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Theorem h1de2i 27143
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 26671 . . . . . . . 8  |-  ( B 
.ih  B )  e.  CC
3 h1de2.1 . . . . . . . 8  |-  A  e. 
~H
42, 3hvmulcli 26604 . . . . . . 7  |-  ( ( B  .ih  B )  .h  A )  e. 
~H
53, 1hicli 26671 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
65, 1hvmulcli 26604 . . . . . . 7  |-  ( ( A  .ih  B )  .h  B )  e. 
~H
7 his2sub 26682 . . . . . . 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 1360 . . . . . 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 26674 . . . . . . . . 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 1360 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( B  .ih  B )  x.  ( A 
.ih  A ) )
113, 3hicli 26671 . . . . . . . . 9  |-  ( A 
.ih  A )  e.  CC
122, 11mulcomi 9595 . . . . . . . 8  |-  ( ( B  .ih  B )  x.  ( A  .ih  A ) )  =  ( ( A  .ih  A
)  x.  ( B 
.ih  B ) )
1310, 12eqtri 2445 . . . . . . 7  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( A  .ih  A )  x.  ( B 
.ih  B ) )
14 ax-his3 26674 . . . . . . . 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 1360 . . . . . . 7  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  A )  =  ( ( A  .ih  B )  x.  ( B 
.ih  A ) )
1613, 15oveq12i 6256 . . . . . 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 2446 . . . . 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 26682 . . . . . . . 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 1360 . . . . . . 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 9595 . . . . . . . . 9  |-  ( ( B  .ih  B )  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  x.  ( B 
.ih  B ) )
21 ax-his3 26674 . . . . . . . . . 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 1360 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( B  .ih  B )  x.  ( A 
.ih  B ) )
23 ax-his3 26674 . . . . . . . . . 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 1360 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  =  ( ( A  .ih  B )  x.  ( B 
.ih  B ) )
2520, 22, 243eqtr4i 2455 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( ( A 
.ih  B )  .h  B )  .ih  B
)
264, 1hicli 26671 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  e.  CC
276, 1hicli 26671 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  e.  CC
2826, 27subeq0i 9900 . . . . . . . 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 212 . . . . . . 7  |-  ( ( ( ( B  .ih  B )  .h  A ) 
.ih  B )  -  ( ( ( A 
.ih  B )  .h  B )  .ih  B
) )  =  0
3019, 29eqtri 2445 . . . . . 6  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  B )  =  0
311h1dei 27140 . . . . . . . . 9  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  <->  ( A  e.  ~H  /\  A. x  e.  ~H  ( ( B 
.ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) ) )
323, 31mpbiran 926 . . . . . . . 8  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  <->  A. x  e.  ~H  ( ( B 
.ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) )
334, 6hvsubcli 26611 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H
34 oveq2 6252 . . . . . . . . . . . 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 2425 . . . . . . . . . . 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 6252 . . . . . . . . . . . 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 2425 . . . . . . . . . . 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 321 . . . . . . . . . 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 3116 . . . . . . . . 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 198 . . . . . . 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 26698 . . . . . . . 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 676 . . . . . . 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 26698 . . . . . . . 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 676 . . . . . . 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 273 . . . . . 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 2469 . . . 4  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( ( A  .ih  A )  x.  ( B 
.ih  B ) )  -  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )  =  0 )
4911, 2mulcli 9594 . . . . 5  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  e.  CC
501, 3hicli 26671 . . . . . 6  |-  ( B 
.ih  A )  e.  CC
515, 50mulcli 9594 . . . . 5  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  e.  CC
5249, 51subeq0i 9900 . . . 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 199 . . 3  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( A  .ih  A
)  x.  ( B 
.ih  B ) )  =  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )
5453eqcomd 2429 . 2  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) ) )
553, 1bcseqi 26710 . 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 199 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 187    = wceq 1437    e. wcel 1872   A.wral 2709   {csn 3936   ` cfv 5539  (class class class)co 6244   CCcc 9483   0cc0 9485    x. cmul 9490    - cmin 9806   ~Hchil 26509    .h csm 26511    .ih csp 26512    -h cmv 26515   _|_cort 26520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563  ax-addf 9564  ax-mulf 9565  ax-hilex 26589  ax-hfvadd 26590  ax-hvcom 26591  ax-hvass 26592  ax-hv0cl 26593  ax-hvaddid 26594  ax-hfvmul 26595  ax-hvmulid 26596  ax-hvmulass 26597  ax-hvdistr1 26598  ax-hvdistr2 26599  ax-hvmul0 26600  ax-hfi 26669  ax-his1 26672  ax-his2 26673  ax-his3 26674  ax-his4 26675  ax-hcompl 26792
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-iin 4240  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-se 4751  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-isom 5548  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-of 6484  df-om 6646  df-1st 6746  df-2nd 6747  df-supp 6865  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7473  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-fsupp 7832  df-fi 7873  df-sup 7904  df-inf 7905  df-oi 7973  df-card 8320  df-cda 8544  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-4 10616  df-5 10617  df-6 10618  df-7 10619  df-8 10620  df-9 10621  df-10 10622  df-n0 10816  df-z 10884  df-dec 10998  df-uz 11106  df-q 11211  df-rp 11249  df-xneg 11355  df-xadd 11356  df-xmul 11357  df-ioo 11585  df-icc 11588  df-fz 11731  df-fzo 11862  df-seq 12159  df-exp 12218  df-hash 12461  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-clim 13490  df-sum 13691  df-struct 15061  df-ndx 15062  df-slot 15063  df-base 15064  df-sets 15065  df-ress 15066  df-plusg 15141  df-mulr 15142  df-starv 15143  df-sca 15144  df-vsca 15145  df-ip 15146  df-tset 15147  df-ple 15148  df-ds 15150  df-unif 15151  df-hom 15152  df-cco 15153  df-rest 15259  df-topn 15260  df-0g 15278  df-gsum 15279  df-topgen 15280  df-pt 15281  df-prds 15284  df-xrs 15338  df-qtop 15344  df-imas 15345  df-xps 15348  df-mre 15430  df-mrc 15431  df-acs 15433  df-mgm 16426  df-sgrp 16465  df-mnd 16475  df-submnd 16521  df-mulg 16614  df-cntz 16909  df-cmn 17370  df-psmet 18900  df-xmet 18901  df-met 18902  df-bl 18903  df-mopn 18904  df-cnfld 18909  df-top 19858  df-bases 19859  df-topon 19860  df-topsp 19861  df-cn 20180  df-cnp 20181  df-lm 20182  df-haus 20268  df-tx 20514  df-hmeo 20707  df-xms 21272  df-ms 21273  df-tms 21274  df-cau 22163  df-grpo 25856  df-gid 25857  df-ginv 25858  df-gdiv 25859  df-ablo 25947  df-vc 26102  df-nv 26148  df-va 26151  df-ba 26152  df-sm 26153  df-0v 26154  df-vs 26155  df-nmcv 26156  df-ims 26157  df-dip 26274  df-hnorm 26558  df-hvsub 26561  df-hlim 26562  df-hcau 26563  df-sh 26797  df-ch 26811  df-oc 26842
This theorem is referenced by:  h1de2bi  27144
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