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Theorem lsslsp 15772
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 15717 . . . . . . 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 2283 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
87, 3lssss 15694 . . . . . . . . 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 13199 . . . . . . . 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 3214 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( Base `  X
) )
13 eqid 2283 . . . . . . 7  |-  ( Base `  X )  =  (
Base `  X )
14 eqid 2283 . . . . . . 7  |-  ( LSubSp `  X )  =  (
LSubSp `  X )
15 lsslsp.n . . . . . . 7  |-  N  =  ( LSpan `  X )
1613, 14, 15lspcl 15733 . . . . . 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 15718 . . . . . 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 15742 . . . 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 15745 . . 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 3189 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( Base `  W
) )
287, 3, 24lspcl 15733 . . . . 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 15745 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  C_  U )
312, 3, 14lsslss 15718 . . . . 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 15742 . . . 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 15745 . . 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 3196 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 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728
This theorem is referenced by:  lss0v  15773  islssfg  26580  lsslindf  26712  islinds3  26716  lcdlsp  31184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lsp 15729
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