polid2i - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > polid2i		Structured version Visualization version Unicode version

Theorem polid2i 28014

Description: Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
polid2.1
polid2.2
polid2.3
polid2.4

**Assertion**
Ref	Expression
polid2i

**Proof of Theorem polid2i**
Step	Hyp	Ref	Expression
1		4cn 11098	. 2
2		polid2.1	. . 3
3		polid2.2	. . 3
4	2, 3	hicli 27938	. 2
5		4ne0 11117	. 2
6		2cn 11091	. . . 4
7		polid2.3	. . . . . 6
8		polid2.4	. . . . . 6
9	7, 8	hicli 27938	. . . . 5
10	4, 9	addcli 10044	. . . 4
11	4, 9	subcli 10357	. . . 4
12	6, 10, 11	adddii 10050	. . 3
13		ppncan 10323	. . . . . . 7 $/\$ $/\$
14	4, 9, 4, 13	mp3an 1424	. . . . . 6
15	4	2timesi 11147	. . . . . 6
16	14, 15	eqtr4i 2647	. . . . 5
17	16	oveq2i 6661	. . . 4
18	6, 6, 4	mulassi 10049	. . . 4
19		2t2e4 11177	. . . . 5
20	19	oveq1i 6660	. . . 4
21	17, 18, 20	3eqtr2ri 2651	. . 3
22	2, 8	hicli 27938	. . . . . . 7
23	7, 3	hicli 27938	. . . . . . 7
24	22, 23	addcli 10044	. . . . . 6
25	24, 10, 10	pnncani 10376	. . . . 5
26	2, 7, 8, 3	normlem8 27974	. . . . . 6
27	2, 7, 8, 3	normlem9 27975	. . . . . 6
28	26, 27	oveq12i 6662	. . . . 5
29	10	2timesi 11147	. . . . 5
30	25, 28, 29	3eqtr4i 2654	. . . 4
31		ax-icn 9995	. . . . . . . . . 10
32	31, 7	hvmulcli 27871	. . . . . . . . 9
33	31, 3	hvmulcli 27871	. . . . . . . . 9
34	2, 32, 8, 33	normlem8 27974	. . . . . . . 8
35	2, 32, 8, 33	normlem9 27975	. . . . . . . 8
36	34, 35	oveq12i 6662	. . . . . . 7
37	32, 33	hicli 27938	. . . . . . . . 9
38	22, 37	addcli 10044	. . . . . . . 8
39	2, 33	hicli 27938	. . . . . . . . 9
40	32, 8	hicli 27938	. . . . . . . . 9
41	39, 40	addcli 10044	. . . . . . . 8
42	38, 41, 41	pnncani 10376	. . . . . . 7
43	41	2timesi 11147	. . . . . . . 8
44		his5 27943	. . . . . . . . . . . 12 $/\$ $/\$
45	31, 2, 3, 44	mp3an 1424	. . . . . . . . . . 11
46		cji 13899	. . . . . . . . . . . 12
47	46	oveq1i 6660	. . . . . . . . . . 11
48	45, 47	eqtri 2644	. . . . . . . . . 10
49		ax-his3 27941	. . . . . . . . . . 11 $/\$ $/\$
50	31, 7, 8, 49	mp3an 1424	. . . . . . . . . 10
51	48, 50	oveq12i 6662	. . . . . . . . 9
52	51	oveq2i 6661	. . . . . . . 8
53	43, 52	eqtr3i 2646	. . . . . . 7
54	36, 42, 53	3eqtri 2648	. . . . . 6
55	54	oveq2i 6661	. . . . 5
56		negicn 10282	. . . . . . . 8
57	56, 4	mulcli 10045	. . . . . . 7
58	31, 9	mulcli 10045	. . . . . . 7
59	57, 58	addcli 10044	. . . . . 6
60	6, 31, 59	mul12i 10231	. . . . 5
61	31, 57, 58	adddii 10050	. . . . . . 7
62	31, 31	mulneg2i 10477	. . . . . . . . . . 11
63		ixi 10656	. . . . . . . . . . . 12
64	63	negeqi 10274	. . . . . . . . . . 11
65		negneg1e1 11128	. . . . . . . . . . 11
66	62, 64, 65	3eqtri 2648	. . . . . . . . . 10
67	66	oveq1i 6660	. . . . . . . . 9
68	31, 56, 4	mulassi 10049	. . . . . . . . 9
69	4	mulid2i 10043	. . . . . . . . 9
70	67, 68, 69	3eqtr3i 2652	. . . . . . . 8
71	63	oveq1i 6660	. . . . . . . . 9
72	31, 31, 9	mulassi 10049	. . . . . . . . 9
73	9	mulm1i 10475	. . . . . . . . 9
74	71, 72, 73	3eqtr3i 2652	. . . . . . . 8
75	70, 74	oveq12i 6662	. . . . . . 7
76	4, 9	negsubi 10359	. . . . . . 7
77	61, 75, 76	3eqtri 2648	. . . . . 6
78	77	oveq2i 6661	. . . . 5
79	55, 60, 78	3eqtr2i 2650	. . . 4
80	30, 79	oveq12i 6662	. . 3
81	12, 21, 80	3eqtr4i 2654	. 2
82	1, 4, 5, 81	mvllmuli 10858	1

Colors of variables: wff setvar class

Syntax hints:

wceq 1483

wcel 1990

cfv 5888 (class class class)co 6650

cc 9934

c1 9937

ci 9938

caddc 9939

cmul 9941

cmin 10266

cneg 10267

cdiv 10684

c2 11070

c4 11072

ccj 13836

chil 27776

cva 27777

csm 27778

csp 27779

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-hfvmul 27862 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941

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-3 11080 df-4 11081 df-cj 13839 df-re 13840 df-im 13841 df-hvsub 27828

This theorem is referenced by: polidi 28015 lnopeq0lem1 28864 lnophmlem2 28876

W3C validator