MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsmssv Unicode version

Theorem lsmssv 14954
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 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 lsmless2.s . . 3  |-  .(+)  =  (
LSSum `  G )
41, 2, 3lsmvalx 14950 . 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 3180 . . . . . . 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 3180 . . . . . . 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 14372 . . . . . 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 2635 . . . 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 2283 . . . . 5  |-  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G
) y ) )  =  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) )
1615fmpt2 6191 . . . 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 5395 . . 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 3212 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 1623    e. wcel 1684   A.wral 2543    C_ wss 3152    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208   Mndcmnd 14361   LSSumclsm 14945
This theorem is referenced by:  lsmsubm  14964  lsmass  14979  lsmcntzr  14989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-mnd 14367  df-lsm 14947
  Copyright terms: Public domain W3C validator