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Theorem lsmless2x 15280
Description: Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmless2.v  |-  B  =  ( Base `  G
)
lsmless2.s  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmless2x  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  -> 
( R  .(+)  T ) 
C_  ( R  .(+)  U ) )

Proof of Theorem lsmless2x
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrexv 3409 . . . . 5  |-  ( T 
C_  U  ->  ( E. z  e.  T  x  =  ( y
( +g  `  G ) z )  ->  E. z  e.  U  x  =  ( y ( +g  `  G ) z ) ) )
21reximdv 2818 . . . 4  |-  ( T 
C_  U  ->  ( E. y  e.  R  E. z  e.  T  x  =  ( y
( +g  `  G ) z )  ->  E. y  e.  R  E. z  e.  U  x  =  ( y ( +g  `  G ) z ) ) )
32adantl 454 . . 3  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  -> 
( E. y  e.  R  E. z  e.  T  x  =  ( y ( +g  `  G
) z )  ->  E. y  e.  R  E. z  e.  U  x  =  ( y
( +g  `  G ) z ) ) )
4 simpl1 961 . . . 4  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  ->  G  e.  V )
5 simpl2 962 . . . 4  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  ->  R  C_  B )
6 simpr 449 . . . . 5  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  ->  T  C_  U )
7 simpl3 963 . . . . 5  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  ->  U  C_  B )
86, 7sstrd 3359 . . . 4  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  ->  T  C_  B )
9 lsmless2.v . . . . 5  |-  B  =  ( Base `  G
)
10 eqid 2437 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
11 lsmless2.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
129, 10, 11lsmelvalx 15275 . . . 4  |-  ( ( G  e.  V  /\  R  C_  B  /\  T  C_  B )  ->  (
x  e.  ( R 
.(+)  T )  <->  E. y  e.  R  E. z  e.  T  x  =  ( y ( +g  `  G ) z ) ) )
134, 5, 8, 12syl3anc 1185 . . 3  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  -> 
( x  e.  ( R  .(+)  T )  <->  E. y  e.  R  E. z  e.  T  x  =  ( y ( +g  `  G ) z ) ) )
149, 10, 11lsmelvalx 15275 . . . 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 ) ) )
1514adantr 453 . . 3  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  -> 
( x  e.  ( R  .(+)  U )  <->  E. y  e.  R  E. z  e.  U  x  =  ( y ( +g  `  G ) z ) ) )
163, 13, 153imtr4d 261 . 2  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  -> 
( x  e.  ( R  .(+)  T )  ->  x  e.  ( R 
.(+)  U ) ) )
1716ssrdv 3355 1  |-  ( ( ( G  e.  V  /\  R  C_  B  /\  U  C_  B )  /\  T  C_  U )  -> 
( R  .(+)  T ) 
C_  ( R  .(+)  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2707    C_ wss 3321   ` cfv 5455  (class class class)co 6082   Basecbs 13470   +g cplusg 13530   LSSumclsm 15269
This theorem is referenced by:  lsmless2  15295  lsmssspx  16161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-lsm 15271
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