bcsiALT - Hilbert Space Explorer

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Theorem bcsiALT 28036

Description: Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)

**Hypotheses**
Ref	Expression
bcs.1
bcs.2

**Assertion**
Ref	Expression
bcsiALT

**Proof of Theorem bcsiALT**
Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3
2		abs0 14025	. . . 4
3		bcs.1	. . . . . 6
4		normge0 27983	. . . . . 6
5	3, 4	ax-mp 5	. . . . 5
6		bcs.2	. . . . . 6
7		normge0 27983	. . . . . 6
8	6, 7	ax-mp 5	. . . . 5
9	3	normcli 27988	. . . . . 6
10	6	normcli 27988	. . . . . 6
11	9, 10	mulge0i 10575	. . . . 5 $/\$
12	5, 8, 11	mp2an 708	. . . 4
13	2, 12	eqbrtri 4674	. . 3
14	1, 13	syl6eqbr 4692	. 2
15		df-ne 2795	. . . 4
16	6, 3	his1i 27957	. . . . . . . 8
17	16	oveq2i 6661	. . . . . . 7
18	17	oveq2i 6661	. . . . . 6
19	3, 6	hicli 27938	. . . . . . 7
20		abslem2 14079	. . . . . . 7 $/\$
21	19, 20	mpan 706	. . . . . 6
22	18, 21	syl5req 2669	. . . . 5
23	19	abs00i 14137	. . . . . . . 8
24	23	necon3bii 2846	. . . . . . 7
25	19	abscli 14134	. . . . . . . . . 10
26	25	recni 10052	. . . . . . . . 9
27	19, 26	divclzi 10760	. . . . . . . 8
28	19, 26	divreczi 10763	. . . . . . . . . 10
29	28	fveq2d 6195	. . . . . . . . 9
30	26	recclzi 10750	. . . . . . . . . . 11
31		absmul 14034	. . . . . . . . . . 11 $/\$
32	19, 30, 31	sylancr 695	. . . . . . . . . 10
33	25	rerecclzi 10789	. . . . . . . . . . . 12
34		0re 10040	. . . . . . . . . . . . . 14
35	33, 34	jctil 560	. . . . . . . . . . . . 13 $/\$
36	19	absgt0i 14138	. . . . . . . . . . . . . . 15
37	24, 36	bitri 264	. . . . . . . . . . . . . 14
38	25	recgt0i 10928	. . . . . . . . . . . . . 14
39	37, 38	sylbi 207	. . . . . . . . . . . . 13
40		ltle 10126	. . . . . . . . . . . . 13 $/\$
41	35, 39, 40	sylc 65	. . . . . . . . . . . 12
42	33, 41	absidd 14161	. . . . . . . . . . 11
43	42	oveq2d 6666	. . . . . . . . . 10
44	32, 43	eqtrd 2656	. . . . . . . . 9
45	26	recidzi 10752	. . . . . . . . 9
46	29, 44, 45	3eqtrd 2660	. . . . . . . 8
47	27, 46	jca 554	. . . . . . 7 $/\$
48	24, 47	sylbir 225	. . . . . 6 $/\$
49	3, 6	normlem7tALT 27976	. . . . . 6 $/\$
50	48, 49	syl 17	. . . . 5
51	22, 50	eqbrtrd 4675	. . . 4
52	15, 51	sylbir 225	. . 3
53	10	recni 10052	. . . . . 6
54	9	recni 10052	. . . . . 6
55		normval 27981	. . . . . . . 8
56	6, 55	ax-mp 5	. . . . . . 7
57		normval 27981	. . . . . . . 8
58	3, 57	ax-mp 5	. . . . . . 7
59	56, 58	oveq12i 6662	. . . . . 6
60	53, 54, 59	mulcomli 10047	. . . . 5
61	60	breq2i 4661	. . . 4
62		2pos 11112	. . . . 5
63		hiidge0 27955	. . . . . . . 8
64		hiidrcl 27952	. . . . . . . . . 10
65	6, 64	ax-mp 5	. . . . . . . . 9
66	65	sqrtcli 14111	. . . . . . . 8
67	6, 63, 66	mp2b 10	. . . . . . 7
68		hiidge0 27955	. . . . . . . 8
69		hiidrcl 27952	. . . . . . . . . 10
70	3, 69	ax-mp 5	. . . . . . . . 9
71	70	sqrtcli 14111	. . . . . . . 8
72	3, 68, 71	mp2b 10	. . . . . . 7
73	67, 72	remulcli 10054	. . . . . 6
74		2re 11090	. . . . . 6
75	25, 73, 74	lemul2i 10947	. . . . 5
76	62, 75	ax-mp 5	. . . 4
77	61, 76	bitri 264	. . 3
78	52, 77	sylibr 224	. 2
79	14, 78	pm2.61i 176	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cdiv 10684

c2 11070

ccj 13836

csqrt 13973

cabs 13974

chil 27776

csp 27779

cno 27780

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-hfvadd 27857 ax-hv0cl 27860 ax-hfvmul 27862 ax-hvmulass 27864 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-hnorm 27825 df-hvsub 27828

This theorem is referenced by: (None)

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