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Theorem lclkrlem2x 31696
Description: Lemma for lclkr 31699. Eliminate by cases the hypotheses of lclkrlem2u 31693, lclkrlem2u 31693 and lclkrlem2w 31695. (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 2545 . . 3  |-  ( ( ( E  .+  G
) `  X )  =/=  ( 0g `  (Scalar `  U ) )  <->  -.  (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) ) )
2 lclkrlem2x.v . . . 4  |-  V  =  ( Base `  U
)
3 eqid 2380 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
4 eqid 2380 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
5 eqid 2380 . . . 4  |-  ( .r
`  (Scalar `  U )
)  =  ( .r
`  (Scalar `  U )
)
6 eqid 2380 . . . 4  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
7 eqid 2380 . . . 4  |-  ( invr `  (Scalar `  U )
)  =  ( invr `  (Scalar `  U )
)
8 eqid 2380 . . . 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 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  X  e.  V )
14 lclkrlem2x.y . . . . 5  |-  ( ph  ->  Y  e.  V )
1514adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  Y  e.  V )
16 lclkrlem2x.e . . . . 5  |-  ( ph  ->  E  e.  F )
1716adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  E  e.  F )
18 lclkrlem2x.g . . . . 5  |-  ( ph  ->  G  e.  F )
1918adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  G  e.  F )
20 eqid 2380 . . . 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 2380 . . . 4  |-  ( LSSum `  U )  =  (
LSSum `  U )
26 lclkrlem2x.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2726adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
28 lclkrlem2x.le . . . . 5  |-  ( ph  ->  ( L `  E
)  =  (  ._|_  `  { X } ) )
2928adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( L `  E
)  =  (  ._|_  `  { X } ) )
30 lclkrlem2x.lg . . . . 5  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { Y } ) )
3130adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( L `  G
)  =  (  ._|_  `  { Y } ) )
32 simpr 448 . . . 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 31693 . . 3  |-  ( (
ph  /\  ( ( E  .+  G ) `  X )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
(  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )
341, 33sylan2br 463 . 2  |-  ( (
ph  /\  -.  (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) ) )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  ( L `
 ( E  .+  G ) ) )
35 df-ne 2545 . . 3  |-  ( ( ( E  .+  G
) `  Y )  =/=  ( 0g `  (Scalar `  U ) )  <->  -.  (
( E  .+  G
) `  Y )  =  ( 0g `  (Scalar `  U ) ) )
3612adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  X  e.  V )
3714adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  Y  e.  V )
3816adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  E  e.  F )
3918adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  ->  G  e.  F )
4026adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4128adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( L `  E
)  =  (  ._|_  `  { X } ) )
4230adantr 452 . . . 4  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
( L `  G
)  =  (  ._|_  `  { Y } ) )
43 simpr 448 . . . 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 31692 . . 3  |-  ( (
ph  /\  ( ( E  .+  G ) `  Y )  =/=  ( 0g `  (Scalar `  U
) ) )  -> 
(  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )
4535, 44sylan2br 463 . 2  |-  ( (
ph  /\  -.  (
( E  .+  G
) `  Y )  =  ( 0g `  (Scalar `  U ) ) )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  ( L `
 ( E  .+  G ) ) )
4612adantr 452 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  X  e.  V )
4714adantr 452 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  Y  e.  V )
4816adantr 452 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  E  e.  F )
4918adantr 452 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  G  e.  F )
5026adantr 452 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
5128adantr 452 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  ( L `  E )  =  ( 
._|_  `  { X }
) )
5230adantr 452 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  ( L `  G )  =  ( 
._|_  `  { Y }
) )
53 simprl 733 . . 3  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  ( ( E  .+  G ) `  X )  =  ( 0g `  (Scalar `  U ) ) )
54 simprr 734 . . 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 31695 . 2  |-  ( (
ph  /\  ( (
( E  .+  G
) `  X )  =  ( 0g `  (Scalar `  U ) )  /\  ( ( E 
.+  G ) `  Y )  =  ( 0g `  (Scalar `  U ) ) ) )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  ( L `
 ( E  .+  G ) ) )
5634, 45, 55pm2.61dda 769 1  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   {csn 3750   ` cfv 5387  (class class class)co 6013   Basecbs 13389   +g cplusg 13449   .rcmulr 13450  Scalarcsca 13452   .scvsca 13453   0gc0g 13643   -gcsg 14608   LSSumclsm 15188   invrcinvr 15696   LSpanclspn 15967  LFnlclfn 29223  LKerclk 29251  LDualcld 29289   HLchlt 29516   LHypclh 30149   DVecHcdvh 31244   ocHcoch 31513
This theorem is referenced by:  lclkrlem2y  31697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-tpos 6408  df-undef 6472  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-0g 13647  df-mre 13731  df-mrc 13732  df-acs 13734  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-mnd 14610  df-submnd 14659  df-grp 14732  df-minusg 14733  df-sbg 14734  df-subg 14861  df-cntz 15036  df-oppg 15062  df-lsm 15190  df-cmn 15334  df-abl 15335  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-invr 15697  df-dvr 15708  df-drng 15757  df-lmod 15872  df-lss 15929  df-lsp 15968  df-lvec 16095  df-lsatoms 29142  df-lshyp 29143  df-lcv 29185  df-lfl 29224  df-lkr 29252  df-ldual 29290  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-llines 29663  df-lplanes 29664  df-lvols 29665  df-lines 29666  df-psubsp 29668  df-pmap 29669  df-padd 29961  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324  df-tgrp 30908  df-tendo 30920  df-edring 30922  df-dveca 31168  df-disoa 31195  df-dvech 31245  df-dib 31305  df-dic 31339  df-dih 31395  df-doch 31514  df-djh 31561
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