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Theorem cdlemn9 34685
Description: Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
Hypotheses
Ref Expression
cdlemn8.b  |-  B  =  ( Base `  K
)
cdlemn8.l  |-  .<_  =  ( le `  K )
cdlemn8.a  |-  A  =  ( Atoms `  K )
cdlemn8.h  |-  H  =  ( LHyp `  K
)
cdlemn8.p  |-  P  =  ( ( oc `  K ) `  W
)
cdlemn8.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
cdlemn8.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn8.e  |-  E  =  ( ( TEndo `  K
) `  W )
cdlemn8.u  |-  U  =  ( ( DVecH `  K
) `  W )
cdlemn8.s  |-  .+  =  ( +g  `  U )
cdlemn8.f  |-  F  =  ( iota_ h  e.  T  ( h `  P
)  =  Q )
cdlemn8.g  |-  G  =  ( iota_ h  e.  T  ( h `  P
)  =  R )
Assertion
Ref Expression
cdlemn9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  (
g `  Q )  =  R )
Distinct variable groups:    .<_ , h    A, h    B, h    h, H   
h, K    T, h    P, h    Q, h    h, W    R, h
Allowed substitution hints:    A( g, s)    B( g, s)    P( g, s)    .+ ( g, h, s)    Q( g, s)    R( g, s)    T( g, s)    U( g, h, s)    E( g, h, s)    F( g, h, s)    G( g, h, s)    H( g, s)    K( g, s)    .<_ ( g, s)    O( g, h, s)    W( g, s)

Proof of Theorem cdlemn9
StepHypRef Expression
1 cdlemn8.b . . . 4  |-  B  =  ( Base `  K
)
2 cdlemn8.l . . . 4  |-  .<_  =  ( le `  K )
3 cdlemn8.a . . . 4  |-  A  =  ( Atoms `  K )
4 cdlemn8.h . . . 4  |-  H  =  ( LHyp `  K
)
5 cdlemn8.p . . . 4  |-  P  =  ( ( oc `  K ) `  W
)
6 cdlemn8.o . . . 4  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
7 cdlemn8.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemn8.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
9 cdlemn8.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
10 cdlemn8.s . . . 4  |-  .+  =  ( +g  `  U )
11 cdlemn8.f . . . 4  |-  F  =  ( iota_ h  e.  T  ( h `  P
)  =  Q )
12 cdlemn8.g . . . 4  |-  G  =  ( iota_ h  e.  T  ( h `  P
)  =  R )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemn8 34684 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  g  =  ( G  o.  `' F ) )
1413fveq1d 5827 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  (
g `  Q )  =  ( ( G  o.  `' F ) `
 Q ) )
15 simp1 1005 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
162, 3, 4, 5lhpocnel2 33496 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
17163ad2ant1 1026 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
18 simp2l 1031 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
192, 3, 4, 7, 11ltrniotacl 34058 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
2015, 17, 18, 19syl3anc 1264 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  F  e.  T )
211, 4, 7ltrn1o 33601 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
2215, 20, 21syl2anc 665 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  F : B -1-1-onto-> B )
23 f1ocnv 5786 . . . 4  |-  ( F : B -1-1-onto-> B  ->  `' F : B -1-1-onto-> B )
24 f1of 5774 . . . 4  |-  ( `' F : B -1-1-onto-> B  ->  `' F : B --> B )
2522, 23, 243syl 18 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  `' F : B --> B )
26 simp2ll 1072 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  Q  e.  A )
271, 3atbase 32767 . . . 4  |-  ( Q  e.  A  ->  Q  e.  B )
2826, 27syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  Q  e.  B )
29 fvco3 5902 . . 3  |-  ( ( `' F : B --> B  /\  Q  e.  B )  ->  ( ( G  o.  `' F ) `  Q
)  =  ( G `
 ( `' F `  Q ) ) )
3025, 28, 29syl2anc 665 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  (
( G  o.  `' F ) `  Q
)  =  ( G `
 ( `' F `  Q ) ) )
312, 3, 4, 7, 11ltrniotacnvval 34061 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( `' F `  Q )  =  P )
3215, 17, 18, 31syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( `' F `  Q )  =  P )
3332fveq2d 5829 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( G `  ( `' F `  Q )
)  =  ( G `
 P ) )
34 simp2r 1032 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
352, 3, 4, 7, 12ltrniotaval 34060 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( G `  P )  =  R )
3615, 17, 34, 35syl3anc 1264 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( G `  P )  =  R )
3733, 36eqtrd 2462 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  ( G `  ( `' F `  Q )
)  =  R )
3814, 30, 373eqtrd 2466 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. G , 
(  _I  |`  T )
>.  =  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )
) )  ->  (
g `  Q )  =  R )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   <.cop 3947   class class class wbr 4366    |-> cmpt 4425    _I cid 4706   `'ccnv 4795    |` cres 4798    o. ccom 4800   -->wf 5540   -1-1-onto->wf1o 5543   ` cfv 5544   iota_crio 6210  (class class class)co 6249   Basecbs 15064   +g cplusg 15133   lecple 15140   occoc 15141   Atomscatm 32741   HLchlt 32828   LHypclh 33461   LTrncltrn 33578   TEndoctendo 34231   DVecHcdvh 34558
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 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-riotaBAD 32437
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-iin 4245  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-undef 6975  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-plusg 15146  df-mulr 15147  df-sca 15149  df-vsca 15150  df-preset 16116  df-poset 16134  df-plt 16147  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-p0 16228  df-p1 16229  df-lat 16235  df-clat 16297  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-llines 32975  df-lplanes 32976  df-lvols 32977  df-lines 32978  df-psubsp 32980  df-pmap 32981  df-padd 33273  df-lhyp 33465  df-laut 33466  df-ldil 33581  df-ltrn 33582  df-trl 33637  df-tendo 34234  df-edring 34236  df-dvech 34559
This theorem is referenced by:  cdlemn11pre  34690
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