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Theorem lkrpssN 29646
Description: Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
lkrpss.f  |-  F  =  (LFnl `  W )
lkrpss.k  |-  K  =  (LKer `  W )
lkrpss.d  |-  D  =  (LDual `  W )
lkrpss.o  |-  .0.  =  ( 0g `  D )
lkrpss.w  |-  ( ph  ->  W  e.  LVec )
lkrpss.g  |-  ( ph  ->  G  e.  F )
lkrpss.h  |-  ( ph  ->  H  e.  F )
Assertion
Ref Expression
lkrpssN  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  .0.  /\  H  =  .0.  )
) )

Proof of Theorem lkrpssN
StepHypRef Expression
1 df-pss 3296 . . 3  |-  ( ( K `  G ) 
C.  ( K `  H )  <->  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )
2 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  C.  ( K `  H )
)
3 eqid 2404 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
4 lkrpss.f . . . . . . . . . 10  |-  F  =  (LFnl `  W )
5 lkrpss.k . . . . . . . . . 10  |-  K  =  (LKer `  W )
6 lkrpss.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LVec )
7 lveclmod 16133 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
86, 7syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
9 lkrpss.h . . . . . . . . . 10  |-  ( ph  ->  H  e.  F )
103, 4, 5, 8, 9lkrssv 29579 . . . . . . . . 9  |-  ( ph  ->  ( K `  H
)  C_  ( Base `  W ) )
1110adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  H )  C_  ( Base `  W ) )
122, 11psssstrd 3416 . . . . . . 7  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  C.  ( Base `  W ) )
1312pssned 3405 . . . . . 6  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  =/=  ( Base `  W ) )
141, 13sylan2br 463 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( K `  G )  =/=  ( Base `  W
) )
15 simplr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  C_  ( K `  H )
)
16 eqid 2404 . . . . . . . . . . 11  |-  (LSHyp `  W )  =  (LSHyp `  W )
176ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  W  e.  LVec )
18 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  e.  (LSHyp `  W )
)  ->  ( K `  G )  e.  (LSHyp `  W ) )
19 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  e.  (LSHyp `  W )
)
2010ad3antrrr 711 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  C_  ( Base `  W
) )
21 simpr 448 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  =  ( Base `  W
) )
22 simpllr 736 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  C_  ( K `  H
) )
2321, 22eqsstr3d 3343 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( Base `  W )  C_  ( K `  H ) )
2420, 23eqssd 3325 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  =  ( Base `  W
) )
253, 16, 4, 5, 6, 9lkrshp4 29591 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( K `  H )  =/=  ( Base `  W )  <->  ( K `  H )  e.  (LSHyp `  W ) ) )
2625ad3antrrr 711 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  (
( K `  H
)  =/=  ( Base `  W )  <->  ( K `  H )  e.  (LSHyp `  W ) ) )
2726necon1bbid 2621 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( -.  ( K `  H
)  e.  (LSHyp `  W )  <->  ( K `  H )  =  (
Base `  W )
) )
2824, 27mpbird 224 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  -.  ( K `  H )  e.  (LSHyp `  W
) )
2919, 28pm2.21dd 101 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  e.  (LSHyp `  W )
)
30 lkrpss.g . . . . . . . . . . . . . 14  |-  ( ph  ->  G  e.  F )
313, 16, 4, 5, 6, 30lkrshpor 29590 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( K `  G )  e.  (LSHyp `  W )  \/  ( K `  G )  =  ( Base `  W
) ) )
3231ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( ( K `  G )  e.  (LSHyp `  W )  \/  ( K `  G
)  =  ( Base `  W ) ) )
3318, 29, 32mpjaodan 762 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  e.  (LSHyp `  W ) )
34 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  H )  e.  (LSHyp `  W ) )
3516, 17, 33, 34lshpcmp 29471 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( ( K `  G )  C_  ( K `  H
)  <->  ( K `  G )  =  ( K `  H ) ) )
3615, 35mpbid 202 . . . . . . . . 9  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  =  ( K `  H ) )
3736ex 424 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  ( ( K `  H )  e.  (LSHyp `  W )  ->  ( K `  G
)  =  ( K `
 H ) ) )
3837necon3ad 2603 . . . . . . 7  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  ( ( K `  G )  =/=  ( K `  H
)  ->  -.  ( K `  H )  e.  (LSHyp `  W )
) )
3938impr 603 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  -.  ( K `  H )  e.  (LSHyp `  W
) )
4025necon1bbid 2621 . . . . . . 7  |-  ( ph  ->  ( -.  ( K `
 H )  e.  (LSHyp `  W )  <->  ( K `  H )  =  ( Base `  W
) ) )
4140adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( -.  ( K `  H
)  e.  (LSHyp `  W )  <->  ( K `  H )  =  (
Base `  W )
) )
4239, 41mpbid 202 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( K `  H )  =  ( Base `  W
) )
4314, 42jca 519 . . . 4  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  (
( K `  G
)  =/=  ( Base `  W )  /\  ( K `  H )  =  ( Base `  W
) ) )
443, 4, 5, 8, 30lkrssv 29579 . . . . . . 7  |-  ( ph  ->  ( K `  G
)  C_  ( Base `  W ) )
4544adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  C_  ( Base `  W
) )
46 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  H )  =  ( Base `  W
) )
4746eqcomd 2409 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( Base `  W )  =  ( K `  H
) )
4845, 47sseqtrd 3344 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  C_  ( K `  H
) )
49 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  =/=  ( Base `  W
) )
5049, 47neeqtrd 2589 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  =/=  ( K `  H
) )
5148, 50jca 519 . . . 4  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  (
( K `  G
)  C_  ( K `  H )  /\  ( K `  G )  =/=  ( K `  H
) ) )
5243, 51impbida 806 . . 3  |-  ( ph  ->  ( ( ( K `
 G )  C_  ( K `  H )  /\  ( K `  G )  =/=  ( K `  H )
)  <->  ( ( K `
 G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) ) )
531, 52syl5bb 249 . 2  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( ( K `  G
)  =/=  ( Base `  W )  /\  ( K `  H )  =  ( Base `  W
) ) ) )
54 lkrpss.d . . . . 5  |-  D  =  (LDual `  W )
55 lkrpss.o . . . . 5  |-  .0.  =  ( 0g `  D )
563, 4, 5, 54, 55, 8, 30lkr0f2 29644 . . . 4  |-  ( ph  ->  ( ( K `  G )  =  (
Base `  W )  <->  G  =  .0.  ) )
5756necon3bid 2602 . . 3  |-  ( ph  ->  ( ( K `  G )  =/=  ( Base `  W )  <->  G  =/=  .0.  ) )
583, 4, 5, 54, 55, 8, 9lkr0f2 29644 . . 3  |-  ( ph  ->  ( ( K `  H )  =  (
Base `  W )  <->  H  =  .0.  ) )
5957, 58anbi12d 692 . 2  |-  ( ph  ->  ( ( ( K `
 G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
)  <->  ( G  =/= 
.0.  /\  H  =  .0.  ) ) )
6053, 59bitrd 245 1  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  .0.  /\  H  =  .0.  )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567    C_ wss 3280    C. wpss 3281   ` cfv 5413   Basecbs 13424   0gc0g 13678   LModclmod 15905   LVecclvec 16129  LSHypclsh 29458  LFnlclfn 29540  LKerclk 29568  LDualcld 29606
This theorem is referenced by:  lkrss2N  29652  lkreqN  29653
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lshyp 29460  df-lfl 29541  df-lkr 29569  df-ldual 29607
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