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Theorem lsslsp 15788
Description: Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap  M `  G and  N `
 G since we are computing a property of  N `  G? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015.
Hypotheses
Ref Expression
lsslsp.x  |-  X  =  ( Ws  U )
lsslsp.m  |-  M  =  ( LSpan `  W )
lsslsp.n  |-  N  =  ( LSpan `  X )
lsslsp.l  |-  L  =  ( LSubSp `  W )
Assertion
Ref Expression
lsslsp  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  =  ( N `  G ) )

Proof of Theorem lsslsp
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  W  e.  LMod )
2 lsslsp.x . . . . . . . 8  |-  X  =  ( Ws  U )
3 lsslsp.l . . . . . . . 8  |-  L  =  ( LSubSp `  W )
42, 3lsslmod 15733 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  X  e.  LMod )
543adant3 975 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  X  e.  LMod )
6 simp3 957 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  U )
7 eqid 2296 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
87, 3lssss 15710 . . . . . . . . 9  |-  ( U  e.  L  ->  U  C_  ( Base `  W
) )
983ad2ant2 977 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  U  C_  ( Base `  W
) )
102, 7ressbas2 13215 . . . . . . . 8  |-  ( U 
C_  ( Base `  W
)  ->  U  =  ( Base `  X )
)
119, 10syl 15 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  U  =  ( Base `  X
) )
126, 11sseqtrd 3227 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( Base `  X
) )
13 eqid 2296 . . . . . . 7  |-  ( Base `  X )  =  (
Base `  X )
14 eqid 2296 . . . . . . 7  |-  ( LSubSp `  X )  =  (
LSubSp `  X )
15 lsslsp.n . . . . . . 7  |-  N  =  ( LSpan `  X )
1613, 14, 15lspcl 15749 . . . . . 6  |-  ( ( X  e.  LMod  /\  G  C_  ( Base `  X
) )  ->  ( N `  G )  e.  ( LSubSp `  X )
)
175, 12, 16syl2anc 642 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  e.  ( LSubSp `  X )
)
182, 3, 14lsslss 15734 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  (
( N `  G
)  e.  ( LSubSp `  X )  <->  ( ( N `  G )  e.  L  /\  ( N `  G )  C_  U ) ) )
19183adant3 975 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  (
( N `  G
)  e.  ( LSubSp `  X )  <->  ( ( N `  G )  e.  L  /\  ( N `  G )  C_  U ) ) )
2017, 19mpbid 201 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  (
( N `  G
)  e.  L  /\  ( N `  G ) 
C_  U ) )
2120simpld 445 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  e.  L )
2213, 15lspssid 15758 . . . 4  |-  ( ( X  e.  LMod  /\  G  C_  ( Base `  X
) )  ->  G  C_  ( N `  G
) )
235, 12, 22syl2anc 642 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( N `  G
) )
24 lsslsp.m . . . 4  |-  M  =  ( LSpan `  W )
253, 24lspssp 15761 . . 3  |-  ( ( W  e.  LMod  /\  ( N `  G )  e.  L  /\  G  C_  ( N `  G ) )  ->  ( M `  G )  C_  ( N `  G )
)
261, 21, 23, 25syl3anc 1182 . 2  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  C_  ( N `  G
) )
276, 9sstrd 3202 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( Base `  W
) )
287, 3, 24lspcl 15749 . . . . 5  |-  ( ( W  e.  LMod  /\  G  C_  ( Base `  W
) )  ->  ( M `  G )  e.  L )
291, 27, 28syl2anc 642 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  e.  L )
303, 24lspssp 15761 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  C_  U )
312, 3, 14lsslss 15734 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  (
( M `  G
)  e.  ( LSubSp `  X )  <->  ( ( M `  G )  e.  L  /\  ( M `  G )  C_  U ) ) )
32313adant3 975 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  (
( M `  G
)  e.  ( LSubSp `  X )  <->  ( ( M `  G )  e.  L  /\  ( M `  G )  C_  U ) ) )
3329, 30, 32mpbir2and 888 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  e.  ( LSubSp `  X )
)
347, 24lspssid 15758 . . . 4  |-  ( ( W  e.  LMod  /\  G  C_  ( Base `  W
) )  ->  G  C_  ( M `  G
) )
351, 27, 34syl2anc 642 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( M `  G
) )
3614, 15lspssp 15761 . . 3  |-  ( ( X  e.  LMod  /\  ( M `  G )  e.  ( LSubSp `  X )  /\  G  C_  ( M `
 G ) )  ->  ( N `  G )  C_  ( M `  G )
)
375, 33, 35, 36syl3anc 1182 . 2  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  C_  ( M `  G
) )
3826, 37eqssd 3209 1  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  =  ( N `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744
This theorem is referenced by:  lss0v  15789  islssfg  27271  lsslindf  27403  islinds3  27407  lcdlsp  32433
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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lsp 15745
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