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Theorem lkrpssN 29898
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 3328 . . 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 2435 . . . . . . . . . 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 16170 . . . . . . . . . . 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 29831 . . . . . . . . 9  |-  ( ph  ->  ( K `  H
)  C_  ( Base `  W ) )
1110adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  H )  C_  ( Base `  W ) )
122, 11psssstrd 3448 . . . . . . 7  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  C.  ( Base `  W ) )
1312pssned 3437 . . . . . 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 2435 . . . . . . . . . . 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 3375 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( Base `  W )  C_  ( K `  H ) )
2420, 23eqssd 3357 . . . . . . . . . . . . . 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 29843 . . . . . . . . . . . . . . . 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 2652 . . . . . . . . . . . . . 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 29842 . . . . . . . . . . . . 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 29723 . . . . . . . . . 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 2634 . . . . . . 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 2652 . . . . . . 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 29831 . . . . . . 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 2440 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( Base `  W )  =  ( K `  H
) )
4845, 47sseqtrd 3376 . . . . 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 2620 . . . . 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 29896 . . . 4  |-  ( ph  ->  ( ( K `  G )  =  (
Base `  W )  <->  G  =  .0.  ) )
5756necon3bid 2633 . . 3  |-  ( ph  ->  ( ( K `  G )  =/=  ( Base `  W )  <->  G  =/=  .0.  ) )
583, 4, 5, 54, 55, 8, 9lkr0f2 29896 . . 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 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312    C. wpss 3313   ` cfv 5446   Basecbs 13461   0gc0g 13715   LModclmod 15942   LVecclvec 16166  LSHypclsh 29710  LFnlclfn 29792  LKerclk 29820  LDualcld 29858
This theorem is referenced by:  lkrss2N  29904  lkreqN  29905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lshyp 29712  df-lfl 29793  df-lkr 29821  df-ldual 29859
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