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Theorem h1de2i 24875
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 24402 . . . . . . . 8  |-  ( B 
.ih  B )  e.  CC
3 h1de2.1 . . . . . . . 8  |-  A  e. 
~H
42, 3hvmulcli 24335 . . . . . . 7  |-  ( ( B  .ih  B )  .h  A )  e. 
~H
53, 1hicli 24402 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
65, 1hvmulcli 24335 . . . . . . 7  |-  ( ( A  .ih  B )  .h  B )  e. 
~H
7 his2sub 24413 . . . . . . 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 1309 . . . . . 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 24405 . . . . . . . . 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 1309 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( B  .ih  B )  x.  ( A 
.ih  A ) )
113, 3hicli 24402 . . . . . . . . 9  |-  ( A 
.ih  A )  e.  CC
122, 11mulcomi 9388 . . . . . . . 8  |-  ( ( B  .ih  B )  x.  ( A  .ih  A ) )  =  ( ( A  .ih  A
)  x.  ( B 
.ih  B ) )
1310, 12eqtri 2461 . . . . . . 7  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( A  .ih  A )  x.  ( B 
.ih  B ) )
14 ax-his3 24405 . . . . . . . 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 1309 . . . . . . 7  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  A )  =  ( ( A  .ih  B )  x.  ( B 
.ih  A ) )
1613, 15oveq12i 6102 . . . . . 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 2462 . . . . 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 24413 . . . . . . . 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 1309 . . . . . . 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 9388 . . . . . . . . 9  |-  ( ( B  .ih  B )  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  x.  ( B 
.ih  B ) )
21 ax-his3 24405 . . . . . . . . . 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 1309 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( B  .ih  B )  x.  ( A 
.ih  B ) )
23 ax-his3 24405 . . . . . . . . . 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 1309 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  =  ( ( A  .ih  B )  x.  ( B 
.ih  B ) )
2520, 22, 243eqtr4i 2471 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( ( A 
.ih  B )  .h  B )  .ih  B
)
264, 1hicli 24402 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  e.  CC
276, 1hicli 24402 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  e.  CC
2826, 27subeq0i 9684 . . . . . . . 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 209 . . . . . . 7  |-  ( ( ( ( B  .ih  B )  .h  A ) 
.ih  B )  -  ( ( ( A 
.ih  B )  .h  B )  .ih  B
) )  =  0
3019, 29eqtri 2461 . . . . . 6  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  B )  =  0
311h1dei 24872 . . . . . . . . 9  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  <->  ( A  e.  ~H  /\  A. x  e.  ~H  ( ( B 
.ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) ) )
323, 31mpbiran 904 . . . . . . . 8  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  <->  A. x  e.  ~H  ( ( B 
.ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) )
334, 6hvsubcli 24342 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H
34 oveq2 6098 . . . . . . . . . . . 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 2449 . . . . . . . . . . 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 6098 . . . . . . . . . . . 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 2449 . . . . . . . . . . 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 320 . . . . . . . . . 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 3066 . . . . . . . . 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 195 . . . . . . 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 24429 . . . . . . . 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 667 . . . . . . 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 24429 . . . . . . . 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 667 . . . . . . 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 270 . . . . . 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 2485 . . . 4  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( ( A  .ih  A )  x.  ( B 
.ih  B ) )  -  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )  =  0 )
4911, 2mulcli 9387 . . . . 5  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  e.  CC
501, 3hicli 24402 . . . . . 6  |-  ( B 
.ih  A )  e.  CC
515, 50mulcli 9387 . . . . 5  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  e.  CC
5249, 51subeq0i 9684 . . . 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 196 . . 3  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( A  .ih  A
)  x.  ( B 
.ih  B ) )  =  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )
5453eqcomd 2446 . 2  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) ) )
553, 1bcseqi 24441 . 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 196 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 184    = wceq 1364    e. wcel 1761   A.wral 2713   {csn 3874   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278    x. cmul 9283    - cmin 9591   ~Hchil 24240    .h csm 24242    .ih csp 24243    -h cmv 24246   _|_cort 24251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358  ax-hilex 24320  ax-hfvadd 24321  ax-hvcom 24322  ax-hvass 24323  ax-hv0cl 24324  ax-hvaddid 24325  ax-hfvmul 24326  ax-hvmulid 24327  ax-hvmulass 24328  ax-hvdistr1 24329  ax-hvdistr2 24330  ax-hvmul0 24331  ax-hfi 24400  ax-his1 24403  ax-his2 24404  ax-his3 24405  ax-his4 24406  ax-hcompl 24523
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cn 18731  df-cnp 18732  df-lm 18733  df-haus 18819  df-tx 19035  df-hmeo 19228  df-xms 19795  df-ms 19796  df-tms 19797  df-cau 20667  df-grpo 23597  df-gid 23598  df-ginv 23599  df-gdiv 23600  df-ablo 23688  df-vc 23843  df-nv 23889  df-va 23892  df-ba 23893  df-sm 23894  df-0v 23895  df-vs 23896  df-nmcv 23897  df-ims 23898  df-dip 24015  df-hnorm 24289  df-hvsub 24292  df-hlim 24293  df-hcau 24294  df-sh 24528  df-ch 24543  df-oc 24574
This theorem is referenced by:  h1de2bi  24876
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