Theorem ncvspi 22956

Description: The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (Revised by AV, 8-Oct-2021.)

Hypotheses

Ref	Expression
ncvsprp.v	|- V = ( Base ` W )
ncvsprp.n	|- N = ( norm ` W )
ncvsprp.s	|- .x. = ( .s ` W )
ncvsdif.p	|- .+ = ( +g ` W )
ncvspi.f	|- F = (Scalar ` W )
ncvspi.k	|- K = ( Base ` F )

Assertion

Ref	Expression
ncvspi	|- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( N ` ( A .+ ( _i .x. B ) ) ) = ( N ` ( B .+ ( -u _i .x. A ) ) ) )

Proof of Theorem ncvspi

Step	Hyp	Ref	Expression
1		elin 3796	. . . . . . 7 |- ( W e. (NrmVec i^i CVec ) <-> ( W e. NrmVec /\ W e. CVec ) )
2		nvcnlm 22500	. . . . . . . . 9 |- ( W e. NrmVec -> W e. NrmMod )
3		nlmngp 22481	. . . . . . . . 9 |- ( W e. NrmMod -> W e. NrmGrp )
4	2, 3	syl 17	. . . . . . . 8 |- ( W e. NrmVec -> W e. NrmGrp )
5	4	adantr 481	. . . . . . 7 |- ( ( W e. NrmVec /\ W e. CVec ) -> W e. NrmGrp )
6	1, 5	sylbi 207	. . . . . 6 |- ( W e. (NrmVec i^i CVec ) -> W e. NrmGrp )
7	6	3ad2ant1 1082	. . . . 5 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> W e. NrmGrp )
8		nvclmod 22502	. . . . . . . . . 10 |- ( W e. NrmVec -> W e. LMod )
9		lmodgrp 18870	. . . . . . . . . 10 |- ( W e. LMod -> W e. Grp )
10	8, 9	syl 17	. . . . . . . . 9 |- ( W e. NrmVec -> W e. Grp )
11	10	adantr 481	. . . . . . . 8 |- ( ( W e. NrmVec /\ W e. CVec ) -> W e. Grp )
12	1, 11	sylbi 207	. . . . . . 7 |- ( W e. (NrmVec i^i CVec ) -> W e. Grp )
13	12	3ad2ant1 1082	. . . . . 6 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> W e. Grp )
14		simp2l 1087	. . . . . 6 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> A e. V )
15		id 22	. . . . . . . . . 10 |- ( W e. CVec -> W e. CVec )
16	15	cvsclm 22926	. . . . . . . . 9 |- ( W e. CVec -> W e. CMod )
17	1, 16	simplbiim 659	. . . . . . . 8 |- ( W e. (NrmVec i^i CVec ) -> W e. CMod )
18	17	3ad2ant1 1082	. . . . . . 7 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> W e. CMod )
19		simp3 1063	. . . . . . 7 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> _i e. K )
20		simp2r 1088	. . . . . . 7 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> B e. V )
21		ncvsprp.v	. . . . . . . 8 |- V = ( Base ` W )
22		ncvspi.f	. . . . . . . 8 |- F = (Scalar ` W )
23		ncvsprp.s	. . . . . . . 8 |- .x. = ( .s ` W )
24		ncvspi.k	. . . . . . . 8 |- K = ( Base ` F )
25	21, 22, 23, 24	clmvscl 22888	. . . . . . 7 |- ( ( W e. CMod /\ _i e. K /\ B e. V ) -> ( _i .x. B ) e. V )
26	18, 19, 20, 25	syl3anc 1326	. . . . . 6 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( _i .x. B ) e. V )
27		ncvsdif.p	. . . . . . 7 |- .+ = ( +g ` W )
28	21, 27	grpcl 17430	. . . . . 6 |- ( ( W e. Grp /\ A e. V /\ ( _i .x. B ) e. V ) -> ( A .+ ( _i .x. B ) ) e. V )
29	13, 14, 26, 28	syl3anc 1326	. . . . 5 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( A .+ ( _i .x. B ) ) e. V )
30		ncvsprp.n	. . . . . 6 |- N = ( norm ` W )
31	21, 30	nmcl 22420	. . . . 5 |- ( ( W e. NrmGrp /\ ( A .+ ( _i .x. B ) ) e. V ) -> ( N ` ( A .+ ( _i .x. B ) ) ) e. RR )
32	7, 29, 31	syl2anc 693	. . . 4 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( N ` ( A .+ ( _i .x. B ) ) ) e. RR )
33	32	recnd 10068	. . 3 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( N ` ( A .+ ( _i .x. B ) ) ) e. CC )
34	33	mulid2d 10058	. 2 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( 1 x. ( N ` ( A .+ ( _i .x. B ) ) ) ) = ( N ` ( A .+ ( _i .x. B ) ) ) )
35		ax-icn 9995	. . . . . 6 |- _i e. CC
36	35	absnegi 14139	. . . . 5 |- ( abs ` -u _i ) = ( abs ` _i )
37		absi 14026	. . . . 5 |- ( abs ` _i ) = 1
38	36, 37	eqtri 2644	. . . 4 |- ( abs ` -u _i ) = 1
39	38	oveq1i 6660	. . 3 |- ( ( abs ` -u _i ) x. ( N ` ( A .+ ( _i .x. B ) ) ) ) = ( 1 x. ( N ` ( A .+ ( _i .x. B ) ) ) )
40		simp1 1061	. . . . 5 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> W e. (NrmVec i^i CVec ) )
41	22, 24	clmneg 22881	. . . . . . . . . . 11 |- ( ( W e. CMod /\ _i e. K ) -> -u _i = ( ( invg ` F ) ` _i ) )
42	16, 41	sylan 488	. . . . . . . . . 10 |- ( ( W e. CVec /\ _i e. K ) -> -u _i = ( ( invg ` F ) ` _i ) )
43	22	clmfgrp 22871	. . . . . . . . . . . 12 |- ( W e. CMod -> F e. Grp )
44	16, 43	syl 17	. . . . . . . . . . 11 |- ( W e. CVec -> F e. Grp )
45		eqid 2622	. . . . . . . . . . . 12 |- ( invg ` F ) = ( invg ` F )
46	24, 45	grpinvcl 17467	. . . . . . . . . . 11 |- ( ( F e. Grp /\ _i e. K ) -> ( ( invg ` F ) ` _i ) e. K )
47	44, 46	sylan 488	. . . . . . . . . 10 |- ( ( W e. CVec /\ _i e. K ) -> ( ( invg ` F ) ` _i ) e. K )
48	42, 47	eqeltrd 2701	. . . . . . . . 9 |- ( ( W e. CVec /\ _i e. K ) -> -u _i e. K )
49	48	ex 450	. . . . . . . 8 |- ( W e. CVec -> ( _i e. K -> -u _i e. K ) )
50	1, 49	simplbiim 659	. . . . . . 7 |- ( W e. (NrmVec i^i CVec ) -> ( _i e. K -> -u _i e. K ) )
51	50	imp 445	. . . . . 6 |- ( ( W e. (NrmVec i^i CVec ) /\ _i e. K ) -> -u _i e. K )
52	51	3adant2 1080	. . . . 5 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> -u _i e. K )
53	21, 30, 23, 22, 24	ncvsprp 22952	. . . . 5 |- ( ( W e. (NrmVec i^i CVec ) /\ -u _i e. K /\ ( A .+ ( _i .x. B ) ) e. V ) -> ( N ` ( -u _i .x. ( A .+ ( _i .x. B ) ) ) ) = ( ( abs ` -u _i ) x. ( N ` ( A .+ ( _i .x. B ) ) ) ) )
54	40, 52, 29, 53	syl3anc 1326	. . . 4 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( N ` ( -u _i .x. ( A .+ ( _i .x. B ) ) ) ) = ( ( abs ` -u _i ) x. ( N ` ( A .+ ( _i .x. B ) ) ) ) )
55	21, 22, 23, 24, 27	clmvsdi 22892	. . . . . . 7 |- ( ( W e. CMod /\ ( -u _i e. K /\ A e. V /\ ( _i .x. B ) e. V ) ) -> ( -u _i .x. ( A .+ ( _i .x. B ) ) ) = ( ( -u _i .x. A ) .+ ( -u _i .x. ( _i .x. B ) ) ) )
56	18, 52, 14, 26, 55	syl13anc 1328	. . . . . 6 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( -u _i .x. ( A .+ ( _i .x. B ) ) ) = ( ( -u _i .x. A ) .+ ( -u _i .x. ( _i .x. B ) ) ) )
57	35, 35	mulneg1i 10476	. . . . . . . . . 10 |- ( -u _i x. _i ) = -u ( _i x. _i )
58		ixi 10656	. . . . . . . . . . . 12 |- ( _i x. _i ) = -u 1
59	58	negeqi 10274	. . . . . . . . . . 11 |- -u ( _i x. _i ) = -u -u 1
60		negneg1e1 11128	. . . . . . . . . . 11 |- -u -u 1 = 1
61	59, 60	eqtri 2644	. . . . . . . . . 10 |- -u ( _i x. _i ) = 1
62	57, 61	eqtri 2644	. . . . . . . . 9 |- ( -u _i x. _i ) = 1
63	62	oveq1i 6660	. . . . . . . 8 |- ( ( -u _i x. _i ) .x. B ) = ( 1 .x. B )
64	21, 22, 23, 24	clmvsass 22889	. . . . . . . . 9 |- ( ( W e. CMod /\ ( -u _i e. K /\ _i e. K /\ B e. V ) ) -> ( ( -u _i x. _i ) .x. B ) = ( -u _i .x. ( _i .x. B ) ) )
65	18, 52, 19, 20, 64	syl13anc 1328	. . . . . . . 8 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( ( -u _i x. _i ) .x. B ) = ( -u _i .x. ( _i .x. B ) ) )
66		simpr 477	. . . . . . . . . . 11 |- ( ( A e. V /\ B e. V ) -> B e. V )
67	17, 66	anim12i 590	. . . . . . . . . 10 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) ) -> ( W e. CMod /\ B e. V ) )
68	67	3adant3 1081	. . . . . . . . 9 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( W e. CMod /\ B e. V ) )
69	21, 23	clmvs1 22893	. . . . . . . . 9 |- ( ( W e. CMod /\ B e. V ) -> ( 1 .x. B ) = B )
70	68, 69	syl 17	. . . . . . . 8 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( 1 .x. B ) = B )
71	63, 65, 70	3eqtr3a 2680	. . . . . . 7 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( -u _i .x. ( _i .x. B ) ) = B )
72	71	oveq2d 6666	. . . . . 6 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( ( -u _i .x. A ) .+ ( -u _i .x. ( _i .x. B ) ) ) = ( ( -u _i .x. A ) .+ B ) )
73		clmabl 22869	. . . . . . . . . 10 |- ( W e. CMod -> W e. Abel )
74	16, 73	syl 17	. . . . . . . . 9 |- ( W e. CVec -> W e. Abel )
75	1, 74	simplbiim 659	. . . . . . . 8 |- ( W e. (NrmVec i^i CVec ) -> W e. Abel )
76	75	3ad2ant1 1082	. . . . . . 7 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> W e. Abel )
77	21, 22, 23, 24	clmvscl 22888	. . . . . . . 8 |- ( ( W e. CMod /\ -u _i e. K /\ A e. V ) -> ( -u _i .x. A ) e. V )
78	18, 52, 14, 77	syl3anc 1326	. . . . . . 7 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( -u _i .x. A ) e. V )
79	21, 27	ablcom 18210	. . . . . . 7 |- ( ( W e. Abel /\ ( -u _i .x. A ) e. V /\ B e. V ) -> ( ( -u _i .x. A ) .+ B ) = ( B .+ ( -u _i .x. A ) ) )
80	76, 78, 20, 79	syl3anc 1326	. . . . . 6 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( ( -u _i .x. A ) .+ B ) = ( B .+ ( -u _i .x. A ) ) )
81	56, 72, 80	3eqtrd 2660	. . . . 5 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( -u _i .x. ( A .+ ( _i .x. B ) ) ) = ( B .+ ( -u _i .x. A ) ) )
82	81	fveq2d 6195	. . . 4 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( N ` ( -u _i .x. ( A .+ ( _i .x. B ) ) ) ) = ( N ` ( B .+ ( -u _i .x. A ) ) ) )
83	54, 82	eqtr3d 2658	. . 3 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( ( abs ` -u _i ) x. ( N ` ( A .+ ( _i .x. B ) ) ) ) = ( N ` ( B .+ ( -u _i .x. A ) ) ) )
84	39, 83	syl5eqr 2670	. 2 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( 1 x. ( N ` ( A .+ ( _i .x. B ) ) ) ) = ( N ` ( B .+ ( -u _i .x. A ) ) ) )
85	34, 84	eqtr3d 2658	1 |- ( ( W e. (NrmVec i^i CVec ) /\ ( A e. V /\ B e. V ) /\ _i e. K ) -> ( N ` ( A .+ ( _i .x. B ) ) ) = ( N ` ( B .+ ( -u _i .x. A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   _ici 9938   x. cmul 9941   -ucneg 10267   abscabs 13974   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   Grpcgrp 17422   invgcminusg 17423   Abelcabl 18194   LModclmod 18863   normcnm 22381  NrmGrpcngp 22382  NrmModcnlm 22385  NrmVeccnvc 22386  CModcclm 22862  CVecccvs 22923

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-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-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-grp 17425  df-minusg 17426  df-subg 17591  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  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-nvc 22392  df-clm 22863  df-cvs 22924

This theorem is referenced by: (None)