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Theorem lsmlub 14990
Description: The least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
Hypothesis
Ref Expression
lsmub1.p  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmlub  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  C_  U  /\  T  C_  U
)  <->  ( S  .(+)  T )  C_  U )
)

Proof of Theorem lsmlub
StepHypRef Expression
1 simp3 957 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  U  e.  (SubGrp `  G
) )
2 lsmub1.p . . . . . 6  |-  .(+)  =  (
LSSum `  G )
32lsmless12 14988 . . . . 5  |-  ( ( ( U  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( S  C_  U  /\  T  C_  U ) )  -> 
( S  .(+)  T ) 
C_  ( U  .(+)  U ) )
43ex 423 . . . 4  |-  ( ( U  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( ( S  C_  U  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( U  .(+)  U ) ) )
51, 1, 4syl2anc 642 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  C_  U  /\  T  C_  U
)  ->  ( S  .(+) 
T )  C_  ( U  .(+)  U ) ) )
62lsmidm 14989 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  ( U  .(+) 
U )  =  U )
763ad2ant3 978 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( U  .(+)  U )  =  U )
87sseq2d 3219 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  T )  C_  ( U  .(+) 
U )  <->  ( S  .(+) 
T )  C_  U
) )
95, 8sylibd 205 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  C_  U  /\  T  C_  U
)  ->  ( S  .(+) 
T )  C_  U
) )
102lsmub1 14983 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )
)  ->  S  C_  ( S  .(+)  T ) )
11103adant3 975 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  S  C_  ( S  .(+)  T ) )
12 sstr2 3199 . . . 4  |-  ( S 
C_  ( S  .(+)  T )  ->  ( ( S  .(+)  T )  C_  U  ->  S  C_  U
) )
1311, 12syl 15 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  T )  C_  U  ->  S 
C_  U ) )
142lsmub2 14984 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )
)  ->  T  C_  ( S  .(+)  T ) )
15143adant3 975 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  T  C_  ( S  .(+)  T ) )
16 sstr2 3199 . . . 4  |-  ( T 
C_  ( S  .(+)  T )  ->  ( ( S  .(+)  T )  C_  U  ->  T  C_  U
) )
1715, 16syl 15 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  T )  C_  U  ->  T 
C_  U ) )
1813, 17jcad 519 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  T )  C_  U  ->  ( S  C_  U  /\  T  C_  U ) ) )
199, 18impbid 183 1  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  C_  U  /\  T  C_  U
)  <->  ( S  .(+)  T )  C_  U )
)
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  SubGrpcsubg 14631   LSSumclsm 14961
This theorem is referenced by:  lsmss1  14991  lsmss2  14993  lsmmod  15000  lsmcntz  15004  dprd2da  15293  dmdprdsplit2lem  15296  dprdsplit  15299  pgpfac1lem1  15325  lsmsp  15855  lspprabs  15864  lsmcv  15910  lrelat  29826  lsatexch  29855  lsatcvatlem  29861  lsatcvat  29862  dihjustlem  32028  dihord1  32030  dihord5apre  32074  lclkrlem2f  32324  lclkrlem2v  32340  lclkrslem2  32350  lcfrlem25  32379  lcfrlem35  32389  mapdlsm  32476  lspindp5  32582
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-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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-lsm 14963
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