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Theorem cdlemn4 34939
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 988 . . 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 33759 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
71, 6syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp2 989 . . . 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 34319 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
121, 7, 8, 11syl3anc 1218 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  F  e.  T )
13 eqid 2443 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
144, 9, 13tendoidcl 34509 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
151, 14syl 16 . . 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 34319 . . 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 34530 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  ( (
TEndo `  K ) `  W ) )
211, 20syl 16 . . 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 2443 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
24 cdlemn4.s . . . 4  |-  .+  =  ( +g  `  U )
25 eqid 2443 . . . 4  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
264, 9, 13, 22, 23, 24, 25dvhopvadd 34834 . . 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 1227 . 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 34478 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  J  e.  T
)  ->  ( F  o.  J )  =  ( J  o.  F ) )
291, 12, 17, 28syl3anc 1218 . . . 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 34938 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( J  o.  F )  =  G )
3229, 31eqtrd 2475 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( F  o.  J )  =  G )
33 eqid 2443 . . . . . . . . 9  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
344, 33, 22, 23dvhsca 34823 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (Scalar `  U )  =  ( ( EDRing `  K ) `  W
) )
3534fveq2d 5716 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  (Scalar `  U ) )  =  ( 0g `  (
( EDRing `  K ) `  W ) ) )
36 eqid 2443 . . . . . . . 8  |-  ( 0g
`  ( ( EDRing `  K ) `  W
) )  =  ( 0g `  ( (
EDRing `  K ) `  W ) )
3718, 4, 9, 33, 19, 36erng0g 34734 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  (
( EDRing `  K ) `  W ) )  =  O )
3835, 37eqtrd 2475 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  (Scalar `  U ) )  =  O )
391, 38syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( 0g `  (Scalar `  U )
)  =  O )
4039oveq2d 6128 . . . 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 34733 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
42 drnggrp 16862 . . . . . . . 8  |-  ( ( ( EDRing `  K ) `  W )  e.  DivRing  -> 
( ( EDRing `  K
) `  W )  e.  Grp )
4341, 42syl 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  Grp )
4434, 43eqeltrd 2517 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (Scalar `  U )  e.  Grp )
451, 44syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  (Scalar `  U
)  e.  Grp )
46 eqid 2443 . . . . . . . 8  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
474, 13, 22, 23, 46dvhbase 34824 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
481, 47syl 16 . . . . . 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 2520 . . . . 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 2443 . . . . . 6  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
5146, 25, 50grprid 15590 . . . . 5  |-  ( ( (Scalar `  U )  e.  Grp  /\  (  _I  |`  T )  e.  (
Base `  (Scalar `  U
) ) )  -> 
( (  _I  |`  T ) ( +g  `  (Scalar `  U ) ) ( 0g `  (Scalar `  U ) ) )  =  (  _I  |`  T ) )
5245, 49, 51syl2anc 661 . . . 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 2477 . . 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 4088 . 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 2476 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   <.cop 3904   class class class wbr 4313    e. cmpt 4371    _I cid 4652    |` cres 4863    o. ccom 4865   ` cfv 5439   iota_crio 6072  (class class class)co 6112   Basecbs 14195   +g cplusg 14259  Scalarcsca 14262   lecple 14266   occoc 14267   0gc0g 14399   Grpcgrp 15431   DivRingcdr 16854   Atomscatm 33004   HLchlt 33091   LHypclh 33724   LTrncltrn 33841   TEndoctendo 34492   EDRingcedring 34493   DVecHcdvh 34819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-riotaBAD 32700
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-undef 6813  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-sca 14275  df-vsca 14276  df-0g 14401  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-clat 15299  df-mnd 15436  df-grp 15566  df-minusg 15567  df-mgp 16614  df-ur 16626  df-rng 16669  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-dvr 16797  df-drng 16856  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-llines 33238  df-lplanes 33239  df-lvols 33240  df-lines 33241  df-psubsp 33243  df-pmap 33244  df-padd 33536  df-lhyp 33728  df-laut 33729  df-ldil 33844  df-ltrn 33845  df-trl 33899  df-tendo 34495  df-edring 34497  df-dvech 34820
This theorem is referenced by:  cdlemn4a  34940
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