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Theorem lsslsp 13705
Description: Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) Terms in the equation were swapped as proposed by NM on 15-Mar-2015. (Revised by AV, 18-Apr-2025.)
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 )  ->  ( N `  G )  =  ( M `  G ) )

Proof of Theorem lsslsp
StepHypRef Expression
1 lsslsp.x . . . . 5  |-  X  =  ( Ws  U )
2 lsslsp.l . . . . 5  |-  L  =  ( LSubSp `  W )
31, 2lsslmod 13656 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  X  e.  LMod )
433adant3 1018 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  X  e.  LMod )
5 simp1 998 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  W  e.  LMod )
6 simp3 1000 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  U )
7 eqid 2188 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
87, 2lssssg 13636 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  U  C_  ( Base `  W
) )
983adant3 1018 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  U  C_  ( Base `  W
) )
106, 9sstrd 3179 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( Base `  W
) )
11 lsslsp.m . . . . . 6  |-  M  =  ( LSpan `  W )
127, 2, 11lspcl 13667 . . . . 5  |-  ( ( W  e.  LMod  /\  G  C_  ( Base `  W
) )  ->  ( M `  G )  e.  L )
135, 10, 12syl2anc 411 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  e.  L )
142, 11lspssp 13679 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  C_  U )
15 eqid 2188 . . . . . 6  |-  ( LSubSp `  X )  =  (
LSubSp `  X )
161, 2, 15lsslss 13657 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  (
( M `  G
)  e.  ( LSubSp `  X )  <->  ( ( M `  G )  e.  L  /\  ( M `  G )  C_  U ) ) )
17163adant3 1018 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  (
( M `  G
)  e.  ( LSubSp `  X )  <->  ( ( M `  G )  e.  L  /\  ( M `  G )  C_  U ) ) )
1813, 14, 17mpbir2and 945 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  e.  ( LSubSp `  X )
)
197, 11lspssid 13676 . . . 4  |-  ( ( W  e.  LMod  /\  G  C_  ( Base `  W
) )  ->  G  C_  ( M `  G
) )
205, 10, 19syl2anc 411 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( M `  G
) )
21 lsslsp.n . . . 4  |-  N  =  ( LSpan `  X )
2215, 21lspssp 13679 . . 3  |-  ( ( X  e.  LMod  /\  ( M `  G )  e.  ( LSubSp `  X )  /\  G  C_  ( M `
 G ) )  ->  ( N `  G )  C_  ( M `  G )
)
234, 18, 20, 22syl3anc 1248 . 2  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  C_  ( M `  G
) )
241a1i 9 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  X  =  ( Ws  U ) )
25 eqidd 2189 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( Base `  W )  =  ( Base `  W
) )
2624, 25, 5, 9ressbas2d 12545 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  U  =  ( Base `  X
) )
276, 26sseqtrd 3207 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( Base `  X
) )
28 eqid 2188 . . . . . . 7  |-  ( Base `  X )  =  (
Base `  X )
2928, 15, 21lspcl 13667 . . . . . 6  |-  ( ( X  e.  LMod  /\  G  C_  ( Base `  X
) )  ->  ( N `  G )  e.  ( LSubSp `  X )
)
304, 27, 29syl2anc 411 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  e.  ( LSubSp `  X )
)
311, 2, 15lsslss 13657 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  (
( N `  G
)  e.  ( LSubSp `  X )  <->  ( ( N `  G )  e.  L  /\  ( N `  G )  C_  U ) ) )
32313adant3 1018 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  (
( N `  G
)  e.  ( LSubSp `  X )  <->  ( ( N `  G )  e.  L  /\  ( N `  G )  C_  U ) ) )
3330, 32mpbid 147 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  (
( N `  G
)  e.  L  /\  ( N `  G ) 
C_  U ) )
3433simpld 112 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  e.  L )
3528, 21lspssid 13676 . . . 4  |-  ( ( X  e.  LMod  /\  G  C_  ( Base `  X
) )  ->  G  C_  ( N `  G
) )
364, 27, 35syl2anc 411 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  G  C_  ( N `  G
) )
372, 11lspssp 13679 . . 3  |-  ( ( W  e.  LMod  /\  ( N `  G )  e.  L  /\  G  C_  ( N `  G ) )  ->  ( M `  G )  C_  ( N `  G )
)
385, 34, 36, 37syl3anc 1248 . 2  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  C_  ( N `  G
) )
3923, 38eqssd 3186 1  |-  ( ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( N `  G )  =  ( M `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 979    = wceq 1363    e. wcel 2159    C_ wss 3143   ` cfv 5230  (class class class)co 5890   Basecbs 12479   ↾s cress 12480   LModclmod 13563   LSubSpclss 13628   LSpanclspn 13662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-addcom 7928  ax-addass 7930  ax-i2m1 7933  ax-0lt1 7934  ax-0id 7936  ax-rnegex 7937  ax-pre-ltirr 7940  ax-pre-lttrn 7942  ax-pre-ltadd 7944
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-pnf 8011  df-mnf 8012  df-ltxr 8014  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-5 8998  df-6 8999  df-ndx 12482  df-slot 12483  df-base 12485  df-sets 12486  df-iress 12487  df-plusg 12567  df-mulr 12568  df-sca 12570  df-vsca 12571  df-0g 12728  df-mgm 12797  df-sgrp 12830  df-mnd 12843  df-grp 12913  df-minusg 12914  df-sbg 12915  df-subg 13074  df-mgp 13235  df-ur 13274  df-ring 13312  df-lmod 13565  df-lssm 13629  df-lsp 13663
This theorem is referenced by:  lss0v  13706
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