cphipval2 - Metamath Proof Explorer

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Theorem cphipval2 23040

Description: Value of the inner product expressed by the norm defined by it. (Contributed by NM, 31-Jan-2007.) (Revised by AV, 18-Oct-2021.)

**Hypotheses**
Ref	Expression
cphipfval.x
cphipfval.p
cphipfval.s
cphipfval.n
cphipfval.i
cphipval2.m
cphipval2.f	Scalar
cphipval2.k

**Assertion**
Ref	Expression
cphipval2	$/\$ $/\$ $/\$

**Proof of Theorem cphipval2**
Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . . . 9 $/\$
2	1	3ad2ant1 1082	. . . . . . . 8 $/\$ $/\$ $/\$
3		cphngp 22973	. . . . . . . . . . 11 NrmGrp
4	3	adantr 481	. . . . . . . . . 10 $/\$ NrmGrp
5		ngpgrp 22403	. . . . . . . . . 10 NrmGrp
6	4, 5	syl 17	. . . . . . . . 9 $/\$
7		cphipfval.x	. . . . . . . . . 10
8		cphipfval.p	. . . . . . . . . 10
9	7, 8	grpcl 17430	. . . . . . . . 9 $/\$ $/\$
10	6, 9	syl3an1 1359	. . . . . . . 8 $/\$ $/\$ $/\$
11		cphipfval.i	. . . . . . . . 9
12		cphipfval.n	. . . . . . . . 9
13	7, 11, 12	nmsq 22994	. . . . . . . 8 $/\$
14	2, 10, 13	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$
15		simp2 1062	. . . . . . . 8 $/\$ $/\$ $/\$
16		simp3 1063	. . . . . . . 8 $/\$ $/\$ $/\$
17	11, 7, 8, 2, 15, 16, 15, 16	cph2di 23007	. . . . . . 7 $/\$ $/\$ $/\$
18	14, 17	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$
19		cphipval2.m	. . . . . . . . . 10
20	7, 19	grpsubcl 17495	. . . . . . . . 9 $/\$ $/\$
21	6, 20	syl3an1 1359	. . . . . . . 8 $/\$ $/\$ $/\$
22	7, 11, 12	nmsq 22994	. . . . . . . 8 $/\$
23	2, 21, 22	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$
24	11, 7, 19, 2, 15, 16, 15, 16	cph2subdi 23010	. . . . . . 7 $/\$ $/\$ $/\$
25	23, 24	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$
26	18, 25	oveq12d 6668	. . . . 5 $/\$ $/\$ $/\$
27	7, 11	reipcl 22997	. . . . . . . . . 10 $/\$
28	27	adantlr 751	. . . . . . . . 9 $/\$ $/\$
29	28	recnd 10068	. . . . . . . 8 $/\$ $/\$
30	29	3adant3 1081	. . . . . . 7 $/\$ $/\$ $/\$
31	7, 11	reipcl 22997	. . . . . . . . . 10 $/\$
32	31	adantlr 751	. . . . . . . . 9 $/\$ $/\$
33	32	recnd 10068	. . . . . . . 8 $/\$ $/\$
34	33	3adant2 1080	. . . . . . 7 $/\$ $/\$ $/\$
35	30, 34	addcld 10059	. . . . . 6 $/\$ $/\$ $/\$
36	7, 11	cphipcl 22991	. . . . . . . 8 $/\$ $/\$
37	1, 36	syl3an1 1359	. . . . . . 7 $/\$ $/\$ $/\$
38	7, 11	cphipcl 22991	. . . . . . . . 9 $/\$ $/\$
39	1, 38	syl3an1 1359	. . . . . . . 8 $/\$ $/\$ $/\$
40	39	3com23 1271	. . . . . . 7 $/\$ $/\$ $/\$
41	37, 40	addcld 10059	. . . . . 6 $/\$ $/\$ $/\$
42	35, 41, 41	pnncand 10431	. . . . 5 $/\$ $/\$ $/\$
43	26, 42	eqtrd 2656	. . . 4 $/\$ $/\$ $/\$
44	6	3ad2ant1 1082	. . . . . . . . . 10 $/\$ $/\$ $/\$
45		cphlmod 22974	. . . . . . . . . . . . . 14
46	45	adantr 481	. . . . . . . . . . . . 13 $/\$
47	46	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$
48		simplr 792	. . . . . . . . . . . 12 $/\$ $/\$
49		simpr 477	. . . . . . . . . . . 12 $/\$ $/\$
50		cphipval2.f	. . . . . . . . . . . . 13 Scalar
51		cphipfval.s	. . . . . . . . . . . . 13
52		cphipval2.k	. . . . . . . . . . . . 13
53	7, 50, 51, 52	lmodvscl 18880	. . . . . . . . . . . 12 $/\$ $/\$
54	47, 48, 49, 53	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$
55	54	3adant2 1080	. . . . . . . . . 10 $/\$ $/\$ $/\$
56	7, 8	grpcl 17430	. . . . . . . . . 10 $/\$ $/\$
57	44, 15, 55, 56	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$
58	7, 11, 12	nmsq 22994	. . . . . . . . 9 $/\$
59	2, 57, 58	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$
60	11, 7, 8, 2, 15, 55, 15, 55	cph2di 23007	. . . . . . . 8 $/\$ $/\$ $/\$
61	59, 60	eqtrd 2656	. . . . . . 7 $/\$ $/\$ $/\$
62	7, 19	grpsubcl 17495	. . . . . . . . . 10 $/\$ $/\$
63	44, 15, 55, 62	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$
64	7, 11, 12	nmsq 22994	. . . . . . . . 9 $/\$
65	2, 63, 64	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$
66	11, 7, 19, 2, 15, 55, 15, 55	cph2subdi 23010	. . . . . . . 8 $/\$ $/\$ $/\$
67	65, 66	eqtrd 2656	. . . . . . 7 $/\$ $/\$ $/\$
68	61, 67	oveq12d 6668	. . . . . 6 $/\$ $/\$ $/\$
69	68	oveq2d 6666	. . . . 5 $/\$ $/\$ $/\$
70	7, 11	cphipcl 22991	. . . . . . . . 9 $/\$ $/\$
71	2, 55, 55, 70	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$
72	30, 71	addcld 10059	. . . . . . 7 $/\$ $/\$ $/\$
73	7, 11	cphipcl 22991	. . . . . . . . 9 $/\$ $/\$
74	2, 15, 55, 73	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$
75	7, 11	cphipcl 22991	. . . . . . . . 9 $/\$ $/\$
76	2, 55, 15, 75	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$
77	74, 76	addcld 10059	. . . . . . 7 $/\$ $/\$ $/\$
78	72, 77, 77	pnncand 10431	. . . . . 6 $/\$ $/\$ $/\$
79	78	oveq2d 6666	. . . . 5 $/\$ $/\$ $/\$
80	7, 51, 11, 50, 52	cphassir 23015	. . . . . . . . 9 $/\$ $/\$ $/\$
81	7, 51, 11, 50, 52	cphassi 23014	. . . . . . . . 9 $/\$ $/\$ $/\$
82	80, 81	oveq12d 6668	. . . . . . . 8 $/\$ $/\$ $/\$
83	82, 82	oveq12d 6668	. . . . . . 7 $/\$ $/\$ $/\$
84	83	oveq2d 6666	. . . . . 6 $/\$ $/\$ $/\$
85		ax-icn 9995	. . . . . . . 8
86	85	a1i 11	. . . . . . 7 $/\$ $/\$ $/\$
87		negicn 10282	. . . . . . . . . 10
88	87	a1i 11	. . . . . . . . 9 $/\$ $/\$ $/\$
89	88, 37	mulcld 10060	. . . . . . . 8 $/\$ $/\$ $/\$
90	86, 40	mulcld 10060	. . . . . . . 8 $/\$ $/\$ $/\$
91	89, 90	addcld 10059	. . . . . . 7 $/\$ $/\$ $/\$
92	86, 91, 91	adddid 10064	. . . . . 6 $/\$ $/\$ $/\$
93	86, 89, 90	adddid 10064	. . . . . . . . 9 $/\$ $/\$ $/\$
94	85, 85	mulneg2i 10477	. . . . . . . . . . . . 13
95		ixi 10656	. . . . . . . . . . . . . 14
96	95	negeqi 10274	. . . . . . . . . . . . 13
97		negneg1e1 11128	. . . . . . . . . . . . 13
98	94, 96, 97	3eqtri 2648	. . . . . . . . . . . 12
99	98	oveq1i 6660	. . . . . . . . . . 11
100	86, 88, 37	mulassd 10063	. . . . . . . . . . 11 $/\$ $/\$ $/\$
101	99, 100	syl5reqr 2671	. . . . . . . . . 10 $/\$ $/\$ $/\$
102	95	oveq1i 6660	. . . . . . . . . . 11
103	86, 86, 40	mulassd 10063	. . . . . . . . . . 11 $/\$ $/\$ $/\$
104	102, 103	syl5reqr 2671	. . . . . . . . . 10 $/\$ $/\$ $/\$
105	101, 104	oveq12d 6668	. . . . . . . . 9 $/\$ $/\$ $/\$
106	93, 105	eqtrd 2656	. . . . . . . 8 $/\$ $/\$ $/\$
107	106, 106	oveq12d 6668	. . . . . . 7 $/\$ $/\$ $/\$
108	37	mulid2d 10058	. . . . . . . . . 10 $/\$ $/\$ $/\$
109	108	oveq1d 6665	. . . . . . . . 9 $/\$ $/\$ $/\$
110		addneg1mul 10472	. . . . . . . . . 10 $/\$
111	37, 40, 110	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
112	109, 111	eqtrd 2656	. . . . . . . 8 $/\$ $/\$ $/\$
113	112, 112	oveq12d 6668	. . . . . . 7 $/\$ $/\$ $/\$
114	107, 113	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$
115	84, 92, 114	3eqtrd 2660	. . . . 5 $/\$ $/\$ $/\$
116	69, 79, 115	3eqtrd 2660	. . . 4 $/\$ $/\$ $/\$
117	43, 116	oveq12d 6668	. . 3 $/\$ $/\$ $/\$
118	117	oveq1d 6665	. 2 $/\$ $/\$ $/\$
119	37, 40	subcld 10392	. . . . 5 $/\$ $/\$ $/\$
120	41, 41, 119, 119	add4d 10264	. . . 4 $/\$ $/\$ $/\$
121	37, 40, 37	ppncand 10432	. . . . 5 $/\$ $/\$ $/\$
122	121, 121	oveq12d 6668	. . . 4 $/\$ $/\$ $/\$
123	120, 122	eqtrd 2656	. . 3 $/\$ $/\$ $/\$
124	123	oveq1d 6665	. 2 $/\$ $/\$ $/\$
125	37	2timesd 11275	. . . . . . 7 $/\$ $/\$ $/\$
126	125	eqcomd 2628	. . . . . 6 $/\$ $/\$ $/\$
127	126, 126	oveq12d 6668	. . . . 5 $/\$ $/\$ $/\$
128		2cnd 11093	. . . . . 6 $/\$ $/\$ $/\$
129	128, 128, 37	adddird 10065	. . . . 5 $/\$ $/\$ $/\$
130		2p2e4 11144	. . . . . . 7
131	130	a1i 11	. . . . . 6 $/\$ $/\$ $/\$
132	131	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$
133	127, 129, 132	3eqtr2d 2662	. . . 4 $/\$ $/\$ $/\$
134	133	oveq1d 6665	. . 3 $/\$ $/\$ $/\$
135		4cn 11098	. . . . 5
136	135	a1i 11	. . . 4 $/\$ $/\$ $/\$
137		4ne0 11117	. . . . 5
138	137	a1i 11	. . . 4 $/\$ $/\$ $/\$
139	37, 136, 138	divcan3d 10806	. . 3 $/\$ $/\$ $/\$
140	134, 139	eqtrd 2656	. 2 $/\$ $/\$ $/\$
141	118, 124, 140	3eqtrrd 2661	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

ci 9938

caddc 9939

cmul 9941

cmin 10266

cneg 10267

cdiv 10684

c2 11070

c4 11072

cexp 12860

cbs 15857

cplusg 15941 Scalarcsca 15944

cvsca 15945

cip 15946

cgrp 17422

csg 17424

clmod 18863

cnm 22381 NrmGrpcngp 22382

ccph 22966

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-rep 4771 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-addf 10015 ax-mulf 10016

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-int 4476 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-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 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-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-topgen 16104 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-ghm 17658 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-staf 18845 df-srng 18846 df-lmod 18865 df-lmhm 19022 df-lvec 19103 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-phl 19971 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-xms 22125 df-ms 22126 df-nm 22387 df-ngp 22388 df-nlm 22391 df-clm 22863 df-cph 22968

This theorem is referenced by: 4cphipval2 23041 cphipval 23042

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