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Theorem lsmlub 16162
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 990 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  U  e.  (SubGrp `  G
) )
2 lsmub1.p . . . . . 6  |-  .(+)  =  (
LSSum `  G )
32lsmless12 16160 . . . . 5  |-  ( ( ( U  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( S  C_  U  /\  T  C_  U ) )  -> 
( S  .(+)  T ) 
C_  ( U  .(+)  U ) )
43ex 434 . . . 4  |-  ( ( U  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( ( S  C_  U  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( U  .(+)  U ) ) )
51, 1, 4syl2anc 661 . . 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 16161 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  ( U  .(+) 
U )  =  U )
763ad2ant3 1011 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( U  .(+)  U )  =  U )
87sseq2d 3384 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  T )  C_  ( U  .(+) 
U )  <->  ( S  .(+) 
T )  C_  U
) )
95, 8sylibd 214 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( ( S  C_  U  /\  T  C_  U
)  ->  ( S  .(+) 
T )  C_  U
) )
102lsmub1 16155 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )
)  ->  S  C_  ( S  .(+)  T ) )
11103adant3 1008 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  S  C_  ( S  .(+)  T ) )
12 sstr2 3363 . . . 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 16156 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )
)  ->  T  C_  ( S  .(+)  T ) )
15143adant3 1008 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  T  C_  ( S  .(+)  T ) )
16 sstr2 3363 . . . 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 533 . 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 191 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3328   ` cfv 5418  (class class class)co 6091  SubGrpcsubg 15675   LSSumclsm 16133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-0g 14380  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-subg 15678  df-lsm 16135
This theorem is referenced by:  lsmss1  16163  lsmss2  16165  lsmmod  16172  lsmcntz  16176  dprd2da  16541  dmdprdsplit2lem  16544  dprdsplit  16547  pgpfac1lem1  16575  lsmsp  17167  lspprabs  17176  lsmcv  17222  lrelat  32659  lsatexch  32688  lsatcvatlem  32694  lsatcvat  32695  dihjustlem  34861  dihord1  34863  dihord5apre  34907  lclkrlem2f  35157  lclkrlem2v  35173  lclkrslem2  35183  lcfrlem25  35212  lcfrlem35  35222  mapdlsm  35309  lspindp5  35415
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