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Theorem lkrpssN 34361
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 3497 . . 3  |-  ( ( K `  G ) 
C.  ( K `  H )  <->  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )
2 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  C.  ( K `  H )
)
3 eqid 2467 . . . . . . . . . 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 17623 . . . . . . . . . . 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 34294 . . . . . . . . 9  |-  ( ph  ->  ( K `  H
)  C_  ( Base `  W ) )
1110adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  H )  C_  ( Base `  W ) )
122, 11psssstrd 3618 . . . . . . 7  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  C.  ( Base `  W ) )
1312pssned 3607 . . . . . 6  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  =/=  ( Base `  W ) )
141, 13sylan2br 476 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( K `  G )  =/=  ( Base `  W
) )
15 simplr 754 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  C_  ( K `  H )
)
16 eqid 2467 . . . . . . . . . . 11  |-  (LSHyp `  W )  =  (LSHyp `  W )
176ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  W  e.  LVec )
18 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  e.  (LSHyp `  W )
)  ->  ( K `  G )  e.  (LSHyp `  W ) )
19 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  e.  (LSHyp `  W )
)
2010ad3antrrr 729 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  C_  ( Base `  W
) )
21 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  =  ( Base `  W
) )
22 simpllr 758 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  C_  ( K `  H
) )
2321, 22eqsstr3d 3544 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( Base `  W )  C_  ( K `  H ) )
2420, 23eqssd 3526 . . . . . . . . . . . . . 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 34306 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( K `  H )  =/=  ( Base `  W )  <->  ( K `  H )  e.  (LSHyp `  W ) ) )
2625ad3antrrr 729 . . . . . . . . . . . . . . 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 2717 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  -.  ( K `  H )  e.  (LSHyp `  W
) )
2919, 28pm2.21dd 174 . . . . . . . . . . . 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 34305 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( K `  G )  e.  (LSHyp `  W )  \/  ( K `  G )  =  ( Base `  W
) ) )
3231ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( ( K `  G )  e.  (LSHyp `  W )  \/  ( K `  G
)  =  ( Base `  W ) ) )
3318, 29, 32mpjaodan 784 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  e.  (LSHyp `  W ) )
34 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  H )  e.  (LSHyp `  W ) )
3516, 17, 33, 34lshpcmp 34186 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( ( K `  G )  C_  ( K `  H
)  <->  ( K `  G )  =  ( K `  H ) ) )
3615, 35mpbid 210 . . . . . . . . 9  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  =  ( K `  H ) )
3736ex 434 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  ( ( K `  H )  e.  (LSHyp `  W )  ->  ( K `  G
)  =  ( K `
 H ) ) )
3837necon3ad 2677 . . . . . . 7  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  ( ( K `  G )  =/=  ( K `  H
)  ->  -.  ( K `  H )  e.  (LSHyp `  W )
) )
3938impr 619 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  -.  ( K `  H )  e.  (LSHyp `  W
) )
4025necon1bbid 2717 . . . . . . 7  |-  ( ph  ->  ( -.  ( K `
 H )  e.  (LSHyp `  W )  <->  ( K `  H )  =  ( Base `  W
) ) )
4140adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( -.  ( K `  H
)  e.  (LSHyp `  W )  <->  ( K `  H )  =  (
Base `  W )
) )
4239, 41mpbid 210 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( K `  H )  =  ( Base `  W
) )
4314, 42jca 532 . . . 4  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  (
( K `  G
)  =/=  ( Base `  W )  /\  ( K `  H )  =  ( Base `  W
) ) )
443, 4, 5, 8, 30lkrssv 34294 . . . . . . 7  |-  ( ph  ->  ( K `  G
)  C_  ( Base `  W ) )
4544adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  C_  ( Base `  W
) )
46 simprr 756 . . . . . . 7  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  H )  =  ( Base `  W
) )
4746eqcomd 2475 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( Base `  W )  =  ( K `  H
) )
4845, 47sseqtrd 3545 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  C_  ( K `  H
) )
49 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  =/=  ( Base `  W
) )
5049, 47neeqtrd 2762 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  =/=  ( K `  H
) )
5148, 50jca 532 . . . 4  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  (
( K `  G
)  C_  ( K `  H )  /\  ( K `  G )  =/=  ( K `  H
) ) )
5243, 51impbida 830 . . 3  |-  ( ph  ->  ( ( ( K `
 G )  C_  ( K `  H )  /\  ( K `  G )  =/=  ( K `  H )
)  <->  ( ( K `
 G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) ) )
531, 52syl5bb 257 . 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 34359 . . . 4  |-  ( ph  ->  ( ( K `  G )  =  (
Base `  W )  <->  G  =  .0.  ) )
5756necon3bid 2725 . . 3  |-  ( ph  ->  ( ( K `  G )  =/=  ( Base `  W )  <->  G  =/=  .0.  ) )
583, 4, 5, 54, 55, 8, 9lkr0f2 34359 . . 3  |-  ( ph  ->  ( ( K `  H )  =  (
Base `  W )  <->  H  =  .0.  ) )
5957, 58anbi12d 710 . 2  |-  ( ph  ->  ( ( ( K `
 G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
)  <->  ( G  =/= 
.0.  /\  H  =  .0.  ) ) )
6053, 59bitrd 253 1  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  .0.  /\  H  =  .0.  )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3481    C. wpss 3482   ` cfv 5594   Basecbs 14507   0gc0g 14712   LModclmod 17383   LVecclvec 17619  LSHypclsh 34173  LFnlclfn 34255  LKerclk 34283  LDualcld 34321
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lshyp 34175  df-lfl 34256  df-lkr 34284  df-ldual 34322
This theorem is referenced by:  lkrss2N  34367  lkreqN  34368
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