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Theorem cdlemn6 35999
Description: Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-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 )
Assertion
Ref Expression
cdlemn6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( <. ( s `  F ) ,  s
>.  .+  <. g ,  O >. )  =  <. (
( s `  F
)  o.  g ) ,  s >. )
Distinct variable groups:    .<_ , h    A, h    B, h    h, H   
h, K    T, h    P, h    Q, h    h, W
Allowed substitution hints:    A( g, s)    B( g, s)    P( g, s)    .+ ( g, h, s)    Q( g, s)    R( g, h, s)    T( g, s)    U( g, h, s)    E( g, h, s)    F( g, h, s)    H( g, s)    K( g, s)    .<_ ( g, s)    O( g, h, s)    W( g, s)

Proof of Theorem cdlemn6
Dummy variables  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp3l 1024 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
s  e.  E )
3 cdlemn8.l . . . . . . 7  |-  .<_  =  ( le `  K )
4 cdlemn8.a . . . . . . 7  |-  A  =  ( Atoms `  K )
5 cdlemn8.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
6 cdlemn8.p . . . . . . 7  |-  P  =  ( ( oc `  K ) `  W
)
73, 4, 5, 6lhpocnel2 34815 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
81, 7syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
9 simp2l 1022 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
10 cdlemn8.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
11 cdlemn8.f . . . . . 6  |-  F  =  ( iota_ h  e.  T  ( h `  P
)  =  Q )
123, 4, 5, 10, 11ltrniotacl 35375 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
131, 8, 9, 12syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  ->  F  e.  T )
14 cdlemn8.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
155, 10, 14tendocl 35563 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( s `  F )  e.  T
)
161, 2, 13, 15syl3anc 1228 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( s `  F
)  e.  T )
17 simp3r 1025 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
g  e.  T )
18 cdlemn8.b . . . . 5  |-  B  =  ( Base `  K
)
19 cdlemn8.o . . . . 5  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
2018, 5, 10, 14, 19tendo0cl 35586 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
211, 20syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  ->  O  e.  E )
22 cdlemn8.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
23 eqid 2467 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
24 cdlemn8.s . . . 4  |-  .+  =  ( +g  `  U )
25 eqid 2467 . . . 4  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
265, 10, 14, 22, 23, 24, 25dvhopvadd 35890 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( s `
 F )  e.  T  /\  s  e.  E )  /\  (
g  e.  T  /\  O  e.  E )
)  ->  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )  =  <. ( ( s `
 F )  o.  g ) ,  ( s ( +g  `  (Scalar `  U ) ) O ) >. )
271, 16, 2, 17, 21, 26syl122anc 1237 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( <. ( s `  F ) ,  s
>.  .+  <. g ,  O >. )  =  <. (
( s `  F
)  o.  g ) ,  ( s ( +g  `  (Scalar `  U ) ) O ) >. )
28 eqid 2467 . . . . . . 7  |-  ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  ( u `  h ) ) ) )  =  ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  ( u `  h ) ) ) )
295, 10, 14, 22, 23, 28, 25dvhfplusr 35881 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  (Scalar `  U ) )  =  ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  (
u `  h )
) ) ) )
301, 29syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( +g  `  (Scalar `  U ) )  =  ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  (
u `  h )
) ) ) )
3130oveqd 6299 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( s ( +g  `  (Scalar `  U )
) O )  =  ( s ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  ( u `  h ) ) ) ) O ) )
3218, 5, 10, 14, 19, 28tendo0plr 35588 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E
)  ->  ( s
( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  (
u `  h )
) ) ) O )  =  s )
331, 2, 32syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( s ( t  e.  E ,  u  e.  E  |->  ( h  e.  T  |->  ( ( t `  h )  o.  ( u `  h ) ) ) ) O )  =  s )
3431, 33eqtrd 2508 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( s ( +g  `  (Scalar `  U )
) O )  =  s )
3534opeq2d 4220 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  ->  <. ( ( s `  F )  o.  g
) ,  ( s ( +g  `  (Scalar `  U ) ) O ) >.  =  <. ( ( s `  F
)  o.  g ) ,  s >. )
3627, 35eqtrd 2508 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( <. ( s `  F ) ,  s
>.  .+  <. g ,  O >. )  =  <. (
( s `  F
)  o.  g ) ,  s >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    |` cres 5001    o. ccom 5003   ` cfv 5586   iota_crio 6242  (class class class)co 6282    |-> cmpt2 6284   Basecbs 14486   +g cplusg 14551  Scalarcsca 14554   lecple 14558   occoc 14559   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548   DVecHcdvh 35875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-undef 6999  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tendo 35551  df-edring 35553  df-dvech 35876
This theorem is referenced by:  cdlemn7  36000  dihordlem6  36010
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