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Theorem lsmless1x 14971
Description: Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmless2.v  |-  B  =  ( Base `  G
)
lsmless2.s  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmless1x  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  -> 
( R  .(+)  U ) 
C_  ( T  .(+)  U ) )

Proof of Theorem lsmless1x
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrexv 3251 . . . 4  |-  ( R 
C_  T  ->  ( E. y  e.  R  E. z  e.  U  x  =  ( y
( +g  `  G ) z )  ->  E. y  e.  T  E. z  e.  U  x  =  ( y ( +g  `  G ) z ) ) )
21adantl 452 . . 3  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  -> 
( E. y  e.  R  E. z  e.  U  x  =  ( y ( +g  `  G
) z )  ->  E. y  e.  T  E. z  e.  U  x  =  ( y
( +g  `  G ) z ) ) )
3 simpl1 958 . . . 4  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  G  e.  V )
4 simpr 447 . . . . 5  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  R  C_  T )
5 simpl2 959 . . . . 5  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  T  C_  B )
64, 5sstrd 3202 . . . 4  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  R  C_  B )
7 simpl3 960 . . . 4  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  U  C_  B )
8 lsmless2.v . . . . 5  |-  B  =  ( Base `  G
)
9 eqid 2296 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
10 lsmless2.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
118, 9, 10lsmelvalx 14967 . . . 4  |-  ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  ->  (
x  e.  ( R 
.(+)  U )  <->  E. y  e.  R  E. z  e.  U  x  =  ( y ( +g  `  G ) z ) ) )
123, 6, 7, 11syl3anc 1182 . . 3  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  -> 
( x  e.  ( R  .(+)  U )  <->  E. y  e.  R  E. z  e.  U  x  =  ( y ( +g  `  G ) z ) ) )
138, 9, 10lsmelvalx 14967 . . . 4  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  (
x  e.  ( T 
.(+)  U )  <->  E. y  e.  T  E. z  e.  U  x  =  ( y ( +g  `  G ) z ) ) )
1413adantr 451 . . 3  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  -> 
( x  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  x  =  ( y ( +g  `  G ) z ) ) )
152, 12, 143imtr4d 259 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  -> 
( x  e.  ( R  .(+)  U )  ->  x  e.  ( T 
.(+)  U ) ) )
1615ssrdv 3198 1  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  -> 
( R  .(+)  U ) 
C_  ( T  .(+)  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   LSSumclsm 14961
This theorem is referenced by:  lsmless1  14986  lsmless12  14988  lsmssspx  15857
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-reu 2563  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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-lsm 14963
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