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Theorem lkrpssN 32641
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 3395 . . 3  |-  ( ( K `  G ) 
C.  ( K `  H )  <->  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )
2 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  C.  ( K `  H )
)
3 eqid 2428 . . . . . . . . . 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 18272 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
86, 7syl 17 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
9 lkrpss.h . . . . . . . . . 10  |-  ( ph  ->  H  e.  F )
103, 4, 5, 8, 9lkrssv 32574 . . . . . . . . 9  |-  ( ph  ->  ( K `  H
)  C_  ( Base `  W ) )
1110adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  H )  C_  ( Base `  W ) )
122, 11psssstrd 3517 . . . . . . 7  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  C.  ( Base `  W ) )
1312pssned 3506 . . . . . 6  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  ( K `  G )  =/=  ( Base `  W ) )
141, 13sylan2br 478 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( K `  G )  =/=  ( Base `  W
) )
15 simplr 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  C_  ( K `  H )
)
16 eqid 2428 . . . . . . . . . . 11  |-  (LSHyp `  W )  =  (LSHyp `  W )
176ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  W  e.  LVec )
18 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  e.  (LSHyp `  W )
)  ->  ( K `  G )  e.  (LSHyp `  W ) )
19 simplr 760 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  e.  (LSHyp `  W )
)
2010ad3antrrr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  H )  C_  ( Base `  W
) )
21 simpr 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  =  ( Base `  W
) )
22 simpllr 767 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( K `  G )  C_  ( K `  H
) )
2321, 22eqsstr3d 3442 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  ( Base `  W )  C_  ( K `  H ) )
2420, 23eqssd 3424 . . . . . . . . . . . . . 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 32586 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( K `  H )  =/=  ( Base `  W )  <->  ( K `  H )  e.  (LSHyp `  W ) ) )
2625ad3antrrr 734 . . . . . . . . . . . . . . 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 2640 . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( K `  G ) 
C_  ( K `  H ) )  /\  ( K `  H )  e.  (LSHyp `  W
) )  /\  ( K `  G )  =  ( Base `  W
) )  ->  -.  ( K `  H )  e.  (LSHyp `  W
) )
2919, 28pm2.21dd 177 . . . . . . . . . . . 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 32585 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( K `  G )  e.  (LSHyp `  W )  \/  ( K `  G )  =  ( Base `  W
) ) )
3231ad2antrr 730 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( ( K `  G )  e.  (LSHyp `  W )  \/  ( K `  G
)  =  ( Base `  W ) ) )
3318, 29, 32mpjaodan 793 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  e.  (LSHyp `  W ) )
34 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  H )  e.  (LSHyp `  W ) )
3516, 17, 33, 34lshpcmp 32466 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( ( K `  G )  C_  ( K `  H
)  <->  ( K `  G )  =  ( K `  H ) ) )
3615, 35mpbid 213 . . . . . . . . 9  |-  ( ( ( ph  /\  ( K `  G )  C_  ( K `  H
) )  /\  ( K `  H )  e.  (LSHyp `  W )
)  ->  ( K `  G )  =  ( K `  H ) )
3736ex 435 . . . . . . . 8  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  ( ( K `  H )  e.  (LSHyp `  W )  ->  ( K `  G
)  =  ( K `
 H ) ) )
3837necon3ad 2614 . . . . . . 7  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  ( ( K `  G )  =/=  ( K `  H
)  ->  -.  ( K `  H )  e.  (LSHyp `  W )
) )
3938impr 623 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  -.  ( K `  H )  e.  (LSHyp `  W
) )
4025necon1bbid 2640 . . . . . . 7  |-  ( ph  ->  ( -.  ( K `
 H )  e.  (LSHyp `  W )  <->  ( K `  H )  =  ( Base `  W
) ) )
4140adantr 466 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( -.  ( K `  H
)  e.  (LSHyp `  W )  <->  ( K `  H )  =  (
Base `  W )
) )
4239, 41mpbid 213 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  ( K `  H )  =  ( Base `  W
) )
4314, 42jca 534 . . . 4  |-  ( (
ph  /\  ( ( K `  G )  C_  ( K `  H
)  /\  ( K `  G )  =/=  ( K `  H )
) )  ->  (
( K `  G
)  =/=  ( Base `  W )  /\  ( K `  H )  =  ( Base `  W
) ) )
443, 4, 5, 8, 30lkrssv 32574 . . . . . . 7  |-  ( ph  ->  ( K `  G
)  C_  ( Base `  W ) )
4544adantr 466 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  C_  ( Base `  W
) )
46 simprr 764 . . . . . . 7  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  H )  =  ( Base `  W
) )
4746eqcomd 2434 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( Base `  W )  =  ( K `  H
) )
4845, 47sseqtrd 3443 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  C_  ( K `  H
) )
49 simprl 762 . . . . . 6  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  =/=  ( Base `  W
) )
5049, 47neeqtrd 2670 . . . . 5  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  ( K `  G )  =/=  ( K `  H
) )
5148, 50jca 534 . . . 4  |-  ( (
ph  /\  ( ( K `  G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) )  ->  (
( K `  G
)  C_  ( K `  H )  /\  ( K `  G )  =/=  ( K `  H
) ) )
5243, 51impbida 840 . . 3  |-  ( ph  ->  ( ( ( K `
 G )  C_  ( K `  H )  /\  ( K `  G )  =/=  ( K `  H )
)  <->  ( ( K `
 G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
) ) )
531, 52syl5bb 260 . 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 32639 . . . 4  |-  ( ph  ->  ( ( K `  G )  =  (
Base `  W )  <->  G  =  .0.  ) )
5756necon3bid 2645 . . 3  |-  ( ph  ->  ( ( K `  G )  =/=  ( Base `  W )  <->  G  =/=  .0.  ) )
583, 4, 5, 54, 55, 8, 9lkr0f2 32639 . . 3  |-  ( ph  ->  ( ( K `  H )  =  (
Base `  W )  <->  H  =  .0.  ) )
5957, 58anbi12d 715 . 2  |-  ( ph  ->  ( ( ( K `
 G )  =/=  ( Base `  W
)  /\  ( K `  H )  =  (
Base `  W )
)  <->  ( G  =/= 
.0.  /\  H  =  .0.  ) ) )
6053, 59bitrd 256 1  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  .0.  /\  H  =  .0.  )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599    C_ wss 3379    C. wpss 3380   ` cfv 5544   Basecbs 15064   0gc0g 15281   LModclmod 18034   LVecclvec 18268  LSHypclsh 32453  LFnlclfn 32535  LKerclk 32563  LDualcld 32601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-of 6489  df-om 6651  df-1st 6751  df-2nd 6752  df-tpos 6928  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-mulr 15147  df-sca 15149  df-vsca 15150  df-0g 15283  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-submnd 16526  df-grp 16616  df-minusg 16617  df-sbg 16618  df-subg 16757  df-cntz 16914  df-lsm 17231  df-cmn 17375  df-abl 17376  df-mgp 17667  df-ur 17679  df-ring 17725  df-oppr 17794  df-dvdsr 17812  df-unit 17813  df-invr 17843  df-drng 17920  df-lmod 18036  df-lss 18099  df-lsp 18138  df-lvec 18269  df-lshyp 32455  df-lfl 32536  df-lkr 32564  df-ldual 32602
This theorem is referenced by:  lkrss2N  32647  lkreqN  32648
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