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Theorem lkrss2N 29981
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 3288 . . 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 2296 . . . . . . 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 29975 . . . . . 6  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  ( 0g
`  D )  /\  H  =  ( 0g `  D ) ) ) )
10 lveclmod 15875 . . . . . . . . . . . 12  |-  ( W  e.  LVec  ->  W  e. 
LMod )
116, 10syl 15 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LMod )
12 lkrss2.s . . . . . . . . . . . 12  |-  S  =  (Scalar `  W )
13 lkrss2.r . . . . . . . . . . . 12  |-  R  =  ( Base `  S
)
14 eqid 2296 . . . . . . . . . . . 12  |-  ( 0g
`  S )  =  ( 0g `  S
)
1512, 13, 14lmod0cl 15672 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  ( 0g
`  S )  e.  R )
1611, 15syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  S
)  e.  R )
1716adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  ( 0g `  S )  e.  R
)
18 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  H  =  ( 0g `  D ) )
19 lkrss2.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  D )
202, 12, 14, 4, 19, 5, 11, 7ldual0vs 29972 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0g `  S )  .x.  G
)  =  ( 0g
`  D ) )
2120adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  ( ( 0g `  S )  .x.  G )  =  ( 0g `  D ) )
2218, 21eqtr4d 2331 . . . . . . . . 9  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  H  =  ( ( 0g `  S )  .x.  G
) )
23 oveq1 5881 . . . . . . . . . . 11  |-  ( r  =  ( 0g `  S )  ->  (
r  .x.  G )  =  ( ( 0g
`  S )  .x.  G ) )
2423eqeq2d 2307 . . . . . . . . . 10  |-  ( r  =  ( 0g `  S )  ->  ( H  =  ( r  .x.  G )  <->  H  =  ( ( 0g `  S )  .x.  G
) ) )
2524rspcev 2897 . . . . . . . . 9  |-  ( ( ( 0g `  S
)  e.  R  /\  H  =  ( ( 0g `  S )  .x.  G ) )  ->  E. r  e.  R  H  =  ( r  .x.  G ) )
2617, 22, 25syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
2726ex 423 . . . . . . 7  |-  ( ph  ->  ( H  =  ( 0g `  D )  ->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
2827adantld 453 . . . . . 6  |-  ( ph  ->  ( ( G  =/=  ( 0g `  D
)  /\  H  =  ( 0g `  D ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) ) )
299, 28sylbid 206 . . . . 5  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  ->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
3029imp 418 . . . 4  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
316adantr 451 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  W  e.  LVec )
327adantr 451 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  G  e.  F )
338adantr 451 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  H  e.  F )
34 simpr 447 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  ( K `  G )  =  ( K `  H ) )
3512, 13, 2, 3, 4, 19, 31, 32, 33, 34eqlkr4 29977 . . . 4  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
3630, 35jaodan 760 . . 3  |-  ( (
ph  /\  ( ( K `  G )  C.  ( K `  H
)  \/  ( K `
 G )  =  ( K `  H
) ) )  ->  E. r  e.  R  H  =  ( r  .x.  G ) )
371, 36sylan2b 461 . 2  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
386adantr 451 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  W  e.  LVec )
397adantr 451 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  G  e.  F )
40 simpr 447 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  r  e.  R )
4112, 13, 2, 3, 4, 19, 38, 39, 40lkrss 29980 . . . . . 6  |-  ( (
ph  /\  r  e.  R )  ->  ( K `  G )  C_  ( K `  (
r  .x.  G )
) )
4241ex 423 . . . . 5  |-  ( ph  ->  ( r  e.  R  ->  ( K `  G
)  C_  ( K `  ( r  .x.  G
) ) ) )
43 fveq2 5541 . . . . . . 7  |-  ( H  =  ( r  .x.  G )  ->  ( K `  H )  =  ( K `  ( r  .x.  G
) ) )
4443sseq2d 3219 . . . . . 6  |-  ( H  =  ( r  .x.  G )  ->  (
( K `  G
)  C_  ( K `  H )  <->  ( K `  G )  C_  ( K `  ( r  .x.  G ) ) ) )
4544biimprcd 216 . . . . 5  |-  ( ( K `  G ) 
C_  ( K `  ( r  .x.  G
) )  ->  ( H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H
) ) )
4642, 45syl6 29 . . . 4  |-  ( ph  ->  ( r  e.  R  ->  ( H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H )
) ) )
4746rexlimdv 2679 . . 3  |-  ( ph  ->  ( E. r  e.  R  H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H )
) )
4847imp 418 . 2  |-  ( (
ph  /\  E. r  e.  R  H  =  ( r  .x.  G
) )  ->  ( K `  G )  C_  ( K `  H
) )
4937, 48impbida 805 1  |-  ( ph  ->  ( ( K `  G )  C_  ( K `  H )  <->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    C_ wss 3165    C. wpss 3166   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   LModclmod 15643   LVecclvec 15871  LFnlclfn 29869  LKerclk 29897  LDualcld 29935
This theorem is referenced by:  lcfrvalsnN  32353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lshyp 29789  df-lfl 29870  df-lkr 29898  df-ldual 29936
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