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Theorem h1de2bi 23048
Description: Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
h1de2.1  |-  A  e. 
~H
h1de2.2  |-  B  e. 
~H
Assertion
Ref Expression
h1de2bi  |-  ( B  =/=  0h  ->  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  A  =  ( ( ( A 
.ih  B )  / 
( B  .ih  B
) )  .h  B
) ) )

Proof of Theorem h1de2bi
StepHypRef Expression
1 h1de2.2 . . . 4  |-  B  e. 
~H
2 his6 22593 . . . 4  |-  ( B  e.  ~H  ->  (
( B  .ih  B
)  =  0  <->  B  =  0h ) )
31, 2ax-mp 8 . . 3  |-  ( ( B  .ih  B )  =  0  <->  B  =  0h )
43necon3bii 2630 . 2  |-  ( ( B  .ih  B )  =/=  0  <->  B  =/=  0h )
5 h1de2.1 . . . . . . . . 9  |-  A  e. 
~H
65, 1h1de2i 23047 . . . . . . . 8  |-  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  ->  (
( B  .ih  B
)  .h  A )  =  ( ( A 
.ih  B )  .h  B ) )
76adantl 453 . . . . . . 7  |-  ( ( ( B  .ih  B
)  =/=  0  /\  A  e.  ( _|_ `  ( _|_ `  { B } ) ) )  ->  ( ( B 
.ih  B )  .h  A )  =  ( ( A  .ih  B
)  .h  B ) )
87oveq2d 6089 . . . . . 6  |-  ( ( ( B  .ih  B
)  =/=  0  /\  A  e.  ( _|_ `  ( _|_ `  { B } ) ) )  ->  ( ( 1  /  ( B  .ih  B ) )  .h  (
( B  .ih  B
)  .h  A ) )  =  ( ( 1  /  ( B 
.ih  B ) )  .h  ( ( A 
.ih  B )  .h  B ) ) )
91, 1hicli 22575 . . . . . . . . . . 11  |-  ( B 
.ih  B )  e.  CC
109recclzi 9731 . . . . . . . . . 10  |-  ( ( B  .ih  B )  =/=  0  ->  (
1  /  ( B 
.ih  B ) )  e.  CC )
11 ax-hvmulass 22502 . . . . . . . . . . 11  |-  ( ( ( 1  /  ( B  .ih  B ) )  e.  CC  /\  ( B  .ih  B )  e.  CC  /\  A  e. 
~H )  ->  (
( ( 1  / 
( B  .ih  B
) )  x.  ( B  .ih  B ) )  .h  A )  =  ( ( 1  / 
( B  .ih  B
) )  .h  (
( B  .ih  B
)  .h  A ) ) )
129, 5, 11mp3an23 1271 . . . . . . . . . 10  |-  ( ( 1  /  ( B 
.ih  B ) )  e.  CC  ->  (
( ( 1  / 
( B  .ih  B
) )  x.  ( B  .ih  B ) )  .h  A )  =  ( ( 1  / 
( B  .ih  B
) )  .h  (
( B  .ih  B
)  .h  A ) ) )
1310, 12syl 16 . . . . . . . . 9  |-  ( ( B  .ih  B )  =/=  0  ->  (
( ( 1  / 
( B  .ih  B
) )  x.  ( B  .ih  B ) )  .h  A )  =  ( ( 1  / 
( B  .ih  B
) )  .h  (
( B  .ih  B
)  .h  A ) ) )
14 ax-1cn 9040 . . . . . . . . . . 11  |-  1  e.  CC
1514, 9divcan1zi 9742 . . . . . . . . . 10  |-  ( ( B  .ih  B )  =/=  0  ->  (
( 1  /  ( B  .ih  B ) )  x.  ( B  .ih  B ) )  =  1 )
1615oveq1d 6088 . . . . . . . . 9  |-  ( ( B  .ih  B )  =/=  0  ->  (
( ( 1  / 
( B  .ih  B
) )  x.  ( B  .ih  B ) )  .h  A )  =  ( 1  .h  A
) )
1713, 16eqtr3d 2469 . . . . . . . 8  |-  ( ( B  .ih  B )  =/=  0  ->  (
( 1  /  ( B  .ih  B ) )  .h  ( ( B 
.ih  B )  .h  A ) )  =  ( 1  .h  A
) )
18 ax-hvmulid 22501 . . . . . . . . 9  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
195, 18ax-mp 8 . . . . . . . 8  |-  ( 1  .h  A )  =  A
2017, 19syl6eq 2483 . . . . . . 7  |-  ( ( B  .ih  B )  =/=  0  ->  (
( 1  /  ( B  .ih  B ) )  .h  ( ( B 
.ih  B )  .h  A ) )  =  A )
2120adantr 452 . . . . . 6  |-  ( ( ( B  .ih  B
)  =/=  0  /\  A  e.  ( _|_ `  ( _|_ `  { B } ) ) )  ->  ( ( 1  /  ( B  .ih  B ) )  .h  (
( B  .ih  B
)  .h  A ) )  =  A )
228, 21eqtr3d 2469 . . . . 5  |-  ( ( ( B  .ih  B
)  =/=  0  /\  A  e.  ( _|_ `  ( _|_ `  { B } ) ) )  ->  ( ( 1  /  ( B  .ih  B ) )  .h  (
( A  .ih  B
)  .h  B ) )  =  A )
235, 1hicli 22575 . . . . . . . . 9  |-  ( A 
.ih  B )  e.  CC
24 ax-hvmulass 22502 . . . . . . . . 9  |-  ( ( ( 1  /  ( B  .ih  B ) )  e.  CC  /\  ( A  .ih  B )  e.  CC  /\  B  e. 
~H )  ->  (
( ( 1  / 
( B  .ih  B
) )  x.  ( A  .ih  B ) )  .h  B )  =  ( ( 1  / 
( B  .ih  B
) )  .h  (
( A  .ih  B
)  .h  B ) ) )
2523, 1, 24mp3an23 1271 . . . . . . . 8  |-  ( ( 1  /  ( B 
.ih  B ) )  e.  CC  ->  (
( ( 1  / 
( B  .ih  B
) )  x.  ( A  .ih  B ) )  .h  B )  =  ( ( 1  / 
( B  .ih  B
) )  .h  (
( A  .ih  B
)  .h  B ) ) )
2610, 25syl 16 . . . . . . 7  |-  ( ( B  .ih  B )  =/=  0  ->  (
( ( 1  / 
( B  .ih  B
) )  x.  ( A  .ih  B ) )  .h  B )  =  ( ( 1  / 
( B  .ih  B
) )  .h  (
( A  .ih  B
)  .h  B ) ) )
27 mulcom 9068 . . . . . . . . . 10  |-  ( ( ( 1  /  ( B  .ih  B ) )  e.  CC  /\  ( A  .ih  B )  e.  CC )  ->  (
( 1  /  ( B  .ih  B ) )  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  x.  ( 1  /  ( B  .ih  B ) ) ) )
2810, 23, 27sylancl 644 . . . . . . . . 9  |-  ( ( B  .ih  B )  =/=  0  ->  (
( 1  /  ( B  .ih  B ) )  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  x.  ( 1  /  ( B  .ih  B ) ) ) )
2923, 9divreczi 9744 . . . . . . . . 9  |-  ( ( B  .ih  B )  =/=  0  ->  (
( A  .ih  B
)  /  ( B 
.ih  B ) )  =  ( ( A 
.ih  B )  x.  ( 1  /  ( B  .ih  B ) ) ) )
3028, 29eqtr4d 2470 . . . . . . . 8  |-  ( ( B  .ih  B )  =/=  0  ->  (
( 1  /  ( B  .ih  B ) )  x.  ( A  .ih  B ) )  =  ( ( A  .ih  B
)  /  ( B 
.ih  B ) ) )
3130oveq1d 6088 . . . . . . 7  |-  ( ( B  .ih  B )  =/=  0  ->  (
( ( 1  / 
( B  .ih  B
) )  x.  ( A  .ih  B ) )  .h  B )  =  ( ( ( A 
.ih  B )  / 
( B  .ih  B
) )  .h  B
) )
3226, 31eqtr3d 2469 . . . . . 6  |-  ( ( B  .ih  B )  =/=  0  ->  (
( 1  /  ( B  .ih  B ) )  .h  ( ( A 
.ih  B )  .h  B ) )  =  ( ( ( A 
.ih  B )  / 
( B  .ih  B
) )  .h  B
) )
3332adantr 452 . . . . 5  |-  ( ( ( B  .ih  B
)  =/=  0  /\  A  e.  ( _|_ `  ( _|_ `  { B } ) ) )  ->  ( ( 1  /  ( B  .ih  B ) )  .h  (
( A  .ih  B
)  .h  B ) )  =  ( ( ( A  .ih  B
)  /  ( B 
.ih  B ) )  .h  B ) )
3422, 33eqtr3d 2469 . . . 4  |-  ( ( ( B  .ih  B
)  =/=  0  /\  A  e.  ( _|_ `  ( _|_ `  { B } ) ) )  ->  A  =  ( ( ( A  .ih  B )  /  ( B 
.ih  B ) )  .h  B ) )
3534ex 424 . . 3  |-  ( ( B  .ih  B )  =/=  0  ->  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  ->  A  =  ( (
( A  .ih  B
)  /  ( B 
.ih  B ) )  .h  B ) ) )
3623, 9divclzi 9741 . . . . 5  |-  ( ( B  .ih  B )  =/=  0  ->  (
( A  .ih  B
)  /  ( B 
.ih  B ) )  e.  CC )
371elexi 2957 . . . . . . . . . . 11  |-  B  e. 
_V
3837snss 3918 . . . . . . . . . 10  |-  ( B  e.  ~H  <->  { B }  C_  ~H )
391, 38mpbi 200 . . . . . . . . 9  |-  { B }  C_  ~H
40 occl 22798 . . . . . . . . 9  |-  ( { B }  C_  ~H  ->  ( _|_ `  { B } )  e.  CH )
4139, 40ax-mp 8 . . . . . . . 8  |-  ( _|_ `  { B } )  e.  CH
4241choccli 22801 . . . . . . 7  |-  ( _|_ `  ( _|_ `  { B } ) )  e. 
CH
4342chshii 22722 . . . . . 6  |-  ( _|_ `  ( _|_ `  { B } ) )  e.  SH
44 h1did 23045 . . . . . . 7  |-  ( B  e.  ~H  ->  B  e.  ( _|_ `  ( _|_ `  { B }
) ) )
451, 44ax-mp 8 . . . . . 6  |-  B  e.  ( _|_ `  ( _|_ `  { B }
) )
46 shmulcl 22712 . . . . . 6  |-  ( ( ( _|_ `  ( _|_ `  { B }
) )  e.  SH  /\  ( ( A  .ih  B )  /  ( B 
.ih  B ) )  e.  CC  /\  B  e.  ( _|_ `  ( _|_ `  { B }
) ) )  -> 
( ( ( A 
.ih  B )  / 
( B  .ih  B
) )  .h  B
)  e.  ( _|_ `  ( _|_ `  { B } ) ) )
4743, 45, 46mp3an13 1270 . . . . 5  |-  ( ( ( A  .ih  B
)  /  ( B 
.ih  B ) )  e.  CC  ->  (
( ( A  .ih  B )  /  ( B 
.ih  B ) )  .h  B )  e.  ( _|_ `  ( _|_ `  { B }
) ) )
4836, 47syl 16 . . . 4  |-  ( ( B  .ih  B )  =/=  0  ->  (
( ( A  .ih  B )  /  ( B 
.ih  B ) )  .h  B )  e.  ( _|_ `  ( _|_ `  { B }
) ) )
49 eleq1 2495 . . . 4  |-  ( A  =  ( ( ( A  .ih  B )  /  ( B  .ih  B ) )  .h  B
)  ->  ( A  e.  ( _|_ `  ( _|_ `  { B }
) )  <->  ( (
( A  .ih  B
)  /  ( B 
.ih  B ) )  .h  B )  e.  ( _|_ `  ( _|_ `  { B }
) ) ) )
5048, 49syl5ibrcom 214 . . 3  |-  ( ( B  .ih  B )  =/=  0  ->  ( A  =  ( (
( A  .ih  B
)  /  ( B 
.ih  B ) )  .h  B )  ->  A  e.  ( _|_ `  ( _|_ `  { B } ) ) ) )
5135, 50impbid 184 . 2  |-  ( ( B  .ih  B )  =/=  0  ->  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  A  =  ( ( ( A 
.ih  B )  / 
( B  .ih  B
) )  .h  B
) ) )
524, 51sylbir 205 1  |-  ( B  =/=  0h  ->  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  A  =  ( ( ( A 
.ih  B )  / 
( B  .ih  B
) )  .h  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   {csn 3806   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987    / cdiv 9669   ~Hchil 22414    .h csm 22416    .ih csp 22417   0hc0v 22419   SHcsh 22423   CHcch 22424   _|_cort 22425
This theorem is referenced by:  h1de2ctlem  23049  elspansn2  23061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvmulass 22502  ax-hvdistr1 22503  ax-hvdistr2 22504  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his2 22577  ax-his3 22578  ax-his4 22579  ax-hcompl 22696
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cn 17283  df-cnp 17284  df-lm 17285  df-haus 17371  df-tx 17586  df-hmeo 17779  df-xms 18342  df-ms 18343  df-tms 18344  df-cau 19201  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-vs 22070  df-nmcv 22071  df-ims 22072  df-dip 22189  df-hnorm 22463  df-hvsub 22466  df-hlim 22467  df-hcau 22468  df-sh 22701  df-ch 22716  df-oc 22746
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