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Theorem cdlemn2a 30654
Description: Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.)
Hypotheses
Ref Expression
cdlemn2a.b  |-  B  =  ( Base `  K
)
cdlemn2a.l  |-  .<_  =  ( le `  K )
cdlemn2a.j  |-  .\/  =  ( join `  K )
cdlemn2a.a  |-  A  =  ( Atoms `  K )
cdlemn2a.h  |-  H  =  ( LHyp `  K
)
cdlemn2a.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn2a.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemn2a.o  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
cdlemn2a.i  |-  I  =  ( ( DIsoB `  K
) `  W )
cdlemn2a.u  |-  U  =  ( ( DVecH `  K
) `  W )
cdlemn2a.n  |-  N  =  ( LSpan `  U )
cdlemn2a.f  |-  F  =  ( iota_ h  e.  T
( h `  Q
)  =  S )
Assertion
Ref Expression
cdlemn2a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( N `  { <. F ,  O >. } )  C_  ( I `  X
) )
Distinct variable groups:    .<_ , h    A, h    B, f    h, H   
f, K    h, K    Q, h    S, h    T, f    T, h    f, W    h, W
Allowed substitution hints:    A( f)    B( h)    Q( f)    R( f, h)    S( f)    U( f, h)    F( f, h)    H( f)    I( f, h)    .\/ ( f, h)   
.<_ ( f)    N( f, h)    O( f, h)    X( f, h)

Proof of Theorem cdlemn2a
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp21 990 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3 simp22 991 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
4 cdlemn2a.l . . . . 5  |-  .<_  =  ( le `  K )
5 cdlemn2a.a . . . . 5  |-  A  =  ( Atoms `  K )
6 cdlemn2a.h . . . . 5  |-  H  =  ( LHyp `  K
)
7 cdlemn2a.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemn2a.f . . . . 5  |-  F  =  ( iota_ h  e.  T
( h `  Q
)  =  S )
94, 5, 6, 7, 8ltrniotacl 30036 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  F  e.  T )
101, 2, 3, 9syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  F  e.  T )
11 cdlemn2a.b . . . 4  |-  B  =  ( Base `  K
)
12 cdlemn2a.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
13 cdlemn2a.o . . . 4  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
14 cdlemn2a.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
15 cdlemn2a.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
16 cdlemn2a.n . . . 4  |-  N  =  ( LSpan `  U )
1711, 6, 7, 12, 13, 14, 15, 16dib1dim2 30626 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  ( N `  { <. F ,  O >. } ) )
181, 10, 17syl2anc 644 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  (
I `  ( R `  F ) )  =  ( N `  { <. F ,  O >. } ) )
19 cdlemn2a.j . . . 4  |-  .\/  =  ( join `  K )
2011, 4, 19, 5, 6, 7, 12, 8cdlemn2 30653 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( R `  F )  .<_  X )
2111, 6, 7, 12trlcl 29621 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  B
)
221, 10, 21syl2anc 644 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( R `  F )  e.  B )
234, 6, 7, 12trlle 29641 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
241, 10, 23syl2anc 644 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( R `  F )  .<_  W )
25 simp23 992 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( X  e.  B  /\  X  .<_  W ) )
2611, 4, 6, 15dibord 30617 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R `
 F )  e.  B  /\  ( R `
 F )  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( ( I `
 ( R `  F ) )  C_  ( I `  X
)  <->  ( R `  F )  .<_  X ) )
271, 22, 24, 25, 26syl121anc 1189 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  (
( I `  ( R `  F )
)  C_  ( I `  X )  <->  ( R `  F )  .<_  X ) )
2820, 27mpbird 225 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  (
I `  ( R `  F ) )  C_  ( I `  X
) )
2918, 28eqsstr3d 3215 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  S  .<_  ( Q  .\/  X
) )  ->  ( N `  { <. F ,  O >. } )  C_  ( I `  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    C_ wss 3154   {csn 3642   <.cop 3645   class class class wbr 4025    e. cmpt 4079    _I cid 4304    |` cres 4691   ` cfv 5222  (class class class)co 5820   iota_crio 6291   Basecbs 13143   lecple 13210   joincjn 14073   LSpanclspn 15723   Atomscatm 28721   HLchlt 28808   LHypclh 29441   LTrncltrn 29558   trLctrl 29615   DVecHcdvh 30536   DIsoBcdib 30596
This theorem is referenced by:  cdlemn5pre  30658
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-fal 1313  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-undef 6292  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-n0 9962  df-z 10021  df-uz 10227  df-fz 10778  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-sca 13219  df-vsca 13220  df-0g 13399  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-mnd 14362  df-grp 14484  df-minusg 14485  df-sbg 14486  df-mgp 15321  df-rng 15335  df-ur 15337  df-oppr 15400  df-dvdsr 15418  df-unit 15419  df-invr 15449  df-dvr 15460  df-drng 15509  df-lmod 15624  df-lss 15685  df-lsp 15724  df-lvec 15851  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-llines 28955  df-lplanes 28956  df-lvols 28957  df-lines 28958  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616  df-tendo 30212  df-edring 30214  df-disoa 30487  df-dvech 30537  df-dib 30597
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