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Theorem dsmm0cl 26875
Description: The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Hypotheses
Ref Expression
dsmmcl.p  |-  P  =  ( S X_s R )
dsmmcl.h  |-  H  =  ( Base `  ( S  (+)m  R ) )
dsmmcl.i  |-  ( ph  ->  I  e.  W )
dsmmcl.s  |-  ( ph  ->  S  e.  V )
dsmmcl.r  |-  ( ph  ->  R : I --> Mnd )
dsmm0cl.z  |-  .0.  =  ( 0g `  P )
Assertion
Ref Expression
dsmm0cl  |-  ( ph  ->  .0.  e.  H )

Proof of Theorem dsmm0cl
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dsmmcl.p . . . 4  |-  P  =  ( S X_s R )
2 dsmmcl.i . . . 4  |-  ( ph  ->  I  e.  W )
3 dsmmcl.s . . . 4  |-  ( ph  ->  S  e.  V )
4 dsmmcl.r . . . 4  |-  ( ph  ->  R : I --> Mnd )
51, 2, 3, 4prdsmndd 14655 . . 3  |-  ( ph  ->  P  e.  Mnd )
6 eqid 2387 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
7 dsmm0cl.z . . . 4  |-  .0.  =  ( 0g `  P )
86, 7mndidcl 14641 . . 3  |-  ( P  e.  Mnd  ->  .0.  e.  ( Base `  P
) )
95, 8syl 16 . 2  |-  ( ph  ->  .0.  e.  ( Base `  P ) )
101, 2, 3, 4prds0g 14656 . . . . . . . . . 10  |-  ( ph  ->  ( 0g  o.  R
)  =  ( 0g
`  P ) )
1110, 7syl6eqr 2437 . . . . . . . . 9  |-  ( ph  ->  ( 0g  o.  R
)  =  .0.  )
1211adantr 452 . . . . . . . 8  |-  ( (
ph  /\  a  e.  I )  ->  ( 0g  o.  R )  =  .0.  )
1312fveq1d 5670 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g  o.  R
) `  a )  =  (  .0.  `  a
) )
14 ffn 5531 . . . . . . . . 9  |-  ( R : I --> Mnd  ->  R  Fn  I )
154, 14syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
16 fvco2 5737 . . . . . . . 8  |-  ( ( R  Fn  I  /\  a  e.  I )  ->  ( ( 0g  o.  R ) `  a
)  =  ( 0g
`  ( R `  a ) ) )
1715, 16sylan 458 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g  o.  R
) `  a )  =  ( 0g `  ( R `  a ) ) )
1813, 17eqtr3d 2421 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  (  .0.  `  a )  =  ( 0g `  ( R `  a )
) )
19 nne 2554 . . . . . 6  |-  ( -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  (  .0.  `  a )  =  ( 0g `  ( R `
 a ) ) )
2018, 19sylibr 204 . . . . 5  |-  ( (
ph  /\  a  e.  I )  ->  -.  (  .0.  `  a )  =/=  ( 0g `  ( R `  a )
) )
2120ralrimiva 2732 . . . 4  |-  ( ph  ->  A. a  e.  I  -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) )
22 rabeq0 3592 . . . 4  |-  ( { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  (/)  <->  A. a  e.  I  -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) )
2321, 22sylibr 204 . . 3  |-  ( ph  ->  { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  (/) )
24 0fin 7272 . . 3  |-  (/)  e.  Fin
2523, 24syl6eqel 2475 . 2  |-  ( ph  ->  { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin )
26 eqid 2387 . . 3  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
27 dsmmcl.h . . 3  |-  H  =  ( Base `  ( S  (+)m  R ) )
281, 26, 6, 27, 2, 15dsmmelbas 26874 . 2  |-  ( ph  ->  (  .0.  e.  H  <->  (  .0.  e.  ( Base `  P )  /\  {
a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
299, 25, 28mpbir2and 889 1  |-  ( ph  ->  .0.  e.  H )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   {crab 2653   (/)c0 3571    o. ccom 4822    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   Fincfn 7045   Basecbs 13396   X_scprds 13596   0gc0g 13650   Mndcmnd 14611    (+)m cdsmm 26866
This theorem is referenced by:  dsmmsubg  26878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-hom 13480  df-cco 13481  df-prds 13598  df-0g 13654  df-mnd 14617  df-dsmm 26867
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