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Theorem cdlemn2a 35180
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 988 . . 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 1021 . . . 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 1022 . . . 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 34562 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  F  e.  T )
101, 2, 3, 9syl3anc 1219 . . 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 35152 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  ( N `  { <. F ,  O >. } ) )
181, 10, 17syl2anc 661 . 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 35179 . . 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 34147 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  B
)
221, 10, 21syl2anc 661 . . . 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 34167 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
241, 10, 23syl2anc 661 . . . 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 1023 . . . 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 35143 . . . 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 1224 . . 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 232 . 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 3500 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3437   {csn 3986   <.cop 3992   class class class wbr 4401    |-> cmpt 4459    _I cid 4740    |` cres 4951   ` cfv 5527   iota_crio 6161  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   LSpanclspn 17176   Atomscatm 33247   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   trLctrl 34141   DVecHcdvh 35062   DIsoBcdib 35122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-riotaBAD 32943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-tpos 6856  df-undef 6903  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-sca 14374  df-vsca 14375  df-0g 14500  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-mnd 15535  df-grp 15665  df-minusg 15666  df-sbg 15667  df-mgp 16715  df-ur 16727  df-rng 16771  df-oppr 16839  df-dvdsr 16857  df-unit 16858  df-invr 16888  df-dvr 16899  df-drng 16958  df-lmod 17074  df-lss 17138  df-lsp 17177  df-lvec 17308  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482  df-lvols 33483  df-lines 33484  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142  df-tendo 34738  df-edring 34740  df-disoa 35013  df-dvech 35063  df-dib 35123
This theorem is referenced by:  cdlemn5pre  35184
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