Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lkrss2N Structured version   Unicode version

Theorem lkrss2N 32819
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 3460 . . 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 2443 . . . . . . 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 32813 . . . . . 6  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  ( 0g
`  D )  /\  H  =  ( 0g `  D ) ) ) )
10 lveclmod 17192 . . . . . . . . . . . 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 2443 . . . . . . . . . . . 12  |-  ( 0g
`  S )  =  ( 0g `  S
)
1512, 13, 14lmod0cl 16979 . . . . . . . . . . 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 32810 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0g `  S )  .x.  G
)  =  ( 0g
`  D ) )
2120adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  ( ( 0g `  S )  .x.  G )  =  ( 0g `  D ) )
2218, 21eqtr4d 2478 . . . . . . . . 9  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  H  =  ( ( 0g `  S )  .x.  G
) )
23 oveq1 6103 . . . . . . . . . . 11  |-  ( r  =  ( 0g `  S )  ->  (
r  .x.  G )  =  ( ( 0g
`  S )  .x.  G ) )
2423eqeq2d 2454 . . . . . . . . . 10  |-  ( r  =  ( 0g `  S )  ->  ( H  =  ( r  .x.  G )  <->  H  =  ( ( 0g `  S )  .x.  G
) ) )
2524rspcev 3078 . . . . . . . . 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 32815 . . . 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 32818 . . . . . 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 5696 . . . . . . 7  |-  ( H  =  ( r  .x.  G )  ->  ( K `  H )  =  ( K `  ( r  .x.  G
) ) )
4443sseq2d 3389 . . . . . 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 2845 . . 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 828 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 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721    C_ wss 3333    C. wpss 3334   ` cfv 5423  (class class class)co 6096   Basecbs 14179  Scalarcsca 14246   .scvsca 14247   0gc0g 14383   LModclmod 16953   LVecclvec 17188  LFnlclfn 32707  LKerclk 32735  LDualcld 32773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-0g 14385  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-subg 15683  df-cntz 15840  df-lsm 16140  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-drng 16839  df-lmod 16955  df-lss 17019  df-lsp 17058  df-lvec 17189  df-lshyp 32627  df-lfl 32708  df-lkr 32736  df-ldual 32774
This theorem is referenced by:  lcfrvalsnN  35191
  Copyright terms: Public domain W3C validator