Theorem ncvs1 22957

Description: From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (Revised by AV, 8-Oct-2021.)

Hypotheses

Ref	Expression
ncvs1.x	|- X = ( Base ` G )
ncvs1.n	|- N = ( norm ` G )
ncvs1.z	|- .0. = ( 0g ` G )
ncvs1.s	|- .x. = ( .s ` G )
ncvs1.f	|- F = (Scalar ` G )
ncvs1.k	|- K = ( Base ` F )

Assertion

Ref	Expression
ncvs1	|- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( N ` ( ( 1 / ( N ` A ) ) .x. A ) ) = 1 )

Proof of Theorem ncvs1

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> G e. (NrmVec i^i CVec ) )
2		simp3 1063	. . . 4 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( 1 / ( N ` A ) ) e. K )
3		elin 3796	. . . . . . . . 9 |- ( G e. (NrmVec i^i CVec ) <-> ( G e. NrmVec /\ G e. CVec ) )
4		nvcnlm 22500	. . . . . . . . . . 11 |- ( G e. NrmVec -> G e. NrmMod )
5		nlmngp 22481	. . . . . . . . . . 11 |- ( G e. NrmMod -> G e. NrmGrp )
6	4, 5	syl 17	. . . . . . . . . 10 |- ( G e. NrmVec -> G e. NrmGrp )
7	6	adantr 481	. . . . . . . . 9 |- ( ( G e. NrmVec /\ G e. CVec ) -> G e. NrmGrp )
8	3, 7	sylbi 207	. . . . . . . 8 |- ( G e. (NrmVec i^i CVec ) -> G e. NrmGrp )
9		simpl 473	. . . . . . . 8 |- ( ( A e. X /\ A =/= .0. ) -> A e. X )
10	8, 9	anim12i 590	. . . . . . 7 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) ) -> ( G e. NrmGrp /\ A e. X ) )
11		ncvs1.x	. . . . . . . 8 |- X = ( Base ` G )
12		ncvs1.n	. . . . . . . 8 |- N = ( norm ` G )
13	11, 12	nmcl 22420	. . . . . . 7 |- ( ( G e. NrmGrp /\ A e. X ) -> ( N ` A ) e. RR )
14	10, 13	syl 17	. . . . . 6 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) ) -> ( N ` A ) e. RR )
15		ncvs1.z	. . . . . . . . . . . 12 |- .0. = ( 0g ` G )
16	11, 12, 15	nmeq0 22422	. . . . . . . . . . 11 |- ( ( G e. NrmGrp /\ A e. X ) -> ( ( N ` A ) = 0 <-> A = .0. ) )
17	16	bicomd 213	. . . . . . . . . 10 |- ( ( G e. NrmGrp /\ A e. X ) -> ( A = .0. <-> ( N ` A ) = 0 ) )
18	8, 17	sylan 488	. . . . . . . . 9 |- ( ( G e. (NrmVec i^i CVec ) /\ A e. X ) -> ( A = .0. <-> ( N ` A ) = 0 ) )
19	18	necon3bid 2838	. . . . . . . 8 |- ( ( G e. (NrmVec i^i CVec ) /\ A e. X ) -> ( A =/= .0. <-> ( N ` A ) =/= 0 ) )
20	19	biimpd 219	. . . . . . 7 |- ( ( G e. (NrmVec i^i CVec ) /\ A e. X ) -> ( A =/= .0. -> ( N ` A ) =/= 0 ) )
21	20	impr 649	. . . . . 6 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) ) -> ( N ` A ) =/= 0 )
22	14, 21	rereccld 10852	. . . . 5 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) ) -> ( 1 / ( N ` A ) ) e. RR )
23	22	3adant3 1081	. . . 4 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( 1 / ( N ` A ) ) e. RR )
24	2, 23	elind 3798	. . 3 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( 1 / ( N ` A ) ) e. ( K i^i RR ) )
25		1re 10039	. . . . . . . 8 |- 1 e. RR
26		0le1 10551	. . . . . . . 8 |- 0 <_ 1
27	25, 26	pm3.2i 471	. . . . . . 7 |- ( 1 e. RR /\ 0 <_ 1 )
28	27	a1i 11	. . . . . 6 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) ) -> ( 1 e. RR /\ 0 <_ 1 ) )
29		simprr 796	. . . . . . 7 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) ) -> A =/= .0. )
30	11, 12, 15	nmgt0 22434	. . . . . . . 8 |- ( ( G e. NrmGrp /\ A e. X ) -> ( A =/= .0. <-> 0 < ( N ` A ) ) )
31	10, 30	syl 17	. . . . . . 7 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) ) -> ( A =/= .0. <-> 0 < ( N ` A ) ) )
32	29, 31	mpbid 222	. . . . . 6 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) ) -> 0 < ( N ` A ) )
33	28, 14, 32	jca32 558	. . . . 5 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) ) -> ( ( 1 e. RR /\ 0 <_ 1 ) /\ ( ( N ` A ) e. RR /\ 0 < ( N ` A ) ) ) )
34	33	3adant3 1081	. . . 4 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( ( 1 e. RR /\ 0 <_ 1 ) /\ ( ( N ` A ) e. RR /\ 0 < ( N ` A ) ) ) )
35		divge0 10892	. . . 4 |- ( ( ( 1 e. RR /\ 0 <_ 1 ) /\ ( ( N ` A ) e. RR /\ 0 < ( N ` A ) ) ) -> 0 <_ ( 1 / ( N ` A ) ) )
36	34, 35	syl 17	. . 3 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> 0 <_ ( 1 / ( N ` A ) ) )
37		simp2l 1087	. . 3 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> A e. X )
38		ncvs1.s	. . . 4 |- .x. = ( .s ` G )
39		ncvs1.f	. . . 4 |- F = (Scalar ` G )
40		ncvs1.k	. . . 4 |- K = ( Base ` F )
41	11, 12, 38, 39, 40	ncvsge0 22953	. . 3 |- ( ( G e. (NrmVec i^i CVec ) /\ ( ( 1 / ( N ` A ) ) e. ( K i^i RR ) /\ 0 <_ ( 1 / ( N ` A ) ) ) /\ A e. X ) -> ( N ` ( ( 1 / ( N ` A ) ) .x. A ) ) = ( ( 1 / ( N ` A ) ) x. ( N ` A ) ) )
42	1, 24, 36, 37, 41	syl121anc 1331	. 2 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( N ` ( ( 1 / ( N ` A ) ) .x. A ) ) = ( ( 1 / ( N ` A ) ) x. ( N ` A ) ) )
43	10	3adant3 1081	. . . . 5 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( G e. NrmGrp /\ A e. X ) )
44	43, 13	syl 17	. . . 4 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( N ` A ) e. RR )
45	44	recnd 10068	. . 3 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( N ` A ) e. CC )
46	21	3adant3 1081	. . 3 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( N ` A ) =/= 0 )
47	45, 46	recid2d 10797	. 2 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( ( 1 / ( N ` A ) ) x. ( N ` A ) ) = 1 )
48	42, 47	eqtrd 2656	1 |- ( ( G e. (NrmVec i^i CVec ) /\ ( A e. X /\ A =/= .0. ) /\ ( 1 / ( N ` A ) ) e. K ) -> ( N ` ( ( 1 / ( N ` A ) ) .x. A ) ) = 1 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   normcnm 22381  NrmGrpcngp 22382  NrmModcnlm 22385  NrmVeccnvc 22386  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-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-subg 17591  df-cmn 18195  df-mgp 18490  df-ring 18549  df-cring 18550  df-subrg 18778  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)