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

Theorem lsmssv 16267
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 2454 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 lsmless2.s . . 3  |-  .(+)  =  (
LSSum `  G )
41, 2, 3lsmvalx 16263 . 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 991 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  G  e.  Mnd )
6 simp2 989 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  T  C_  B )
76sselda 3467 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  x  e.  T )  ->  x  e.  B )
87adantrr 716 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  x  e.  B )
9 simp3 990 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  ->  U  C_  B )
109sselda 3467 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  y  e.  U )  ->  y  e.  B )
1110adantrl 715 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  y  e.  B )
121, 2mndcl 15543 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
135, 8, 11, 12syl3anc 1219 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  T  C_  B  /\  U  C_  B )  /\  ( x  e.  T  /\  y  e.  U
) )  ->  (
x ( +g  `  G
) y )  e.  B )
1413ralrimivva 2914 . . . 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 2454 . . . . 5  |-  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G
) y ) )  =  ( x  e.  T ,  y  e.  U  |->  ( x ( +g  `  G ) y ) )
1615fmpt2 6754 . . . 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 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 3501 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799    C_ wss 3439    X. cxp 4949   ran crn 4952   -->wf 5525   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   Basecbs 14296   +g cplusg 14361   Mndcmnd 15532   LSSumclsm 16258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-mnd 15538  df-lsm 16260
This theorem is referenced by:  lsmsubm  16277  lsmass  16292  lsmcntzr  16302
  Copyright terms: Public domain W3C validator