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Theorem lsmlub 15289
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 959 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  U  e.  (SubGrp `  G
) )
2 lsmub1.p . . . . . 6  |-  .(+)  =  (
LSSum `  G )
32lsmless12 15287 . . . . 5  |-  ( ( ( U  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( S  C_  U  /\  T  C_  U ) )  -> 
( S  .(+)  T ) 
C_  ( U  .(+)  U ) )
43ex 424 . . . 4  |-  ( ( U  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( ( S  C_  U  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( U  .(+)  U ) ) )
51, 1, 4syl2anc 643 . . 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 15288 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  ( U  .(+) 
U )  =  U )
763ad2ant3 980 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( U  .(+)  U )  =  U )
87sseq2d 3368 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  T )  C_  ( U  .(+) 
U )  <->  ( S  .(+) 
T )  C_  U
) )
95, 8sylibd 206 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  C_  U  /\  T  C_  U
)  ->  ( S  .(+) 
T )  C_  U
) )
102lsmub1 15282 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )
)  ->  S  C_  ( S  .(+)  T ) )
11103adant3 977 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  S  C_  ( S  .(+)  T ) )
12 sstr2 3347 . . . 4  |-  ( S 
C_  ( S  .(+)  T )  ->  ( ( S  .(+)  T )  C_  U  ->  S  C_  U
) )
1311, 12syl 16 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  T )  C_  U  ->  S 
C_  U ) )
142lsmub2 15283 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )
)  ->  T  C_  ( S  .(+)  T ) )
15143adant3 977 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  T  C_  ( S  .(+)  T ) )
16 sstr2 3347 . . . 4  |-  ( T 
C_  ( S  .(+)  T )  ->  ( ( S  .(+)  T )  C_  U  ->  T  C_  U
) )
1715, 16syl 16 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  T )  C_  U  ->  T 
C_  U ) )
1813, 17jcad 520 . 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 184 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312   ` cfv 5446  (class class class)co 6073  SubGrpcsubg 14930   LSSumclsm 15260
This theorem is referenced by:  lsmss1  15290  lsmss2  15292  lsmmod  15299  lsmcntz  15303  dprd2da  15592  dmdprdsplit2lem  15595  dprdsplit  15598  pgpfac1lem1  15624  lsmsp  16150  lspprabs  16159  lsmcv  16205  lrelat  29749  lsatexch  29778  lsatcvatlem  29784  lsatcvat  29785  dihjustlem  31951  dihord1  31953  dihord5apre  31997  lclkrlem2f  32247  lclkrlem2v  32263  lclkrslem2  32273  lcfrlem25  32302  lcfrlem35  32312  mapdlsm  32399  lspindp5  32505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-subg 14933  df-lsm 15262
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