)
19	11	sqege0 4704	. . . . . 6 ⊢ 0 ≤ ((abs ‘(C ·i D))↑2)
20	18, 19	jctir 241	. . . . 5 ⊢ (¬ D = 0v → (0 ≤ R ∧ 0 ≤ ((abs ‘(C ·i D))↑2)))
21	1, 13, 20	sylc 62	. . . 4 ⊢ (¬ D = 0v → 0 ≤ (R · ((abs ‘(C ·i D))↑2)))
22	21	adantr 306	. . 3 ⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → 0 ≤ (R · ((abs ‘(C ·i D))↑2)))
23		pjthlem9.5	. . . . . 6 ⊢ S = (R · (C ·i D))
24	6, 7, 2, 3, 23, 5	pjthlem9 5233	. . . . 5 ⊢ (¬ D = 0v → ((norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A)) ↔ ((norm ‘C)↑2) ≤ ((norm ‘(C −v (S ·s D)))↑2)))
25	24	biimpa 324	. . . 4 ⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → ((norm ‘C)↑2) ≤ ((norm ‘(C −v (S ·s D)))↑2))
26	2, 3, 9, 23	pjthlem8 5232	. . . . 5 ⊢ (¬ D = 0v → ((norm ‘(C −v (S ·s D)))↑2) = (((norm ‘C)↑2) − (R · ((abs ‘(C ·i D))↑2))))
27	26	adantr 306	. . . 4 ⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → ((norm ‘(C −v (S ·s D)))↑2) = (((norm ‘C)↑2) − (R · ((abs ‘(C ·i D))↑2))))
28	25, 27	breqtrd 2081	. . 3 ⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → ((norm ‘C)↑2) ≤ (((norm ‘C)↑2) − (R · ((abs ‘(C ·i D))↑2))))
29	22, 28	jca 236	. 2 ⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → (0 ≤ (R · ((abs ‘(C ·i D))↑2)) ∧ ((norm ‘C)↑2) ≤ (((norm ‘C)↑2) − (R · ((abs ‘(C ·i D))↑2)))))
30		axmulrcl 4069	. . . . . 6 ⊢ ((R ∈ ℝ ∧ ((abs ‘(C ·i D))↑2) ∈ ℝ) → (R · ((abs ‘(C ·i D))↑2)) ∈ ℝ)
31	13, 30	syl 12	. . . . 5 ⊢ (¬ D = 0v → (R · ((abs ‘(C ·i D))↑2)) ∈ ℝ)
32	9	normcl 5081	. . . . . 6 ⊢ (norm ‘C) ∈ ℝ
33	32	sqrecl 4699	. . . . 5 ⊢ ((norm ‘C)↑2) ∈ ℝ
34	31, 33	jctir 241	. . . 4 ⊢ (¬ D = 0v → ((R · ((abs ‘(C ·i D))↑2)) ∈ ℝ ∧ ((norm ‘C)↑2) ∈ ℝ))
35		lesub0t 4374	. . . 4 ⊢ (((R · ((abs ‘(C ·i D))↑2)) ∈ ℝ ∧ ((norm ‘C)↑2) ∈ ℝ) → ((0 ≤ (R · ((abs ‘(C ·i D))↑2)) ∧ ((norm ‘C)↑2) ≤ (((norm ‘C)↑2) − (R · ((abs ‘(C ·i D))↑2)))) ↔ (R · ((abs ‘(C ·i D))↑2)) = 0))
36	34, 35	syl 12	. . 3 ⊢ (¬ D = 0v → ((0 ≤ (R · ((abs ‘(C ·i D))↑2)) ∧ ((norm ‘C)↑2) ≤ (((norm ‘C)↑2) − (R · ((abs ‘(C ·i D))↑2)))) ↔ (R · ((abs ‘(C ·i D))↑2)) = 0))
37	36	adantr 306	. 2 ⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → ((0 ≤ (R · ((abs ‘(C ·i D))↑2)) ∧ ((norm ‘C)↑2) ≤ (((norm ‘C)↑2) − (R · ((abs ‘(C ·i D))↑2)))) ↔ (R · ((abs ‘(C ·i D))↑2)) = 0))
38	29, 37	mpbid 170	1 ⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → (R · ((abs ‘(C ·i D))↑2)) = 0)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   class class class wbr 2054   ‘cfv 2422  (class class class)co 3001  ℝcr 4027  0cc0 4028  1c1 4029   · cmulc 4032   < clt 4033   − cmin 4089   / cdiv 4091   ≤ cle 4092  2c2 4454  ↑cexp 4675  abscabs 4789   ℋ chil 4958   +_v cva 4959   ·_s csm 4960  0_vc0v 4961   −_v cmv 4962   ·_i csp 4963  normcno 4964

This theorem is referenced by:  pjthlem11 5235

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-hvaddcl 4984  ax-hvcom 4985  ax-hvass 4986  ax-hvzercl 4987  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  ax-his3 5047  ax-his4 5048

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676  df-sqr 4728  df-re 4790  df-im 4791  df-cj 4792  df-abs 4793  df-hvsub 4996  df-hnorm 5074

metamath.org