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Theorem lsmssv 14970
Description: Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmless2.v  |-  B  =  ( Base `  G
)
lsmless2.s  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmssv  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  C_  B )

Proof of Theorem lsmssv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmless2.v . . 3  |-  B  =  ( Base `  G
)
2 eqid 2296 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 lsmless2.s . . 3  |-  .(+)  =  (
LSSum `  G )
41, 2, 3lsmvalx 14966 . 2  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) ) )
5 simpl1 958 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  G  e.  Mnd )
6 simp2 956 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  T  C_  B )
76sselda 3193 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  x  e.  T )  ->  x  e.  B )
87adantrr 697 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  x  e.  B )
9 simp3 957 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  U  C_  B )
109sselda 3193 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  y  e.  U )  ->  y  e.  B )
1110adantrl 696 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  y  e.  B )
121, 2mndcl 14388 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
135, 8, 11, 12syl3anc 1182 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  (
x ( +g  `  G
) y )  e.  B )
1413ralrimivva 2648 . . . 4  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  A. x  e.  T  A. y  e.  U  ( x
( +g  `  G ) y )  e.  B
)
15 eqid 2296 . . . . 5  |-  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G
) y ) )  =  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) )
1615fmpt2 6207 . . . 4  |-  ( A. x  e.  T  A. y  e.  U  (
x ( +g  `  G
) y )  e.  B  <->  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) ) : ( T  X.  U
) --> B )
1714, 16sylib 188 . . 3  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  (
x  e.  T , 
y  e.  U  |->  ( x ( +g  `  G
) y ) ) : ( T  X.  U ) --> B )
18 frn 5411 . . 3  |-  ( ( x  e.  T , 
y  e.  U  |->  ( x ( +g  `  G
) y ) ) : ( T  X.  U ) --> B  ->  ran  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) )  C_  B )
1917, 18syl 15 . 2  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  ran  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) )  C_  B )
204, 19eqsstrd 3225 1  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164   +g cplusg 13224   Mndcmnd 14377   LSSumclsm 14961
This theorem is referenced by:  lsmsubm  14980  lsmass  14995  lsmcntzr  15005
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-mnd 14383  df-lsm 14963
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