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Theorem lsmless1x 15206
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 3352 . . . 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 453 . . 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 960 . . . 4  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  G  e.  V )
4 simpr 448 . . . . 5  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  R  C_  T )
5 simpl2 961 . . . . 5  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  T  C_  B )
64, 5sstrd 3302 . . . 4  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  ->  R  C_  B )
7 simpl3 962 . . . 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 2388 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
10 lsmless2.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
118, 9, 10lsmelvalx 15202 . . . 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 1184 . . 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 15202 . . . 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 452 . . 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 260 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  R  C_  T )  -> 
( x  e.  ( R  .(+)  U )  ->  x  e.  ( T 
.(+)  U ) ) )
1615ssrdv 3298 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2651    C_ wss 3264   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   LSSumclsm 15196
This theorem is referenced by:  lsmless1  15221  lsmless12  15223  lsmssspx  16088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-lsm 15198
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