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Theorem cdlemn4 34218
Description: Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
cdlemn4.b  |-  B  =  ( Base `  K
)
cdlemn4.l  |-  .<_  =  ( le `  K )
cdlemn4.a  |-  A  =  ( Atoms `  K )
cdlemn4.p  |-  P  =  ( ( oc `  K ) `  W
)
cdlemn4.h  |-  H  =  ( LHyp `  K
)
cdlemn4.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn4.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
cdlemn4.u  |-  U  =  ( ( DVecH `  K
) `  W )
cdlemn4.f  |-  F  =  ( iota_ h  e.  T  ( h `  P
)  =  Q )
cdlemn4.g  |-  G  =  ( iota_ h  e.  T  ( h `  P
)  =  R )
cdlemn4.j  |-  J  =  ( iota_ h  e.  T  ( h `  Q
)  =  R )
cdlemn4.s  |-  .+  =  ( +g  `  U )
Assertion
Ref Expression
cdlemn4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  <. G , 
(  _I  |`  T )
>.  =  ( <. F ,  (  _I  |`  T )
>.  .+  <. J ,  O >. ) )
Distinct variable groups:    .<_ , h    A, h    B, h    h, H   
h, K    P, h    Q, h    R, h    T, h   
h, W
Allowed substitution hints:    .+ ( h)    U( h)    F( h)    G( h)    J( h)    O( h)

Proof of Theorem cdlemn4
StepHypRef Expression
1 simp1 997 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 cdlemn4.l . . . . . 6  |-  .<_  =  ( le `  K )
3 cdlemn4.a . . . . . 6  |-  A  =  ( Atoms `  K )
4 cdlemn4.h . . . . . 6  |-  H  =  ( LHyp `  K
)
5 cdlemn4.p . . . . . 6  |-  P  =  ( ( oc `  K ) `  W
)
62, 3, 4, 5lhpocnel2 33036 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
71, 6syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp2 998 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
9 cdlemn4.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
10 cdlemn4.f . . . . 5  |-  F  =  ( iota_ h  e.  T  ( h `  P
)  =  Q )
112, 3, 4, 9, 10ltrniotacl 33598 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
121, 7, 8, 11syl3anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  F  e.  T )
13 eqid 2402 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
144, 9, 13tendoidcl 33788 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
151, 14syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W ) )
16 cdlemn4.j . . . 4  |-  J  =  ( iota_ h  e.  T  ( h `  Q
)  =  R )
172, 3, 4, 9, 16ltrniotacl 33598 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  J  e.  T )
18 cdlemn4.b . . . . 5  |-  B  =  ( Base `  K
)
19 cdlemn4.o . . . . 5  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
2018, 4, 9, 13, 19tendo0cl 33809 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  ( (
TEndo `  K ) `  W ) )
211, 20syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  O  e.  ( ( TEndo `  K
) `  W )
)
22 cdlemn4.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
23 eqid 2402 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
24 cdlemn4.s . . . 4  |-  .+  =  ( +g  `  U )
25 eqid 2402 . . . 4  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
264, 9, 13, 22, 23, 24, 25dvhopvadd 34113 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W ) )  /\  ( J  e.  T  /\  O  e.  (
( TEndo `  K ) `  W ) ) )  ->  ( <. F , 
(  _I  |`  T )
>.  .+  <. J ,  O >. )  =  <. ( F  o.  J ) ,  ( (  _I  |`  T ) ( +g  `  (Scalar `  U )
) O ) >.
)
271, 12, 15, 17, 21, 26syl122anc 1239 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( <. F ,  (  _I  |`  T )
>.  .+  <. J ,  O >. )  =  <. ( F  o.  J ) ,  ( (  _I  |`  T ) ( +g  `  (Scalar `  U )
) O ) >.
)
284, 9ltrncom 33757 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  J  e.  T
)  ->  ( F  o.  J )  =  ( J  o.  F ) )
291, 12, 17, 28syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( F  o.  J )  =  ( J  o.  F ) )
30 cdlemn4.g . . . . 5  |-  G  =  ( iota_ h  e.  T  ( h `  P
)  =  R )
312, 3, 5, 4, 9, 10, 30, 16cdlemn3 34217 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( J  o.  F )  =  G )
3229, 31eqtrd 2443 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( F  o.  J )  =  G )
33 eqid 2402 . . . . . . . . 9  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
344, 33, 22, 23dvhsca 34102 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (Scalar `  U )  =  ( ( EDRing `  K ) `  W
) )
3534fveq2d 5853 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  (Scalar `  U ) )  =  ( 0g `  (
( EDRing `  K ) `  W ) ) )
36 eqid 2402 . . . . . . . 8  |-  ( 0g
`  ( ( EDRing `  K ) `  W
) )  =  ( 0g `  ( (
EDRing `  K ) `  W ) )
3718, 4, 9, 33, 19, 36erng0g 34013 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  (
( EDRing `  K ) `  W ) )  =  O )
3835, 37eqtrd 2443 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  (Scalar `  U ) )  =  O )
391, 38syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( 0g `  (Scalar `  U )
)  =  O )
4039oveq2d 6294 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( (  _I  |`  T ) ( +g  `  (Scalar `  U ) ) ( 0g `  (Scalar `  U ) ) )  =  ( (  _I  |`  T ) ( +g  `  (Scalar `  U )
) O ) )
414, 33erngdv 34012 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
42 drnggrp 17724 . . . . . . . 8  |-  ( ( ( EDRing `  K ) `  W )  e.  DivRing  -> 
( ( EDRing `  K
) `  W )  e.  Grp )
4341, 42syl 17 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  Grp )
4434, 43eqeltrd 2490 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (Scalar `  U )  e.  Grp )
451, 44syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  (Scalar `  U
)  e.  Grp )
46 eqid 2402 . . . . . . . 8  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
474, 13, 22, 23, 46dvhbase 34103 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
481, 47syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( Base `  (Scalar `  U )
)  =  ( (
TEndo `  K ) `  W ) )
4915, 48eleqtrrd 2493 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  (  _I  |`  T )  e.  (
Base `  (Scalar `  U
) ) )
50 eqid 2402 . . . . . 6  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
5146, 25, 50grprid 16405 . . . . 5  |-  ( ( (Scalar `  U )  e.  Grp  /\  (  _I  |`  T )  e.  (
Base `  (Scalar `  U
) ) )  -> 
( (  _I  |`  T ) ( +g  `  (Scalar `  U ) ) ( 0g `  (Scalar `  U ) ) )  =  (  _I  |`  T ) )
5245, 49, 51syl2anc 659 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( (  _I  |`  T ) ( +g  `  (Scalar `  U ) ) ( 0g `  (Scalar `  U ) ) )  =  (  _I  |`  T ) )
5340, 52eqtr3d 2445 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( (  _I  |`  T ) ( +g  `  (Scalar `  U ) ) O )  =  (  _I  |`  T ) )
5432, 53opeq12d 4167 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  <. ( F  o.  J ) ,  ( (  _I  |`  T ) ( +g  `  (Scalar `  U ) ) O ) >.  =  <. G ,  (  _I  |`  T )
>. )
5527, 54eqtr2d 2444 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  <. G , 
(  _I  |`  T )
>.  =  ( <. F ,  (  _I  |`  T )
>.  .+  <. J ,  O >. ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   <.cop 3978   class class class wbr 4395    |-> cmpt 4453    _I cid 4733    |` cres 4825    o. ccom 4827   ` cfv 5569   iota_crio 6239  (class class class)co 6278   Basecbs 14841   +g cplusg 14909  Scalarcsca 14912   lecple 14916   occoc 14917   0gc0g 15054   Grpcgrp 16377   DivRingcdr 17716   Atomscatm 32281   HLchlt 32368   LHypclh 33001   LTrncltrn 33118   TEndoctendo 33771   EDRingcedring 33772   DVecHcdvh 34098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-riotaBAD 31977
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-tpos 6958  df-undef 7005  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-sca 14925  df-vsca 14926  df-0g 15056  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-mgp 17462  df-ur 17474  df-ring 17520  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641  df-dvr 17652  df-drng 17718  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516  df-lvols 32517  df-lines 32518  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177  df-tendo 33774  df-edring 33776  df-dvech 34099
This theorem is referenced by:  cdlemn4a  34219
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