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Theorem lkrss2N 29967
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 3446 . . 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 2436 . . . . . . 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 29961 . . . . . 6  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  ( 0g
`  D )  /\  H  =  ( 0g `  D ) ) ) )
10 lveclmod 16178 . . . . . . . . . . . 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 2436 . . . . . . . . . . . 12  |-  ( 0g
`  S )  =  ( 0g `  S
)
1512, 13, 14lmod0cl 15976 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  ( 0g
`  S )  e.  R )
1611, 15syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  S
)  e.  R )
1716adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  ( 0g `  S )  e.  R
)
18 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  H  =  ( 0g `  D ) )
19 lkrss2.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  D )
202, 12, 14, 4, 19, 5, 11, 7ldual0vs 29958 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0g `  S )  .x.  G
)  =  ( 0g
`  D ) )
2120adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  ( ( 0g `  S )  .x.  G )  =  ( 0g `  D ) )
2218, 21eqtr4d 2471 . . . . . . . . 9  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  H  =  ( ( 0g `  S )  .x.  G
) )
23 oveq1 6088 . . . . . . . . . . 11  |-  ( r  =  ( 0g `  S )  ->  (
r  .x.  G )  =  ( ( 0g
`  S )  .x.  G ) )
2423eqeq2d 2447 . . . . . . . . . 10  |-  ( r  =  ( 0g `  S )  ->  ( H  =  ( r  .x.  G )  <->  H  =  ( ( 0g `  S )  .x.  G
) ) )
2524rspcev 3052 . . . . . . . . 9  |-  ( ( ( 0g `  S
)  e.  R  /\  H  =  ( ( 0g `  S )  .x.  G ) )  ->  E. r  e.  R  H  =  ( r  .x.  G ) )
2617, 22, 25syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
2726ex 424 . . . . . . 7  |-  ( ph  ->  ( H  =  ( 0g `  D )  ->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
2827adantld 454 . . . . . 6  |-  ( ph  ->  ( ( G  =/=  ( 0g `  D
)  /\  H  =  ( 0g `  D ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) ) )
299, 28sylbid 207 . . . . 5  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  ->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
3029imp 419 . . . 4  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
316adantr 452 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  W  e.  LVec )
327adantr 452 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  G  e.  F )
338adantr 452 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  H  e.  F )
34 simpr 448 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  ( K `  G )  =  ( K `  H ) )
3512, 13, 2, 3, 4, 19, 31, 32, 33, 34eqlkr4 29963 . . . 4  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
3630, 35jaodan 761 . . 3  |-  ( (
ph  /\  ( ( K `  G )  C.  ( K `  H
)  \/  ( K `
 G )  =  ( K `  H
) ) )  ->  E. r  e.  R  H  =  ( r  .x.  G ) )
371, 36sylan2b 462 . 2  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
386adantr 452 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  W  e.  LVec )
397adantr 452 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  G  e.  F )
40 simpr 448 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  r  e.  R )
4112, 13, 2, 3, 4, 19, 38, 39, 40lkrss 29966 . . . . . 6  |-  ( (
ph  /\  r  e.  R )  ->  ( K `  G )  C_  ( K `  (
r  .x.  G )
) )
4241ex 424 . . . . 5  |-  ( ph  ->  ( r  e.  R  ->  ( K `  G
)  C_  ( K `  ( r  .x.  G
) ) ) )
43 fveq2 5728 . . . . . . 7  |-  ( H  =  ( r  .x.  G )  ->  ( K `  H )  =  ( K `  ( r  .x.  G
) ) )
4443sseq2d 3376 . . . . . 6  |-  ( H  =  ( r  .x.  G )  ->  (
( K `  G
)  C_  ( K `  H )  <->  ( K `  G )  C_  ( K `  ( r  .x.  G ) ) ) )
4544biimprcd 217 . . . . 5  |-  ( ( K `  G ) 
C_  ( K `  ( r  .x.  G
) )  ->  ( H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H
) ) )
4642, 45syl6 31 . . . 4  |-  ( ph  ->  ( r  e.  R  ->  ( H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H )
) ) )
4746rexlimdv 2829 . . 3  |-  ( ph  ->  ( E. r  e.  R  H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H )
) )
4847imp 419 . 2  |-  ( (
ph  /\  E. r  e.  R  H  =  ( r  .x.  G
) )  ->  ( K `  G )  C_  ( K `  H
) )
4937, 48impbida 806 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 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    C_ wss 3320    C. wpss 3321   ` cfv 5454  (class class class)co 6081   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   LModclmod 15950   LVecclvec 16174  LFnlclfn 29855  LKerclk 29883  LDualcld 29921
This theorem is referenced by:  lcfrvalsnN  32339
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lvec 16175  df-lshyp 29775  df-lfl 29856  df-lkr 29884  df-ldual 29922
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