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Theorem lkrss2N 34259
Description: Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
lkrss2.s  |-  S  =  (Scalar `  W )
lkrss2.r  |-  R  =  ( Base `  S
)
lkrss2.f  |-  F  =  (LFnl `  W )
lkrss2.k  |-  K  =  (LKer `  W )
lkrss2.d  |-  D  =  (LDual `  W )
lkrss2.t  |-  .x.  =  ( .s `  D )
lkrss2.w  |-  ( ph  ->  W  e.  LVec )
lkrss2.g  |-  ( ph  ->  G  e.  F )
lkrss2.h  |-  ( ph  ->  H  e.  F )
Assertion
Ref Expression
lkrss2N  |-  ( ph  ->  ( ( K `  G )  C_  ( K `  H )  <->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
Distinct variable groups:    F, r    G, r    H, r    K, r    R, r    S, r    W, r    ph, r    .x. , r
Allowed substitution hint:    D( r)

Proof of Theorem lkrss2N
StepHypRef Expression
1 sspss 3608 . . 3  |-  ( ( K `  G ) 
C_  ( K `  H )  <->  ( ( K `  G )  C.  ( K `  H
)  \/  ( K `
 G )  =  ( K `  H
) ) )
2 lkrss2.f . . . . . . 7  |-  F  =  (LFnl `  W )
3 lkrss2.k . . . . . . 7  |-  K  =  (LKer `  W )
4 lkrss2.d . . . . . . 7  |-  D  =  (LDual `  W )
5 eqid 2467 . . . . . . 7  |-  ( 0g
`  D )  =  ( 0g `  D
)
6 lkrss2.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
7 lkrss2.g . . . . . . 7  |-  ( ph  ->  G  e.  F )
8 lkrss2.h . . . . . . 7  |-  ( ph  ->  H  e.  F )
92, 3, 4, 5, 6, 7, 8lkrpssN 34253 . . . . . 6  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  ( 0g
`  D )  /\  H  =  ( 0g `  D ) ) ) )
10 lveclmod 17600 . . . . . . . . . . . 12  |-  ( W  e.  LVec  ->  W  e. 
LMod )
116, 10syl 16 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LMod )
12 lkrss2.s . . . . . . . . . . . 12  |-  S  =  (Scalar `  W )
13 lkrss2.r . . . . . . . . . . . 12  |-  R  =  ( Base `  S
)
14 eqid 2467 . . . . . . . . . . . 12  |-  ( 0g
`  S )  =  ( 0g `  S
)
1512, 13, 14lmod0cl 17386 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  ( 0g
`  S )  e.  R )
1611, 15syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  S
)  e.  R )
1716adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  ( 0g `  S )  e.  R
)
18 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  H  =  ( 0g `  D ) )
19 lkrss2.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  D )
202, 12, 14, 4, 19, 5, 11, 7ldual0vs 34250 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0g `  S )  .x.  G
)  =  ( 0g
`  D ) )
2120adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  ( ( 0g `  S )  .x.  G )  =  ( 0g `  D ) )
2218, 21eqtr4d 2511 . . . . . . . . 9  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  H  =  ( ( 0g `  S )  .x.  G
) )
23 oveq1 6301 . . . . . . . . . . 11  |-  ( r  =  ( 0g `  S )  ->  (
r  .x.  G )  =  ( ( 0g
`  S )  .x.  G ) )
2423eqeq2d 2481 . . . . . . . . . 10  |-  ( r  =  ( 0g `  S )  ->  ( H  =  ( r  .x.  G )  <->  H  =  ( ( 0g `  S )  .x.  G
) ) )
2524rspcev 3219 . . . . . . . . 9  |-  ( ( ( 0g `  S
)  e.  R  /\  H  =  ( ( 0g `  S )  .x.  G ) )  ->  E. r  e.  R  H  =  ( r  .x.  G ) )
2617, 22, 25syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
2726ex 434 . . . . . . 7  |-  ( ph  ->  ( H  =  ( 0g `  D )  ->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
2827adantld 467 . . . . . 6  |-  ( ph  ->  ( ( G  =/=  ( 0g `  D
)  /\  H  =  ( 0g `  D ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) ) )
299, 28sylbid 215 . . . . 5  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  ->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
3029imp 429 . . . 4  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
316adantr 465 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  W  e.  LVec )
327adantr 465 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  G  e.  F )
338adantr 465 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  H  e.  F )
34 simpr 461 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  ( K `  G )  =  ( K `  H ) )
3512, 13, 2, 3, 4, 19, 31, 32, 33, 34eqlkr4 34255 . . . 4  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
3630, 35jaodan 783 . . 3  |-  ( (
ph  /\  ( ( K `  G )  C.  ( K `  H
)  \/  ( K `
 G )  =  ( K `  H
) ) )  ->  E. r  e.  R  H  =  ( r  .x.  G ) )
371, 36sylan2b 475 . 2  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
386adantr 465 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  W  e.  LVec )
397adantr 465 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  G  e.  F )
40 simpr 461 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  r  e.  R )
4112, 13, 2, 3, 4, 19, 38, 39, 40lkrss 34258 . . . . . 6  |-  ( (
ph  /\  r  e.  R )  ->  ( K `  G )  C_  ( K `  (
r  .x.  G )
) )
4241ex 434 . . . . 5  |-  ( ph  ->  ( r  e.  R  ->  ( K `  G
)  C_  ( K `  ( r  .x.  G
) ) ) )
43 fveq2 5871 . . . . . . 7  |-  ( H  =  ( r  .x.  G )  ->  ( K `  H )  =  ( K `  ( r  .x.  G
) ) )
4443sseq2d 3537 . . . . . 6  |-  ( H  =  ( r  .x.  G )  ->  (
( K `  G
)  C_  ( K `  H )  <->  ( K `  G )  C_  ( K `  ( r  .x.  G ) ) ) )
4544biimprcd 225 . . . . 5  |-  ( ( K `  G ) 
C_  ( K `  ( r  .x.  G
) )  ->  ( H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H
) ) )
4642, 45syl6 33 . . . 4  |-  ( ph  ->  ( r  e.  R  ->  ( H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H )
) ) )
4746rexlimdv 2957 . . 3  |-  ( ph  ->  ( E. r  e.  R  H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H )
) )
4847imp 429 . 2  |-  ( (
ph  /\  E. r  e.  R  H  =  ( r  .x.  G
) )  ->  ( K `  G )  C_  ( K `  H
) )
4937, 48impbida 830 1  |-  ( ph  ->  ( ( K `  G )  C_  ( K `  H )  <->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818    C_ wss 3481    C. wpss 3482   ` cfv 5593  (class class class)co 6294   Basecbs 14502  Scalarcsca 14570   .scvsca 14571   0gc0g 14707   LModclmod 17360   LVecclvec 17596  LFnlclfn 34147  LKerclk 34175  LDualcld 34213
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-1st 6794  df-2nd 6795  df-tpos 6965  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-sca 14583  df-vsca 14584  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-submnd 15820  df-grp 15906  df-minusg 15907  df-sbg 15908  df-subg 16047  df-cntz 16204  df-lsm 16506  df-cmn 16650  df-abl 16651  df-mgp 16991  df-ur 17003  df-ring 17049  df-oppr 17121  df-dvdsr 17139  df-unit 17140  df-invr 17170  df-drng 17246  df-lmod 17362  df-lss 17427  df-lsp 17466  df-lvec 17597  df-lshyp 34067  df-lfl 34148  df-lkr 34176  df-ldual 34214
This theorem is referenced by:  lcfrvalsnN  36631
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