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Theorem dsmm0cl 27206
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 14405 . . 3  |-  ( ph  ->  P  e.  Mnd )
6 eqid 2283 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
7 dsmm0cl.z . . . 4  |-  .0.  =  ( 0g `  P )
86, 7mndidcl 14391 . . 3  |-  ( P  e.  Mnd  ->  .0.  e.  ( Base `  P
) )
95, 8syl 15 . 2  |-  ( ph  ->  .0.  e.  ( Base `  P ) )
101, 2, 3, 4prds0g 14406 . . . . . . . . . 10  |-  ( ph  ->  ( 0g  o.  R
)  =  ( 0g
`  P ) )
1110, 7syl6eqr 2333 . . . . . . . . 9  |-  ( ph  ->  ( 0g  o.  R
)  =  .0.  )
1211adantr 451 . . . . . . . 8  |-  ( (
ph  /\  a  e.  I )  ->  ( 0g  o.  R )  =  .0.  )
1312fveq1d 5527 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g  o.  R
) `  a )  =  (  .0.  `  a
) )
14 ffn 5389 . . . . . . . . 9  |-  ( R : I --> Mnd  ->  R  Fn  I )
154, 14syl 15 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
16 fvco2 5594 . . . . . . . 8  |-  ( ( R  Fn  I  /\  a  e.  I )  ->  ( ( 0g  o.  R ) `  a
)  =  ( 0g
`  ( R `  a ) ) )
1715, 16sylan 457 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g  o.  R
) `  a )  =  ( 0g `  ( R `  a ) ) )
1813, 17eqtr3d 2317 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  (  .0.  `  a )  =  ( 0g `  ( R `  a )
) )
19 nne 2450 . . . . . 6  |-  ( -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  (  .0.  `  a )  =  ( 0g `  ( R `
 a ) ) )
2018, 19sylibr 203 . . . . 5  |-  ( (
ph  /\  a  e.  I )  ->  -.  (  .0.  `  a )  =/=  ( 0g `  ( R `  a )
) )
2120ralrimiva 2626 . . . 4  |-  ( ph  ->  A. a  e.  I  -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) )
22 rabeq0 3476 . . . 4  |-  ( { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  (/)  <->  A. a  e.  I  -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) )
2321, 22sylibr 203 . . 3  |-  ( ph  ->  { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  (/) )
24 0fin 7087 . . 3  |-  (/)  e.  Fin
2523, 24syl6eqel 2371 . 2  |-  ( ph  ->  { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin )
26 eqid 2283 . . 3  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
27 dsmmcl.h . . 3  |-  H  =  ( Base `  ( S  (+)m  R ) )
281, 26, 6, 27, 2, 15dsmmelbas 27205 . 2  |-  ( ph  ->  (  .0.  e.  H  <->  (  .0.  e.  ( Base `  P )  /\  {
a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
299, 25, 28mpbir2and 888 1  |-  ( ph  ->  .0.  e.  H )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   (/)c0 3455    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   X_scprds 13346   0gc0g 13400   Mndcmnd 14361    (+)m cdsmm 27197
This theorem is referenced by:  dsmmsubg  27209
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  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-0g 13404  df-mnd 14367  df-dsmm 27198
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