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Theorem h1de2i 27254
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 26782 . . . . . . . 8  |-  ( B 
.ih  B )  e.  CC
3 h1de2.1 . . . . . . . 8  |-  A  e. 
~H
42, 3hvmulcli 26715 . . . . . . 7  |-  ( ( B  .ih  B )  .h  A )  e. 
~H
53, 1hicli 26782 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
65, 1hvmulcli 26715 . . . . . . 7  |-  ( ( A  .ih  B )  .h  B )  e. 
~H
7 his2sub 26793 . . . . . . 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 1373 . . . . . 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 26785 . . . . . . . . 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 1373 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( B  .ih  B )  x.  ( A 
.ih  A ) )
113, 3hicli 26782 . . . . . . . . 9  |-  ( A 
.ih  A )  e.  CC
122, 11mulcomi 9674 . . . . . . . 8  |-  ( ( B  .ih  B )  x.  ( A  .ih  A ) )  =  ( ( A  .ih  A
)  x.  ( B 
.ih  B ) )
1310, 12eqtri 2483 . . . . . . 7  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( A  .ih  A )  x.  ( B 
.ih  B ) )
14 ax-his3 26785 . . . . . . . 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 1373 . . . . . . 7  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  A )  =  ( ( A  .ih  B )  x.  ( B 
.ih  A ) )
1613, 15oveq12i 6326 . . . . . 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 2484 . . . . 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 26793 . . . . . . . 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 1373 . . . . . . 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 9674 . . . . . . . . 9  |-  ( ( B  .ih  B )  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  x.  ( B 
.ih  B ) )
21 ax-his3 26785 . . . . . . . . . 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 1373 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( B  .ih  B )  x.  ( A 
.ih  B ) )
23 ax-his3 26785 . . . . . . . . . 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 1373 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  =  ( ( A  .ih  B )  x.  ( B 
.ih  B ) )
2520, 22, 243eqtr4i 2493 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( ( A 
.ih  B )  .h  B )  .ih  B
)
264, 1hicli 26782 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  e.  CC
276, 1hicli 26782 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  e.  CC
2826, 27subeq0i 9979 . . . . . . . 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 214 . . . . . . 7  |-  ( ( ( ( B  .ih  B )  .h  A ) 
.ih  B )  -  ( ( ( A 
.ih  B )  .h  B )  .ih  B
) )  =  0
3019, 29eqtri 2483 . . . . . 6  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  B )  =  0
311h1dei 27251 . . . . . . . . 9  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  <->  ( A  e.  ~H  /\  A. x  e.  ~H  ( ( B 
.ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) ) )
323, 31mpbiran 934 . . . . . . . 8  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  <->  A. x  e.  ~H  ( ( B 
.ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) )
334, 6hvsubcli 26722 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H
34 oveq2 6322 . . . . . . . . . . . 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 2463 . . . . . . . . . . 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 6322 . . . . . . . . . . . 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 2463 . . . . . . . . . . 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 326 . . . . . . . . . 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 3157 . . . . . . . . 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 5 . . . . . . . 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 200 . . . . . . 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 26809 . . . . . . . 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 683 . . . . . . 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 26809 . . . . . . . 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 683 . . . . . . 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 278 . . . . . 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 20 . . . . 5  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  .ih  A )  =  0 )
4817, 47syl5eq 2507 . . . 4  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( ( A  .ih  A )  x.  ( B 
.ih  B ) )  -  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )  =  0 )
4911, 2mulcli 9673 . . . . 5  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  e.  CC
501, 3hicli 26782 . . . . . 6  |-  ( B 
.ih  A )  e.  CC
515, 50mulcli 9673 . . . . 5  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  e.  CC
5249, 51subeq0i 9979 . . . 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 201 . . 3  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( A  .ih  A
)  x.  ( B 
.ih  B ) )  =  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )
5453eqcomd 2467 . 2  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) ) )
553, 1bcseqi 26821 . 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 201 1  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( B  .ih  B
)  .h  A )  =  ( ( A 
.ih  B )  .h  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1454    e. wcel 1897   A.wral 2748   {csn 3979   ` cfv 5600  (class class class)co 6314   CCcc 9562   0cc0 9564    x. cmul 9569    - cmin 9885   ~Hchil 26620    .h csm 26622    .ih csp 26623    -h cmv 26626   _|_cort 26631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642  ax-addf 9643  ax-mulf 9644  ax-hilex 26700  ax-hfvadd 26701  ax-hvcom 26702  ax-hvass 26703  ax-hv0cl 26704  ax-hvaddid 26705  ax-hfvmul 26706  ax-hvmulid 26707  ax-hvmulass 26708  ax-hvdistr1 26709  ax-hvdistr2 26710  ax-hvmul0 26711  ax-hfi 26780  ax-his1 26783  ax-his2 26784  ax-his3 26785  ax-his4 26786  ax-hcompl 26903
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-2o 7208  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-fi 7950  df-sup 7981  df-inf 7982  df-oi 8050  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-q 11293  df-rp 11331  df-xneg 11437  df-xadd 11438  df-xmul 11439  df-ioo 11667  df-icc 11670  df-fz 11813  df-fzo 11946  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-clim 13600  df-sum 13801  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-mulr 15252  df-starv 15253  df-sca 15254  df-vsca 15255  df-ip 15256  df-tset 15257  df-ple 15258  df-ds 15260  df-unif 15261  df-hom 15262  df-cco 15263  df-rest 15369  df-topn 15370  df-0g 15388  df-gsum 15389  df-topgen 15390  df-pt 15391  df-prds 15394  df-xrs 15448  df-qtop 15454  df-imas 15455  df-xps 15458  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-mulg 16724  df-cntz 17019  df-cmn 17480  df-psmet 19010  df-xmet 19011  df-met 19012  df-bl 19013  df-mopn 19014  df-cnfld 19019  df-top 19969  df-bases 19970  df-topon 19971  df-topsp 19972  df-cn 20291  df-cnp 20292  df-lm 20293  df-haus 20379  df-tx 20625  df-hmeo 20818  df-xms 21383  df-ms 21384  df-tms 21385  df-cau 22274  df-grpo 25967  df-gid 25968  df-ginv 25969  df-gdiv 25970  df-ablo 26058  df-vc 26213  df-nv 26259  df-va 26262  df-ba 26263  df-sm 26264  df-0v 26265  df-vs 26266  df-nmcv 26267  df-ims 26268  df-dip 26385  df-hnorm 26669  df-hvsub 26672  df-hlim 26673  df-hcau 26674  df-sh 26908  df-ch 26922  df-oc 26953
This theorem is referenced by:  h1de2bi  27255
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