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Theorem lsslsp 17220
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 988 . . 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 17165 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  X  e.  LMod )
543adant3 1008 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  X  e.  LMod )
6 simp3 990 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  U )
7 eqid 2454 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
87, 3lssss 17142 . . . . . . . . 9  |-  ( U  e.  L  ->  U  C_  ( Base `  W
) )
983ad2ant2 1010 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  U  C_  ( Base `  W
) )
102, 7ressbas2 14349 . . . . . . . 8  |-  ( U 
C_  ( Base `  W
)  ->  U  =  ( Base `  X )
)
119, 10syl 16 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  U  =  ( Base `  X
) )
126, 11sseqtrd 3501 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( Base `  X
) )
13 eqid 2454 . . . . . . 7  |-  ( Base `  X )  =  (
Base `  X )
14 eqid 2454 . . . . . . 7  |-  ( LSubSp `  X )  =  (
LSubSp `  X )
15 lsslsp.n . . . . . . 7  |-  N  =  ( LSpan `  X )
1613, 14, 15lspcl 17181 . . . . . 6  |-  ( ( X  e.  LMod  /\  G  C_  ( Base `  X
) )  ->  ( N `  G )  e.  ( LSubSp `  X )
)
175, 12, 16syl2anc 661 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  e.  ( LSubSp `  X )
)
182, 3, 14lsslss 17166 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  (
( N `  G
)  e.  ( LSubSp `  X )  <->  ( ( N `  G )  e.  L  /\  ( N `  G )  C_  U ) ) )
19183adant3 1008 . . . . 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 210 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  (
( N `  G
)  e.  L  /\  ( N `  G ) 
C_  U ) )
2120simpld 459 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  e.  L )
2213, 15lspssid 17190 . . . 4  |-  ( ( X  e.  LMod  /\  G  C_  ( Base `  X
) )  ->  G  C_  ( N `  G
) )
235, 12, 22syl2anc 661 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( N `  G
) )
24 lsslsp.m . . . 4  |-  M  =  ( LSpan `  W )
253, 24lspssp 17193 . . 3  |-  ( ( W  e.  LMod  /\  ( N `  G )  e.  L  /\  G  C_  ( N `  G ) )  ->  ( M `  G )  C_  ( N `  G )
)
261, 21, 23, 25syl3anc 1219 . 2  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  C_  ( N `  G
) )
276, 9sstrd 3475 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( Base `  W
) )
287, 3, 24lspcl 17181 . . . . 5  |-  ( ( W  e.  LMod  /\  G  C_  ( Base `  W
) )  ->  ( M `  G )  e.  L )
291, 27, 28syl2anc 661 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  e.  L )
303, 24lspssp 17193 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  C_  U )
312, 3, 14lsslss 17166 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  (
( M `  G
)  e.  ( LSubSp `  X )  <->  ( ( M `  G )  e.  L  /\  ( M `  G )  C_  U ) ) )
32313adant3 1008 . . . 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 913 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  e.  ( LSubSp `  X )
)
347, 24lspssid 17190 . . . 4  |-  ( ( W  e.  LMod  /\  G  C_  ( Base `  W
) )  ->  G  C_  ( M `  G
) )
351, 27, 34syl2anc 661 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( M `  G
) )
3614, 15lspssp 17193 . . 3  |-  ( ( X  e.  LMod  /\  ( M `  G )  e.  ( LSubSp `  X )  /\  G  C_  ( M `
 G ) )  ->  ( N `  G )  C_  ( M `  G )
)
375, 33, 35, 36syl3anc 1219 . 2  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  C_  ( M `  G
) )
3826, 37eqssd 3482 1  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  =  ( N `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3437   ` cfv 5527  (class class class)co 6201   Basecbs 14293   ↾s cress 14294   LModclmod 17072   LSubSpclss 17137   LSpanclspn 17176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-sca 14374  df-vsca 14375  df-0g 14500  df-mnd 15535  df-grp 15665  df-minusg 15666  df-sbg 15667  df-subg 15798  df-mgp 16715  df-ur 16727  df-rng 16771  df-lmod 17074  df-lss 17138  df-lsp 17177
This theorem is referenced by:  lss0v  17221  lsslindf  18385  islinds3  18389  islssfg  29572  lcdlsp  35605
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