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Theorem bcseqi 27977

Description: Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 28037. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
normlem7t.1
normlem7t.2

**Assertion**
Ref	Expression
bcseqi

**Proof of Theorem bcseqi**
Step	Hyp	Ref	Expression
1		normlem7t.2	. . . . . . . 8
2	1, 1	hicli 27938	. . . . . . 7
3		normlem7t.1	. . . . . . 7
4	2, 3	hvmulcli 27871	. . . . . 6
5	3, 1	hicli 27938	. . . . . . 7
6	5, 1	hvmulcli 27871	. . . . . 6
7	4, 6, 4, 6	normlem9 27975	. . . . 5
8		oveq1 6657	. . . . . . . . . 10
9	8	eqcomd 2628	. . . . . . . . 9
10		his5 27943	. . . . . . . . . . 11 $/\$ $/\$
11	2, 4, 3, 10	mp3an 1424	. . . . . . . . . 10
12		hiidrcl 27952	. . . . . . . . . . . 12
13		cjre 13879	. . . . . . . . . . . 12
14	1, 12, 13	mp2b 10	. . . . . . . . . . 11
15		ax-his3 27941	. . . . . . . . . . . 12 $/\$ $/\$
16	2, 3, 3, 15	mp3an 1424	. . . . . . . . . . 11
17	14, 16	oveq12i 6662	. . . . . . . . . 10
18	3, 3	hicli 27938	. . . . . . . . . . . . 13
19	2, 18	mulcli 10045	. . . . . . . . . . . 12
20	2, 19	mulcomi 10046	. . . . . . . . . . 11
21	18, 2	mulcomi 10046	. . . . . . . . . . . 12
22	21	oveq1i 6660	. . . . . . . . . . 11
23	20, 22	eqtr4i 2647	. . . . . . . . . 10
24	11, 17, 23	3eqtri 2648	. . . . . . . . 9
25		his5 27943	. . . . . . . . . . 11 $/\$ $/\$
26	5, 4, 1, 25	mp3an 1424	. . . . . . . . . 10
27	1, 3	his1i 27957	. . . . . . . . . . . 12
28	27	eqcomi 2631	. . . . . . . . . . 11
29		ax-his3 27941	. . . . . . . . . . . 12 $/\$ $/\$
30	2, 3, 1, 29	mp3an 1424	. . . . . . . . . . 11
31	28, 30	oveq12i 6662	. . . . . . . . . 10
32	1, 3	hicli 27938	. . . . . . . . . . . 12
33	2, 5	mulcli 10045	. . . . . . . . . . . 12
34	32, 33	mulcomi 10046	. . . . . . . . . . 11
35	2, 5, 32	mulassi 10049	. . . . . . . . . . 11
36	5, 32	mulcli 10045	. . . . . . . . . . . 12
37	2, 36	mulcomi 10046	. . . . . . . . . . 11
38	34, 35, 37	3eqtri 2648	. . . . . . . . . 10
39	26, 31, 38	3eqtri 2648	. . . . . . . . 9
40	9, 24, 39	3eqtr4g 2681	. . . . . . . 8
41		ax-his3 27941	. . . . . . . . . . . 12 $/\$ $/\$
42	5, 1, 3, 41	mp3an 1424	. . . . . . . . . . 11
43	14, 42	oveq12i 6662	. . . . . . . . . 10
44		his5 27943	. . . . . . . . . . 11 $/\$ $/\$
45	2, 6, 3, 44	mp3an 1424	. . . . . . . . . 10
46		his5 27943	. . . . . . . . . . . 12 $/\$ $/\$
47	5, 6, 1, 46	mp3an 1424	. . . . . . . . . . 11
48		ax-his3 27941	. . . . . . . . . . . . 13 $/\$ $/\$
49	5, 1, 1, 48	mp3an 1424	. . . . . . . . . . . 12
50	28, 49	oveq12i 6662	. . . . . . . . . . 11
51	5, 2	mulcli 10045	. . . . . . . . . . . . 13
52	32, 51	mulcomi 10046	. . . . . . . . . . . 12
53	5, 2, 32	mul32i 10232	. . . . . . . . . . . 12
54	36, 2	mulcomi 10046	. . . . . . . . . . . 12
55	52, 53, 54	3eqtri 2648	. . . . . . . . . . 11
56	47, 50, 55	3eqtri 2648	. . . . . . . . . 10
57	43, 45, 56	3eqtr4ri 2655	. . . . . . . . 9
58	57	a1i 11	. . . . . . . 8
59	40, 58	oveq12d 6668	. . . . . . 7
60	59	oveq1d 6665	. . . . . 6
61	4, 6	hicli 27938	. . . . . . . 8
62	6, 4	hicli 27938	. . . . . . . 8
63	61, 62	addcli 10044	. . . . . . 7
64	63	subidi 10352	. . . . . 6
65	60, 64	syl6eq 2672	. . . . 5
66	7, 65	syl5eq 2668	. . . 4
67	4, 6	hvsubcli 27878	. . . . 5
68		his6 27956	. . . . 5
69	67, 68	ax-mp 5	. . . 4
70	66, 69	sylib 208	. . 3
71	4, 6	hvsubeq0i 27920	. . 3
72	70, 71	sylib 208	. 2
73		oveq1 6657	. . . 4
74	21, 16	eqtr4i 2647	. . . 4
75	42	eqcomi 2631	. . . 4
76	73, 74, 75	3eqtr4g 2681	. . 3
77	76	eqcomd 2628	. 2
78	72, 77	impbii 199	1

Colors of variables: wff setvar class

Syntax hints:

wb 196

wceq 1483

wcel 1990

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

caddc 9939

cmul 9941

cmin 10266

ccj 13836

chil 27776

csm 27778

csp 27779

c0v 27781

cmv 27782

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-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-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvdistr2 27866 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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 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-2 11079 df-cj 13839 df-re 13840 df-im 13841 df-hvsub 27828

This theorem is referenced by: h1de2i 28412

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