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Theorem lclkrlem2x 31720
Description: Lemma for lclkr 31723. Eliminate by cases the hypotheses of lclkrlem2u 31717, lclkrlem2u 31717 and lclkrlem2w 31719. (Contributed by NM, 18-Jan-2015.)
Hypotheses
Ref Expression
lclkrlem2x.l  |-  L  =  (LKer `  U )
lclkrlem2x.h  |-  H  =  ( LHyp `  K
)
lclkrlem2x.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lclkrlem2x.u  |-  U  =  ( ( DVecH `  K
) `  W )
lclkrlem2x.v  |-  V  =  ( Base `  U
)
lclkrlem2x.f  |-  F  =  (LFnl `  U )
lclkrlem2x.d  |-  D  =  (LDual `  U )
lclkrlem2x.p  |-  .+  =  ( +g  `  D )
lclkrlem2x.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lclkrlem2x.x  |-  ( ph  ->  X  e.  V )
lclkrlem2x.y  |-  ( ph  ->  Y  e.  V )
lclkrlem2x.e  |-  ( ph  ->  E  e.  F )
lclkrlem2x.g  |-  ( ph  ->  G  e.  F )
lclkrlem2x.le  |-  ( ph  ->  ( L `  E
)  =  (  ._|_  `  { X } ) )
lclkrlem2x.lg  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { Y } ) )
Assertion
Ref Expression
lclkrlem2x  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )

Proof of Theorem lclkrlem2x
StepHypRef Expression
1 df-ne 2448 . . 3  |-  ( ( ( E  .+  G
) `  X )  =/=  ( 0g `  (Scalar `  U ) )  <->  -.  (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) ) )
2 lclkrlem2x.v . . . 4  |-  V  =  ( Base `  U
)
3 eqid 2283 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
4 eqid 2283 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
5 eqid 2283 . . . 4  |-  ( .r
`  (Scalar `  U )
)  =  ( .r
`  (Scalar `  U )
)
6 eqid 2283 . . . 4  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
7 eqid 2283 . . . 4  |-  ( invr `  (Scalar `  U )
)  =  ( invr `  (Scalar `  U )
)
8 eqid 2283 . . . 4  |-  ( -g `  U )  =  (
-g `  U )
9 lclkrlem2x.f . . . 4  |-  F  =  (LFnl `  U )
10 lclkrlem2x.d . . . 4  |-  D  =  (LDual `  U )
11 lclkrlem2x.p . . . 4  |-  .+  =  ( +g  `  D )
12 lclkrlem2x.x . . . . 5  |-  ( ph  ->  X  e.  V )
1312adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  X  e.  V )
14 lclkrlem2x.y . . . . 5  |-  ( ph  ->  Y  e.  V )
1514adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  Y  e.  V )
16 lclkrlem2x.e . . . . 5  |-  ( ph  ->  E  e.  F )
1716adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  E  e.  F )
18 lclkrlem2x.g . . . . 5  |-  ( ph  ->  G  e.  F )
1918adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  G  e.  F )
20 eqid 2283 . . . 4  |-  ( LSpan `  U )  =  (
LSpan `  U )
21 lclkrlem2x.l . . . 4  |-  L  =  (LKer `  U )
22 lclkrlem2x.h . . . 4  |-  H  =  ( LHyp `  K
)
23 lclkrlem2x.o . . . 4  |-  ._|_  =  ( ( ocH `  K
) `  W )
24 lclkrlem2x.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
25 eqid 2283 . . . 4  |-  ( LSSum `  U )  =  (
LSSum `  U )
26 lclkrlem2x.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2726adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
28 lclkrlem2x.le . . . . 5  |-  ( ph  ->  ( L `  E
)  =  (  ._|_  `  { X } ) )
2928adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( L `  E
)  =  (  ._|_  `  { X } ) )
30 lclkrlem2x.lg . . . . 5  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { Y } ) )
3130adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( L `  G
)  =  (  ._|_  `  { Y } ) )
32 simpr 447 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( ( E  .+  G ) `  X
)  =/=  ( 0g
`  (Scalar `  U )
) )
332, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 20, 21, 22, 23, 24, 25, 27, 29, 31, 32lclkrlem2u 31717 . . 3  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
(  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )
341, 33sylan2br 462 . 2  |-  ( (
ph  /\  -.  (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) ) )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  ( L `
 ( E  .+  G ) ) )
35 df-ne 2448 . . 3  |-  ( ( ( E  .+  G
) `  Y )  =/=  ( 0g `  (Scalar `  U ) )  <->  -.  (
( E  .+  G
) `  Y )  =  ( 0g `  (Scalar `  U ) ) )
3612adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  X  e.  V )
3714adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  Y  e.  V )
3816adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  E  e.  F )
3918adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  G  e.  F )
4026adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4128adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( L `  E
)  =  (  ._|_  `  { X } ) )
4230adantr 451 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( L `  G
)  =  (  ._|_  `  { Y } ) )
43 simpr 447 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( ( E  .+  G ) `  Y
)  =/=  ( 0g
`  (Scalar `  U )
) )
442, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 40, 41, 42, 43lclkrlem2t 31716 . . 3  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
(  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )
4535, 44sylan2br 462 . 2  |-  ( (
ph  /\  -.  (
( E  .+  G
) `  Y )  =  ( 0g `  (Scalar `  U ) ) )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  ( L `
 ( E  .+  G ) ) )
4612adantr 451 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  X  e.  V )
4714adantr 451 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  Y  e.  V )
4816adantr 451 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  E  e.  F )
4918adantr 451 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  G  e.  F )
5026adantr 451 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
5128adantr 451 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  ( L `  E )  =  ( 
._|_  `  { X }
) )
5230adantr 451 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  ( L `  G )  =  ( 
._|_  `  { Y }
) )
53 simprl 732 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  ( ( E  .+  G ) `  X )  =  ( 0g `  (Scalar `  U ) ) )
54 simprr 733 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  ( ( E  .+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) )
552, 3, 4, 5, 6, 7, 8, 9, 10, 11, 46, 47, 48, 49, 20, 21, 22, 23, 24, 25, 50, 51, 52, 53, 54lclkrlem2w 31719 . 2  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  ( L `
 ( E  .+  G ) ) )
5634, 45, 55pm2.61dda 768 1  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   -gcsg 14365   LSSumclsm 14945   invrcinvr 15453   LSpanclspn 15728  LFnlclfn 29247  LKerclk 29275  LDualcld 29313   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268   ocHcoch 31537
This theorem is referenced by:  lclkrlem2y  31721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209  df-lfl 29248  df-lkr 29276  df-ldual 29314  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tgrp 30932  df-tendo 30944  df-edring 30946  df-dveca 31192  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585
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