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Theorem lkrss2N 35291
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 3589 . . 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 2454 . . . . . . 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 35285 . . . . . 6  |-  ( ph  ->  ( ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  ( 0g
`  D )  /\  H  =  ( 0g `  D ) ) ) )
10 lveclmod 17947 . . . . . . . . . . . 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 2454 . . . . . . . . . . . 12  |-  ( 0g
`  S )  =  ( 0g `  S
)
1512, 13, 14lmod0cl 17733 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  ( 0g
`  S )  e.  R )
1611, 15syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  S
)  e.  R )
1716adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  ( 0g `  S )  e.  R
)
18 simpr 459 . . . . . . . . . 10  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  H  =  ( 0g `  D ) )
19 lkrss2.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  D )
202, 12, 14, 4, 19, 5, 11, 7ldual0vs 35282 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0g `  S )  .x.  G
)  =  ( 0g
`  D ) )
2120adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  ( ( 0g `  S )  .x.  G )  =  ( 0g `  D ) )
2218, 21eqtr4d 2498 . . . . . . . . 9  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  H  =  ( ( 0g `  S )  .x.  G
) )
23 oveq1 6277 . . . . . . . . . . 11  |-  ( r  =  ( 0g `  S )  ->  (
r  .x.  G )  =  ( ( 0g
`  S )  .x.  G ) )
2423eqeq2d 2468 . . . . . . . . . 10  |-  ( r  =  ( 0g `  S )  ->  ( H  =  ( r  .x.  G )  <->  H  =  ( ( 0g `  S )  .x.  G
) ) )
2524rspcev 3207 . . . . . . . . 9  |-  ( ( ( 0g `  S
)  e.  R  /\  H  =  ( ( 0g `  S )  .x.  G ) )  ->  E. r  e.  R  H  =  ( r  .x.  G ) )
2617, 22, 25syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  H  =  ( 0g `  D ) )  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
2726ex 432 . . . . . . 7  |-  ( ph  ->  ( H  =  ( 0g `  D )  ->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
2827adantld 465 . . . . . 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 427 . . . 4  |-  ( (
ph  /\  ( K `  G )  C.  ( K `  H )
)  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
316adantr 463 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  W  e.  LVec )
327adantr 463 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  G  e.  F )
338adantr 463 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  H  e.  F )
34 simpr 459 . . . . 5  |-  ( (
ph  /\  ( K `  G )  =  ( K `  H ) )  ->  ( K `  G )  =  ( K `  H ) )
3512, 13, 2, 3, 4, 19, 31, 32, 33, 34eqlkr4 35287 . . . 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 473 . 2  |-  ( (
ph  /\  ( K `  G )  C_  ( K `  H )
)  ->  E. r  e.  R  H  =  ( r  .x.  G
) )
386adantr 463 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  W  e.  LVec )
397adantr 463 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  G  e.  F )
40 simpr 459 . . . . . . 7  |-  ( (
ph  /\  r  e.  R )  ->  r  e.  R )
4112, 13, 2, 3, 4, 19, 38, 39, 40lkrss 35290 . . . . . 6  |-  ( (
ph  /\  r  e.  R )  ->  ( K `  G )  C_  ( K `  (
r  .x.  G )
) )
4241ex 432 . . . . 5  |-  ( ph  ->  ( r  e.  R  ->  ( K `  G
)  C_  ( K `  ( r  .x.  G
) ) ) )
43 fveq2 5848 . . . . . . 7  |-  ( H  =  ( r  .x.  G )  ->  ( K `  H )  =  ( K `  ( r  .x.  G
) ) )
4443sseq2d 3517 . . . . . 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 2944 . . 3  |-  ( ph  ->  ( E. r  e.  R  H  =  ( r  .x.  G )  ->  ( K `  G )  C_  ( K `  H )
) )
4847imp 427 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    C_ wss 3461    C. wpss 3462   ` cfv 5570  (class class class)co 6270   Basecbs 14716  Scalarcsca 14787   .scvsca 14788   0gc0g 14929   LModclmod 17707   LVecclvec 17943  LFnlclfn 35179  LKerclk 35207  LDualcld 35245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-drng 17593  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lvec 17944  df-lshyp 35099  df-lfl 35180  df-lkr 35208  df-ldual 35246
This theorem is referenced by:  lcfrvalsnN  37665
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