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Theorem lkrpssN 29353
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 3168 . . 3  |-  ( ( K `  G ) 
C.  ( K `  H )  <->  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )
2 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  C.  ( K `  H )
)
3 eqid 2283 . . . . . . . . . 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 15859 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
86, 7syl 15 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
9 lkrpss.h . . . . . . . . . 10  |-  ( ph  ->  H  e.  F )
103, 4, 5, 8, 9lkrssv 29286 . . . . . . . . 9  |-  ( ph  ->  ( K `  H
)  C_  ( Base `  W ) )
1110adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  H )  C_  ( Base `  W ) )
12 psssstr 3282 . . . . . . . 8  |-  ( ( ( K `  G
)  C.  ( K `  H )  /\  ( K `  H )  C_  ( Base `  W
) )  ->  ( K `  G )  C.  ( Base `  W
) )
132, 11, 12syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  C.  ( Base `  W ) )
14 pssne 3272 . . . . . . 7  |-  ( ( K `  G ) 
C.  ( Base `  W
)  ->  ( K `  G )  =/=  ( Base `  W ) )
1513, 14syl 15 . . . . . 6  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  =/=  ( Base `  W ) )
161, 15sylan2br 462 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( K `  G )  =/=  ( Base `  W
) )
17 simplr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  C_  ( K `  H )
)
18 eqid 2283 . . . . . . . . . . 11  |-  (LSHyp `  W )  =  (LSHyp `  W )
196ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  W  e.  LVec )
20 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  e.  (LSHyp `  W )
)  ->  ( K `  G )  e.  (LSHyp `  W ) )
21 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  e.  (LSHyp `  W )
)
2210ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  C_  ( Base `  W
) )
23 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  =  ( Base `  W
) )
24 simpllr 735 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  C_  ( K `  H
) )
2523, 24eqsstr3d 3213 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( Base `  W )  C_  ( K `  H ) )
2622, 25eqssd 3196 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  =  ( Base `  W
) )
273, 18, 4, 5, 6, 9lkrshp4 29298 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( K `  H )  =/=  ( Base `  W )  <->  ( K `  H )  e.  (LSHyp `  W ) ) )
2827ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  (
( K `  H
)  =/=  ( Base `  W )  <->  ( K `  H )  e.  (LSHyp `  W ) ) )
2928necon1bbid 2500 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( -.  ( K `  H
)  e.  (LSHyp `  W )  <->  ( K `  H )  =  (
Base `  W )
) )
3026, 29mpbird 223 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  -.  ( K `  H )  e.  (LSHyp `  W
) )
3130pm2.21d 98 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  (
( K `  H
)  e.  (LSHyp `  W )  ->  ( K `  G )  e.  (LSHyp `  W )
) )
3221, 31mpd 14 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  e.  (LSHyp `  W )
)
33 lkrpss.g . . . . . . . . . . . . . 14  |-  ( ph  ->  G  e.  F )
343, 18, 4, 5, 6, 33lkrshpor 29297 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( K `  G )  e.  (LSHyp `  W )  \/  ( K `  G )  =  ( Base `  W
) ) )
3534ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( ( K `  G )  e.  (LSHyp `  W )  \/  ( K `  G
)  =  ( Base `  W ) ) )
3620, 32, 35mpjaodan 761 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  e.  (LSHyp `  W ) )
37 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  H )  e.  (LSHyp `  W ) )
3818, 19, 36, 37lshpcmp 29178 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( ( K `  G )  C_  ( K `  H
)  <->  ( K `  G )  =  ( K `  H ) ) )
3917, 38mpbid 201 . . . . . . . . 9  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  =  ( K `  H ) )
4039ex 423 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  ( ( K `  H )  e.  (LSHyp `  W )  ->  ( K `  G
)  =  ( K `
 H ) ) )
4140necon3ad 2482 . . . . . . 7  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  ( ( K `  G )  =/=  ( K `  H
)  ->  -.  ( K `  H )  e.  (LSHyp `  W )
) )
4241impr 602 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  -.  ( K `  H )  e.  (LSHyp `  W
) )
4327necon1bbid 2500 . . . . . . 7  |-  ( ph  ->  ( -.  ( K `
 H )  e.  (LSHyp `  W )  <->  ( K `  H )  =  ( Base `  W
) ) )
4443adantr 451 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( -.  ( K `  H
)  e.  (LSHyp `  W )  <->  ( K `  H )  =  (
Base `  W )
) )
4542, 44mpbid 201 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( K `  H )  =  ( Base `  W
) )
4616, 45jca 518 . . . 4  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  (
( K `  G
)  =/=  ( Base `  W )  /\  ( K `  H )  =  ( Base `  W
) ) )
473, 4, 5, 8, 33lkrssv 29286 . . . . . . 7  |-  ( ph  ->  ( K `  G
)  C_  ( Base `  W ) )
4847adantr 451 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  C_  ( Base `  W
) )
49 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  H )  =  ( Base `  W
) )
5049eqcomd 2288 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( Base `  W )  =  ( K `  H
) )
5148, 50sseqtrd 3214 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  C_  ( K `  H
) )
52 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  =/=  ( Base `  W
) )
5352, 50neeqtrd 2468 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  =/=  ( K `  H
) )
5451, 53jca 518 . . . 4  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  (
( K `  G
)  C_  ( K `  H )  /\  ( K `  G )  =/=  ( K `  H
) ) )
5546, 54impbida 805 . . 3  |-  ( ph  ->  ( ( ( K `
 G )  C_  ( K `  H )  /\  ( K `  G )  =/=  ( K `  H )
)  <->  ( ( K `
 G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) ) )
561, 55syl5bb 248 . 2  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( ( K `  G
)  =/=  ( Base `  W )  /\  ( K `  H )  =  ( Base `  W
) ) ) )
57 lkrpss.d . . . . 5  |-  D  =  (LDual `  W )
58 lkrpss.o . . . . 5  |-  .0.  =  ( 0g `  D )
593, 4, 5, 57, 58, 8, 33lkr0f2 29351 . . . 4  |-  ( ph  ->  ( ( K `  G )  =  (
Base `  W )  <->  G  =  .0.  ) )
6059necon3bid 2481 . . 3  |-  ( ph  ->  ( ( K `  G )  =/=  ( Base `  W )  <->  G  =/=  .0.  ) )
613, 4, 5, 57, 58, 8, 9lkr0f2 29351 . . 3  |-  ( ph  ->  ( ( K `  H )  =  (
Base `  W )  <->  H  =  .0.  ) )
6260, 61anbi12d 691 . 2  |-  ( ph  ->  ( ( ( K `
 G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
)  <->  ( G  =/= 
.0.  /\  H  =  .0.  ) ) )
6356, 62bitrd 244 1  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  .0.  /\  H  =  .0.  )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152    C. wpss 3153   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LVecclvec 15855  LSHypclsh 29165  LFnlclfn 29247  LKerclk 29275  LDualcld 29313
This theorem is referenced by:  lkrss2N  29359  lkreqN  29360
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-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-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-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-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  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-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lshyp 29167  df-lfl 29248  df-lkr 29276  df-ldual 29314
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