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Theorem h1de2i 23057
Description: Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.)
Hypotheses
Ref Expression
h1de2.1  |-  A  e. 
~H
h1de2.2  |-  B  e. 
~H
Assertion
Ref Expression
h1de2i  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( B  .ih  B
)  .h  A )  =  ( ( A 
.ih  B )  .h  B ) )

Proof of Theorem h1de2i
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 h1de2.2 . . . . . . . . 9  |-  B  e. 
~H
21, 1hicli 22585 . . . . . . . 8  |-  ( B 
.ih  B )  e.  CC
3 h1de2.1 . . . . . . . 8  |-  A  e. 
~H
42, 3hvmulcli 22519 . . . . . . 7  |-  ( ( B  .ih  B )  .h  A )  e. 
~H
53, 1hicli 22585 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
65, 1hvmulcli 22519 . . . . . . 7  |-  ( ( A  .ih  B )  .h  B )  e. 
~H
7 his2sub 22596 . . . . . . 7  |-  ( ( ( ( B  .ih  B )  .h  A )  e.  ~H  /\  (
( A  .ih  B
)  .h  B )  e.  ~H  /\  A  e.  ~H )  ->  (
( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  .ih  A )  =  ( ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  -  ( ( ( A 
.ih  B )  .h  B )  .ih  A
) ) )
84, 6, 3, 7mp3an 1280 . . . . . 6  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  A )  =  ( ( ( ( B  .ih  B )  .h  A ) 
.ih  A )  -  ( ( ( A 
.ih  B )  .h  B )  .ih  A
) )
9 ax-his3 22588 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  (
( ( B  .ih  B )  .h  A ) 
.ih  A )  =  ( ( B  .ih  B )  x.  ( A 
.ih  A ) ) )
102, 3, 3, 9mp3an 1280 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( B  .ih  B )  x.  ( A 
.ih  A ) )
113, 3hicli 22585 . . . . . . . . 9  |-  ( A 
.ih  A )  e.  CC
122, 11mulcomi 9098 . . . . . . . 8  |-  ( ( B  .ih  B )  x.  ( A  .ih  A ) )  =  ( ( A  .ih  A
)  x.  ( B 
.ih  B ) )
1310, 12eqtri 2458 . . . . . . 7  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( A  .ih  A )  x.  ( B 
.ih  B ) )
14 ax-his3 22588 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  e.  CC  /\  B  e.  ~H  /\  A  e.  ~H )  ->  (
( ( A  .ih  B )  .h  B ) 
.ih  A )  =  ( ( A  .ih  B )  x.  ( B 
.ih  A ) ) )
155, 1, 3, 14mp3an 1280 . . . . . . 7  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  A )  =  ( ( A  .ih  B )  x.  ( B 
.ih  A ) )
1613, 15oveq12i 6095 . . . . . 6  |-  ( ( ( ( B  .ih  B )  .h  A ) 
.ih  A )  -  ( ( ( A 
.ih  B )  .h  B )  .ih  A
) )  =  ( ( ( A  .ih  A )  x.  ( B 
.ih  B ) )  -  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )
178, 16eqtr2i 2459 . . . . 5  |-  ( ( ( A  .ih  A
)  x.  ( B 
.ih  B ) )  -  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )  =  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) )  .ih  A )
18 his2sub 22596 . . . . . . . 8  |-  ( ( ( ( B  .ih  B )  .h  A )  e.  ~H  /\  (
( A  .ih  B
)  .h  B )  e.  ~H  /\  B  e.  ~H )  ->  (
( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  .ih  B )  =  ( ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  -  ( ( ( A 
.ih  B )  .h  B )  .ih  B
) ) )
194, 6, 1, 18mp3an 1280 . . . . . . 7  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  B )  =  ( ( ( ( B  .ih  B )  .h  A ) 
.ih  B )  -  ( ( ( A 
.ih  B )  .h  B )  .ih  B
) )
202, 5mulcomi 9098 . . . . . . . . 9  |-  ( ( B  .ih  B )  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  x.  ( B 
.ih  B ) )
21 ax-his3 22588 . . . . . . . . . 10  |-  ( ( ( B  .ih  B
)  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( ( B  .ih  B )  .h  A ) 
.ih  B )  =  ( ( B  .ih  B )  x.  ( A 
.ih  B ) ) )
222, 3, 1, 21mp3an 1280 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( B  .ih  B )  x.  ( A 
.ih  B ) )
23 ax-his3 22588 . . . . . . . . . 10  |-  ( ( ( A  .ih  B
)  e.  CC  /\  B  e.  ~H  /\  B  e.  ~H )  ->  (
( ( A  .ih  B )  .h  B ) 
.ih  B )  =  ( ( A  .ih  B )  x.  ( B 
.ih  B ) ) )
245, 1, 1, 23mp3an 1280 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  =  ( ( A  .ih  B )  x.  ( B 
.ih  B ) )
2520, 22, 243eqtr4i 2468 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( ( A 
.ih  B )  .h  B )  .ih  B
)
264, 1hicli 22585 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  e.  CC
276, 1hicli 22585 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  e.  CC
2826, 27subeq0i 9382 . . . . . . . 8  |-  ( ( ( ( ( B 
.ih  B )  .h  A )  .ih  B
)  -  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B ) )  =  0  <->  ( (
( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( ( A 
.ih  B )  .h  B )  .ih  B
) )
2925, 28mpbir 202 . . . . . . 7  |-  ( ( ( ( B  .ih  B )  .h  A ) 
.ih  B )  -  ( ( ( A 
.ih  B )  .h  B )  .ih  B
) )  =  0
3019, 29eqtri 2458 . . . . . 6  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  B )  =  0
311h1dei 23054 . . . . . . . . 9  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  <->  ( A  e.  ~H  /\  A. x  e.  ~H  ( ( B 
.ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) ) )
323, 31mpbiran 886 . . . . . . . 8  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  <->  A. x  e.  ~H  ( ( B 
.ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) )
334, 6hvsubcli 22526 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H
34 oveq2 6091 . . . . . . . . . . . 12  |-  ( x  =  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) )  ->  ( B  .ih  x )  =  ( B  .ih  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) ) ) )
3534eqeq1d 2446 . . . . . . . . . . 11  |-  ( x  =  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) )  ->  ( ( B  .ih  x )  =  0  <->  ( B  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  0 ) )
36 oveq2 6091 . . . . . . . . . . . 12  |-  ( x  =  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) )  ->  ( A  .ih  x )  =  ( A  .ih  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) ) ) )
3736eqeq1d 2446 . . . . . . . . . . 11  |-  ( x  =  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) )  ->  ( ( A  .ih  x )  =  0  <->  ( A  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  0 ) )
3835, 37imbi12d 313 . . . . . . . . . 10  |-  ( x  =  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) )  ->  ( (
( B  .ih  x
)  =  0  -> 
( A  .ih  x
)  =  0 )  <-> 
( ( B  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  0  ->  ( A  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  0 ) ) )
3938rspcv 3050 . . . . . . . . 9  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H  ->  ( A. x  e.  ~H  ( ( B 
.ih  x )  =  0  ->  ( A  .ih  x )  =  0 )  ->  ( ( B  .ih  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) ) )  =  0  ->  ( A  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  0 ) ) )
4033, 39ax-mp 8 . . . . . . . 8  |-  ( A. x  e.  ~H  (
( B  .ih  x
)  =  0  -> 
( A  .ih  x
)  =  0 )  ->  ( ( B 
.ih  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) ) )  =  0  ->  ( A  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  0 ) )
4132, 40sylbi 189 . . . . . . 7  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( B  .ih  (
( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) ) )  =  0  ->  ( A  .ih  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) ) )  =  0 ) )
42 orthcom 22612 . . . . . . . 8  |-  ( ( ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  B )  =  0  <->  ( B  .ih  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) ) )  =  0 ) )
4333, 1, 42mp2an 655 . . . . . . 7  |-  ( ( ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  .ih  B )  =  0  <->  ( B  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  0 )
44 orthcom 22612 . . . . . . . 8  |-  ( ( ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  e.  ~H  /\  A  e.  ~H )  ->  ( ( ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  A )  =  0  <->  ( A  .ih  ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) ) )  =  0 ) )
4533, 3, 44mp2an 655 . . . . . . 7  |-  ( ( ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  .ih  A )  =  0  <->  ( A  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  0 )
4641, 43, 453imtr4g 263 . . . . . 6  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A  .ih  B )  .h  B ) )  .ih  B )  =  0  ->  (
( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  .ih  A )  =  0 ) )
4730, 46mpi 17 . . . . 5  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  .ih  A )  =  0 )
4817, 47syl5eq 2482 . . . 4  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( ( A  .ih  A )  x.  ( B 
.ih  B ) )  -  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )  =  0 )
4911, 2mulcli 9097 . . . . 5  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  e.  CC
501, 3hicli 22585 . . . . . 6  |-  ( B 
.ih  A )  e.  CC
515, 50mulcli 9097 . . . . 5  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  e.  CC
5249, 51subeq0i 9382 . . . 4  |-  ( ( ( ( A  .ih  A )  x.  ( B 
.ih  B ) )  -  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )  =  0  <->  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  =  ( ( A  .ih  B
)  x.  ( B 
.ih  A ) ) )
5348, 52sylib 190 . . 3  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( A  .ih  A
)  x.  ( B 
.ih  B ) )  =  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )
5453eqcomd 2443 . 2  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) ) )
553, 1bcseqi 22624 . 2  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  <->  ( ( B  .ih  B )  .h  A )  =  ( ( A  .ih  B
)  .h  B ) )
5654, 55sylib 190 1  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( B  .ih  B
)  .h  A )  =  ( ( A 
.ih  B )  .h  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   A.wral 2707   {csn 3816   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    x. cmul 8997    - cmin 9293   ~Hchil 22424    .h csm 22426    .ih csp 22427    -h cmv 22430   _|_cort 22435
This theorem is referenced by:  h1de2bi  23058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072  ax-hilex 22504  ax-hfvadd 22505  ax-hvcom 22506  ax-hvass 22507  ax-hv0cl 22508  ax-hvaddid 22509  ax-hfvmul 22510  ax-hvmulid 22511  ax-hvmulass 22512  ax-hvdistr1 22513  ax-hvdistr2 22514  ax-hvmul0 22515  ax-hfi 22583  ax-his1 22586  ax-his2 22587  ax-his3 22588  ax-his4 22589  ax-hcompl 22706
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cn 17293  df-cnp 17294  df-lm 17295  df-haus 17381  df-tx 17596  df-hmeo 17789  df-xms 18352  df-ms 18353  df-tms 18354  df-cau 19211  df-grpo 21781  df-gid 21782  df-ginv 21783  df-gdiv 21784  df-ablo 21872  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-vs 22080  df-nmcv 22081  df-ims 22082  df-dip 22199  df-hnorm 22473  df-hvsub 22476  df-hlim 22477  df-hcau 22478  df-sh 22711  df-ch 22726  df-oc 22756
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