Theorem bcsiALT 10679

Description: Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98.

Hypotheses

Ref	Expression
bcs.1	|- A e. ~H
bcs.2	|- B e. ~H

Assertion

Ref	Expression
bcsiALT	|- (abs` (A .ih B)) <_ ((normh` A) x. (normh` B))

Proof of Theorem bcsiALT

Step	Hyp	Ref	Expression
1		fveq2 4681	. . 3 |- ((A .ih B) = 0 -> (abs` (A .ih B)) = (abs` 0))
2		abs0 8129	. . . 4 |- (abs` 0) = 0
3		bcs.1	. . . . . 6 |- A e. ~H
4		normge0 10625	. . . . . 6 |- (A e. ~H -> 0 <_ (normh` A))
5	3, 4	ax-mp 7	. . . . 5 |- 0 <_ (normh` A)
6		bcs.2	. . . . . 6 |- B e. ~H
7		normge0 10625	. . . . . 6 |- (B e. ~H -> 0 <_ (normh` B))
8	6, 7	ax-mp 7	. . . . 5 |- 0 <_ (normh` B)
9	3	normcli 10631	. . . . . 6 |- (normh` A) e. RR
10	6	normcli 10631	. . . . . 6 |- (normh` B) e. RR
11	9, 10	mulge0i 6787	. . . . 5 |- ((0 <_ (normh` A) /\ 0 <_ (normh` B)) -> 0 <_ ((normh` A) x. (normh` B)))
12	5, 8, 11	mp2an 761	. . . 4 |- 0 <_ ((normh` A) x. (normh` B))
13	2, 12	eqbrtri 3356	. . 3 |- (abs` 0) <_ ((normh` A) x. (normh` B))
14	1, 13	syl6eqbr 3374	. 2 |- ((A .ih B) = 0 -> (abs` (A .ih B)) <_ ((normh` A) x. (normh` B)))
15		df-ne 2019	. . . 4 |- ((A .ih B) =/= 0 <-> -. (A .ih B) = 0)
16	3, 6	hicli 10581	. . . . . . 7 |- (A .ih B) e. CC
17	16	abslem2 8161	. . . . . 6 |- ((A .ih B) =/= 0 -> (((*` ((A .ih B) / (abs` (A .ih B)))) x. (A .ih B)) + (((A .ih B) / (abs` (A .ih B))) x. (*` (A .ih B)))) = (2 x. (abs` (A .ih B))))
18	6, 3	his1i 10599	. . . . . . . 8 |- (B .ih A) = (*` (A .ih B))
19	18	opreq2i 4893	. . . . . . 7 |- (((A .ih B) / (abs` (A .ih B))) x. (B .ih A)) = (((A .ih B) / (abs` (A .ih B))) x. (*` (A .ih B)))
20	19	opreq2i 4893	. . . . . 6 |- (((*` ((A .ih B) / (abs` (A .ih B)))) x. (A .ih B)) + (((A .ih B) / (abs` (A .ih B))) x. (B .ih A))) = (((*` ((A .ih B) / (abs` (A .ih B)))) x. (A .ih B)) + (((A .ih B) / (abs` (A .ih B))) x. (*` (A .ih B))))
21	17, 20	syl5req 1941	. . . . 5 |- ((A .ih B) =/= 0 -> (2 x. (abs` (A .ih B))) = (((*` ((A .ih B) / (abs` (A .ih B)))) x. (A .ih B)) + (((A .ih B) / (abs` (A .ih B))) x. (B .ih A))))
22	16	abs00i 8093	. . . . . . . 8 |- ((abs` (A .ih B)) = 0 <-> (A .ih B) = 0)
23	22	necon3bii 2032	. . . . . . 7 |- ((abs` (A .ih B)) =/= 0 <-> (A .ih B) =/= 0)
24	16	abscli 8090	. . . . . . . . . 10 |- (abs` (A .ih B)) e. RR
25	24	recni 6467	. . . . . . . . 9 |- (abs` (A .ih B)) e. CC
26	16, 25	divclzi 6900	. . . . . . . 8 |- ((abs` (A .ih B)) =/= 0 -> ((A .ih B) / (abs` (A .ih B))) e. CC)
27	16, 25	divreczi 6921	. . . . . . . . . 10 |- ((abs` (A .ih B)) =/= 0 -> ((A .ih B) / (abs` (A .ih B))) = ((A .ih B) x. (1 / (abs` (A .ih B)))))
28	27	fveq2d 4685	. . . . . . . . 9 |- ((abs` (A .ih B)) =/= 0 -> (abs` ((A .ih B) / (abs` (A .ih B)))) = (abs` ((A .ih B) x. (1 / (abs` (A .ih B))))))
29	25	recclzi 6903	. . . . . . . . . . . 12 |- ((abs` (A .ih B)) =/= 0 -> (1 / (abs` (A .ih B))) e. CC)
30	29, 16	jctil 316	. . . . . . . . . . 11 |- ((abs` (A .ih B)) =/= 0 -> ((A .ih B) e. CC /\ (1 / (abs` (A .ih B))) e. CC))
31		absmul 8109	. . . . . . . . . . 11 |- (((A .ih B) e. CC /\ (1 / (abs` (A .ih B))) e. CC) -> (abs` ((A .ih B) x. (1 / (abs` (A .ih B))))) = ((abs` (A .ih B)) x. (abs` (1 / (abs` (A .ih B))))))
32	30, 31	syl 12	. . . . . . . . . 10 |- ((abs` (A .ih B)) =/= 0 -> (abs` ((A .ih B) x. (1 / (abs` (A .ih B))))) = ((abs` (A .ih B)) x. (abs` (1 / (abs` (A .ih B))))))
33	24	rerecclzi 6980	. . . . . . . . . . . 12 |- ((abs` (A .ih B)) =/= 0 -> (1 / (abs` (A .ih B))) e. RR)
34		0re 6603	. . . . . . . . . . . . . 14 |- 0 e. RR
35	33, 34	jctil 316	. . . . . . . . . . . . 13 |- ((abs` (A .ih B)) =/= 0 -> (0 e. RR /\ (1 / (abs` (A .ih B))) e. RR))
36	16	absgt0i 8094	. . . . . . . . . . . . . . 15 |- ((A .ih B) =/= 0 <-> 0 < (abs` (A .ih B)))
37	23, 36	bitri 190	. . . . . . . . . . . . . 14 |- ((abs` (A .ih B)) =/= 0 <-> 0 < (abs` (A .ih B)))
38	24	recgt0i 7044	. . . . . . . . . . . . . 14 |- (0 < (abs` (A .ih B)) -> 0 < (1 / (abs` (A .ih B))))
39	37, 38	sylbi 216	. . . . . . . . . . . . 13 |- ((abs` (A .ih B)) =/= 0 -> 0 < (1 / (abs` (A .ih B))))
40		ltle 6690	. . . . . . . . . . . . 13 |- ((0 e. RR /\ (1 / (abs` (A .ih B))) e. RR) -> (0 < (1 / (abs` (A .ih B))) -> 0 <_ (1 / (abs` (A .ih B)))))
41	35, 39, 40	sylc 83	. . . . . . . . . . . 12 |- ((abs` (A .ih B)) =/= 0 -> 0 <_ (1 / (abs` (A .ih B))))
42		absid 8113	. . . . . . . . . . . 12 |- (((1 / (abs` (A .ih B))) e. RR /\ 0 <_ (1 / (abs` (A .ih B)))) -> (abs` (1 / (abs` (A .ih B)))) = (1 / (abs` (A .ih B))))
43	33, 41, 42	syl11anc 524	. . . . . . . . . . 11 |- ((abs` (A .ih B)) =/= 0 -> (abs` (1 / (abs` (A .ih B)))) = (1 / (abs` (A .ih B))))
44	43	opreq2d 4898	. . . . . . . . . 10 |- ((abs` (A .ih B)) =/= 0 -> ((abs` (A .ih B)) x. (abs` (1 / (abs` (A .ih B))))) = ((abs` (A .ih B)) x. (1 / (abs` (A .ih B)))))
45	32, 44	eqtrd 1925	. . . . . . . . 9 |- ((abs` (A .ih B)) =/= 0 -> (abs` ((A .ih B) x. (1 / (abs` (A .ih B))))) = ((abs` (A .ih B)) x. (1 / (abs` (A .ih B)))))
46	25	recidzi 6917	. . . . . . . . 9 |- ((abs` (A .ih B)) =/= 0 -> ((abs` (A .ih B)) x. (1 / (abs` (A .ih B)))) = 1)
47	28, 45, 46	3eqtrd 1929	. . . . . . . 8 |- ((abs` (A .ih B)) =/= 0 -> (abs` ((A .ih B) / (abs` (A .ih B)))) = 1)
48	26, 47	jca 310	. . . . . . 7 |- ((abs` (A .ih B)) =/= 0 -> (((A .ih B) / (abs` (A .ih B))) e. CC /\ (abs` ((A .ih B) / (abs` (A .ih B)))) = 1))
49	23, 48	sylbir 218	. . . . . 6 |- ((A .ih B) =/= 0 -> (((A .ih B) / (abs` (A .ih B))) e. CC /\ (abs` ((A .ih B) / (abs` (A .ih B)))) = 1))
50	3, 6	normlem7tALT 10618	. . . . . 6 |- ((((A .ih B) / (abs` (A .ih B))) e. CC /\ (abs` ((A .ih B) / (abs` (A .ih B)))) = 1) -> (((*` ((A .ih B) / (abs` (A .ih B)))) x. (A .ih B)) + (((A .ih B) / (abs` (A .ih B))) x. (B .ih A))) <_ (2 x. ((sqr` (B .ih B)) x. (sqr` (A .ih A)))))
51	49, 50	syl 12	. . . . 5 |- ((A .ih B) =/= 0 -> (((*` ((A .ih B) / (abs` (A .ih B)))) x. (A .ih B)) + (((A .ih B) / (abs` (A .ih B))) x. (B .ih A))) <_ (2 x. ((sqr` (B .ih B)) x. (sqr` (A .ih A)))))
52	21, 51	eqbrtrd 3357	. . . 4 |- ((A .ih B) =/= 0 -> (2 x. (abs` (A .ih B))) <_ (2 x. ((sqr` (B .ih B)) x. (sqr` (A .ih A)))))
53	15, 52	sylbir 218	. . 3 |- (-. (A .ih B) = 0 -> (2 x. (abs` (A .ih B))) <_ (2 x. ((sqr` (B .ih B)) x. (sqr` (A .ih A)))))
54	9	recni 6467	. . . . . . 7 |- (normh` A) e. CC
55	10	recni 6467	. . . . . . 7 |- (normh` B) e. CC
56	54, 55	mulcomi 6476	. . . . . 6 |- ((normh` A) x. (normh` B)) = ((normh` B) x. (normh` A))
57		normval 10623	. . . . . . . 8 |- (B e. ~H -> (normh` B) = (sqr` (B .ih B)))
58	6, 57	ax-mp 7	. . . . . . 7 |- (normh` B) = (sqr` (B .ih B))
59		normval 10623	. . . . . . . 8 |- (A e. ~H -> (normh` A) = (sqr` (A .ih A)))
60	3, 59	ax-mp 7	. . . . . . 7 |- (normh` A) = (sqr` (A .ih A))
61	58, 60	opreq12i 4894	. . . . . 6 |- ((normh` B) x. (normh` A)) = ((sqr` (B .ih B)) x. (sqr` (A .ih A)))
62	56, 61	eqtri 1908	. . . . 5 |- ((normh` A) x. (normh` B)) = ((sqr` (B .ih B)) x. (sqr` (A .ih A)))
63	62	breq2i 3346	. . . 4 |- ((abs` (A .ih B)) <_ ((normh` A) x. (normh` B)) <-> (abs` (A .ih B)) <_ ((sqr` (B .ih B)) x. (sqr` (A .ih A))))
64		2pos 7173	. . . . 5 |- 0 < 2
65		hiidge0 10597	. . . . . . . . 9 |- (B e. ~H -> 0 <_ (B .ih B))
66	6, 65	ax-mp 7	. . . . . . . 8 |- 0 <_ (B .ih B)
67		hiidrcl 10594	. . . . . . . . . 10 |- (B e. ~H -> (B .ih B) e. RR)
68	6, 67	ax-mp 7	. . . . . . . . 9 |- (B .ih B) e. RR
69	68	sqrcli 7950	. . . . . . . 8 |- (0 <_ (B .ih B) -> (sqr` (B .ih B)) e. RR)
70	66, 69	ax-mp 7	. . . . . . 7 |- (sqr` (B .ih B)) e. RR
71		hiidge0 10597	. . . . . . . . 9 |- (A e. ~H -> 0 <_ (A .ih A))
72	3, 71	ax-mp 7	. . . . . . . 8 |- 0 <_ (A .ih A)
73		hiidrcl 10594	. . . . . . . . . 10 |- (A e. ~H -> (A .ih A) e. RR)
74	3, 73	ax-mp 7	. . . . . . . . 9 |- (A .ih A) e. RR
75	74	sqrcli 7950	. . . . . . . 8 |- (0 <_ (A .ih A) -> (sqr` (A .ih A)) e. RR)
76	72, 75	ax-mp 7	. . . . . . 7 |- (sqr` (A .ih A)) e. RR
77	70, 76	remulcli 6488	. . . . . 6 |- ((sqr` (B .ih B)) x. (sqr` (A .ih A))) e. RR
78		2re 7163	. . . . . 6 |- 2 e. RR
79	24, 77, 78	lemul2i 7018	. . . . 5 |- (0 < 2 -> ((abs` (A .ih B)) <_ ((sqr` (B .ih B)) x. (sqr` (A .ih A))) <-> (2 x. (abs` (A .ih B))) <_ (2 x. ((sqr` (B .ih B)) x. (sqr` (A .ih A))))))
80	64, 79	ax-mp 7	. . . 4 |- ((abs` (A .ih B)) <_ ((sqr` (B .ih B)) x. (sqr` (A .ih A))) <-> (2 x. (abs` (A .ih B))) <_ (2 x. ((sqr` (B .ih B)) x. (sqr` (A .ih A)))))
81	63, 80	bitri 190	. . 3 |- ((abs` (A .ih B)) <_ ((normh` A) x. (normh` B)) <-> (2 x. (abs` (A .ih B))) <_ (2 x. ((sqr` (B .ih B)) x. (sqr` (A .ih A)))))
82	53, 81	sylibr 217	. 2 |- (-. (A .ih B) = 0 -> (abs` (A .ih B)) <_ ((normh` A) x. (normh` B)))
83	14, 82	pm2.61i 140	1 |- (abs` (A .ih B)) <_ ((normh` A) x. (normh` B))

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   / cdiv 6447   <_ cle 6448   < clt 6653  2c2 7145  sqrcsqr 7919  *ccj 7999  abscabs 8000  ~Hchil 10420   .ih csp 10425  normhcno 10426

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-hfvadd 10502  ax-hv0cl 10505  ax-hfvmul 10507  ax-hvmulass 10509  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584  ax-his4 10585

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-hnorm 10469  df-hvsub 10472

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