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Theorem lsmssv 16879
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 2402 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 lsmless2.s . . 3  |-  .(+)  =  (
LSSum `  G )
41, 2, 3lsmvalx 16875 . 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 1000 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  G  e.  Mnd )
6 simp2 998 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  T  C_  B )
76sselda 3441 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  x  e.  T )  ->  x  e.  B )
87adantrr 715 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  x  e.  B )
9 simp3 999 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  U  C_  B )
109sselda 3441 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  y  e.  U )  ->  y  e.  B )
1110adantrl 714 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  y  e.  B )
121, 2mndcl 16145 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
135, 8, 11, 12syl3anc 1230 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  (
x ( +g  `  G
) y )  e.  B )
1413ralrimivva 2824 . . . 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 2402 . . . . 5  |-  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G
) y ) )  =  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) )
1615fmpt2 6805 . . . 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 196 . . 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 5676 . . 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 17 . 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 3475 1  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753    C_ wss 3413    X. cxp 4940   ran crn 4943   -->wf 5521   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236   Basecbs 14733   +g cplusg 14801   Mndcmnd 16135   LSSumclsm 16870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-lsm 16872
This theorem is referenced by:  lsmsubm  16889  lsmass  16904  lsmcntzr  16914
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