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Theorem lsmssv 15277
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 2436 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 lsmless2.s . . 3  |-  .(+)  =  (
LSSum `  G )
41, 2, 3lsmvalx 15273 . 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 960 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  G  e.  Mnd )
6 simp2 958 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  T  C_  B )
76sselda 3348 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  x  e.  T )  ->  x  e.  B )
87adantrr 698 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  x  e.  B )
9 simp3 959 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  U  C_  B )
109sselda 3348 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  y  e.  U )  ->  y  e.  B )
1110adantrl 697 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  y  e.  B )
121, 2mndcl 14695 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
135, 8, 11, 12syl3anc 1184 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  (
x ( +g  `  G
) y )  e.  B )
1413ralrimivva 2798 . . . 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 2436 . . . . 5  |-  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G
) y ) )  =  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) )
1615fmpt2 6418 . . . 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 189 . . 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 5597 . . 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 16 . 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 3382 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320    X. cxp 4876   ran crn 4879   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   Basecbs 13469   +g cplusg 13529   Mndcmnd 14684   LSSumclsm 15268
This theorem is referenced by:  lsmsubm  15287  lsmass  15302  lsmcntzr  15312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-mnd 14690  df-lsm 15270
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