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Theorem dsmmacl 27207
Description: The finite hull is closed under addition. (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 )
dsmmacl.j  |-  ( ph  ->  J  e.  H )
dsmmacl.k  |-  ( ph  ->  K  e.  H )
dsmmacl.a  |-  .+  =  ( +g  `  P )
Assertion
Ref Expression
dsmmacl  |-  ( ph  ->  ( J  .+  K
)  e.  H )

Proof of Theorem dsmmacl
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dsmmcl.p . . 3  |-  P  =  ( S X_s R )
2 eqid 2283 . . 3  |-  ( Base `  P )  =  (
Base `  P )
3 dsmmacl.a . . 3  |-  .+  =  ( +g  `  P )
4 dsmmcl.s . . 3  |-  ( ph  ->  S  e.  V )
5 dsmmcl.i . . 3  |-  ( ph  ->  I  e.  W )
6 dsmmcl.r . . 3  |-  ( ph  ->  R : I --> Mnd )
7 dsmmacl.j . . . . 5  |-  ( ph  ->  J  e.  H )
8 eqid 2283 . . . . . 6  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
9 dsmmcl.h . . . . . 6  |-  H  =  ( Base `  ( S  (+)m  R ) )
10 ffn 5389 . . . . . . 7  |-  ( R : I --> Mnd  ->  R  Fn  I )
116, 10syl 15 . . . . . 6  |-  ( ph  ->  R  Fn  I )
121, 8, 2, 9, 5, 11dsmmelbas 27205 . . . . 5  |-  ( ph  ->  ( J  e.  H  <->  ( J  e.  ( Base `  P )  /\  {
a  e.  I  |  ( J `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
137, 12mpbid 201 . . . 4  |-  ( ph  ->  ( J  e.  (
Base `  P )  /\  { a  e.  I  |  ( J `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) )
1413simpld 445 . . 3  |-  ( ph  ->  J  e.  ( Base `  P ) )
15 dsmmacl.k . . . . 5  |-  ( ph  ->  K  e.  H )
161, 8, 2, 9, 5, 11dsmmelbas 27205 . . . . 5  |-  ( ph  ->  ( K  e.  H  <->  ( K  e.  ( Base `  P )  /\  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
1715, 16mpbid 201 . . . 4  |-  ( ph  ->  ( K  e.  (
Base `  P )  /\  { a  e.  I  |  ( K `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) )
1817simpld 445 . . 3  |-  ( ph  ->  K  e.  ( Base `  P ) )
191, 2, 3, 4, 5, 6, 14, 18prdsplusgcl 14403 . 2  |-  ( ph  ->  ( J  .+  K
)  e.  ( Base `  P ) )
204adantr 451 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  S  e.  V )
215adantr 451 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  I  e.  W )
2211adantr 451 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  R  Fn  I )
2314adantr 451 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  J  e.  ( Base `  P
) )
2418adantr 451 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  K  e.  ( Base `  P
) )
25 simpr 447 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  a  e.  I )
261, 2, 20, 21, 22, 23, 24, 3, 25prdsplusgfval 13373 . . . . 5  |-  ( (
ph  /\  a  e.  I )  ->  (
( J  .+  K
) `  a )  =  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) ) )
2726neeq1d 2459 . . . 4  |-  ( (
ph  /\  a  e.  I )  ->  (
( ( J  .+  K ) `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  ( ( J `  a )
( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) ) )
2827rabbidva 2779 . . 3  |-  ( ph  ->  { a  e.  I  |  ( ( J 
.+  K ) `  a )  =/=  ( 0g `  ( R `  a ) ) }  =  { a  e.  I  |  ( ( J `  a ) ( +g  `  ( R `  a )
) ( K `  a ) )  =/=  ( 0g `  ( R `  a )
) } )
2913simprd 449 . . . . 5  |-  ( ph  ->  { a  e.  I  |  ( J `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
3017simprd 449 . . . . 5  |-  ( ph  ->  { a  e.  I  |  ( K `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
31 unfi 7124 . . . . 5  |-  ( ( { a  e.  I  |  ( J `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin  /\  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin )  -> 
( { a  e.  I  |  ( J `
 a )  =/=  ( 0g `  ( R `  a )
) }  u.  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) } )  e.  Fin )
3229, 30, 31syl2anc 642 . . . 4  |-  ( ph  ->  ( { a  e.  I  |  ( J `
 a )  =/=  ( 0g `  ( R `  a )
) }  u.  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) } )  e.  Fin )
33 neorian 2533 . . . . . . . . . 10  |-  ( ( ( J `  a
)  =/=  ( 0g
`  ( R `  a ) )  \/  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) )  <->  -.  ( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) ) )
3433bicomi 193 . . . . . . . . 9  |-  ( -.  ( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) )  <->  ( ( J `
 a )  =/=  ( 0g `  ( R `  a )
)  \/  ( K `
 a )  =/=  ( 0g `  ( R `  a )
) ) )
3534con1bii 321 . . . . . . . 8  |-  ( -.  ( ( J `  a )  =/=  ( 0g `  ( R `  a ) )  \/  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) )  <-> 
( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) ) )
36 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( R : I --> Mnd  /\  a  e.  I )  ->  ( R `  a
)  e.  Mnd )
376, 36sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  ( R `  a )  e.  Mnd )
38 eqid 2283 . . . . . . . . . . . 12  |-  ( Base `  ( R `  a
) )  =  (
Base `  ( R `  a ) )
39 eqid 2283 . . . . . . . . . . . 12  |-  ( 0g
`  ( R `  a ) )  =  ( 0g `  ( R `  a )
)
4038, 39mndidcl 14391 . . . . . . . . . . 11  |-  ( ( R `  a )  e.  Mnd  ->  ( 0g `  ( R `  a ) )  e.  ( Base `  ( R `  a )
) )
4137, 40syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  ( 0g `  ( R `  a ) )  e.  ( Base `  ( R `  a )
) )
42 eqid 2283 . . . . . . . . . . 11  |-  ( +g  `  ( R `  a
) )  =  ( +g  `  ( R `
 a ) )
4338, 42, 39mndlid 14393 . . . . . . . . . 10  |-  ( ( ( R `  a
)  e.  Mnd  /\  ( 0g `  ( R `
 a ) )  e.  ( Base `  ( R `  a )
) )  ->  (
( 0g `  ( R `  a )
) ( +g  `  ( R `  a )
) ( 0g `  ( R `  a ) ) )  =  ( 0g `  ( R `
 a ) ) )
4437, 41, 43syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g `  ( R `  a )
) ( +g  `  ( R `  a )
) ( 0g `  ( R `  a ) ) )  =  ( 0g `  ( R `
 a ) ) )
45 oveq12 5867 . . . . . . . . . 10  |-  ( ( ( J `  a
)  =  ( 0g
`  ( R `  a ) )  /\  ( K `  a )  =  ( 0g `  ( R `  a ) ) )  ->  (
( J `  a
) ( +g  `  ( R `  a )
) ( K `  a ) )  =  ( ( 0g `  ( R `  a ) ) ( +g  `  ( R `  a )
) ( 0g `  ( R `  a ) ) ) )
4645eqeq1d 2291 . . . . . . . . 9  |-  ( ( ( J `  a
)  =  ( 0g
`  ( R `  a ) )  /\  ( K `  a )  =  ( 0g `  ( R `  a ) ) )  ->  (
( ( J `  a ) ( +g  `  ( R `  a
) ) ( K `
 a ) )  =  ( 0g `  ( R `  a ) )  <->  ( ( 0g
`  ( R `  a ) ) ( +g  `  ( R `
 a ) ) ( 0g `  ( R `  a )
) )  =  ( 0g `  ( R `
 a ) ) ) )
4744, 46syl5ibrcom 213 . . . . . . . 8  |-  ( (
ph  /\  a  e.  I )  ->  (
( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) )  ->  ( ( J `  a )
( +g  `  ( R `
 a ) ) ( K `  a
) )  =  ( 0g `  ( R `
 a ) ) ) )
4835, 47syl5bi 208 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  ( -.  ( ( J `  a )  =/=  ( 0g `  ( R `  a ) )  \/  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) )  ->  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) )  =  ( 0g `  ( R `
 a ) ) ) )
4948necon1ad 2513 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  (
( ( J `  a ) ( +g  `  ( R `  a
) ) ( K `
 a ) )  =/=  ( 0g `  ( R `  a ) )  ->  ( ( J `  a )  =/=  ( 0g `  ( R `  a )
)  \/  ( K `
 a )  =/=  ( 0g `  ( R `  a )
) ) ) )
5049ss2rabdv 3254 . . . . 5  |-  ( ph  ->  { a  e.  I  |  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) } 
C_  { a  e.  I  |  ( ( J `  a )  =/=  ( 0g `  ( R `  a ) )  \/  ( K `
 a )  =/=  ( 0g `  ( R `  a )
) ) } )
51 unrab 3439 . . . . 5  |-  ( { a  e.  I  |  ( J `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  u.  { a  e.  I  |  ( K `
 a )  =/=  ( 0g `  ( R `  a )
) } )  =  { a  e.  I  |  ( ( J `
 a )  =/=  ( 0g `  ( R `  a )
)  \/  ( K `
 a )  =/=  ( 0g `  ( R `  a )
) ) }
5250, 51syl6sseqr 3225 . . . 4  |-  ( ph  ->  { a  e.  I  |  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) } 
C_  ( { a  e.  I  |  ( J `  a )  =/=  ( 0g `  ( R `  a ) ) }  u.  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) } ) )
53 ssfi 7083 . . . 4  |-  ( ( ( { a  e.  I  |  ( J `
 a )  =/=  ( 0g `  ( R `  a )
) }  u.  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) } )  e.  Fin  /\  { a  e.  I  |  ( ( J `  a ) ( +g  `  ( R `  a
) ) ( K `
 a ) )  =/=  ( 0g `  ( R `  a ) ) }  C_  ( { a  e.  I  |  ( J `  a )  =/=  ( 0g `  ( R `  a ) ) }  u.  { a  e.  I  |  ( K `
 a )  =/=  ( 0g `  ( R `  a )
) } ) )  ->  { a  e.  I  |  ( ( J `  a ) ( +g  `  ( R `  a )
) ( K `  a ) )  =/=  ( 0g `  ( R `  a )
) }  e.  Fin )
5432, 52, 53syl2anc 642 . . 3  |-  ( ph  ->  { a  e.  I  |  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
5528, 54eqeltrd 2357 . 2  |-  ( ph  ->  { a  e.  I  |  ( ( J 
.+  K ) `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
561, 8, 2, 9, 5, 11dsmmelbas 27205 . 2  |-  ( ph  ->  ( ( J  .+  K )  e.  H  <->  ( ( J  .+  K
)  e.  ( Base `  P )  /\  {
a  e.  I  |  ( ( J  .+  K ) `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
5719, 55, 56mpbir2and 888 1  |-  ( ph  ->  ( J  .+  K
)  e.  H )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    u. cun 3150    C_ wss 3152    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   +g cplusg 13208   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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