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Theorem dsmmacl 27310
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 2296 . . 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 2296 . . . . . 6  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
9 dsmmcl.h . . . . . 6  |-  H  =  ( Base `  ( S  (+)m  R ) )
10 ffn 5405 . . . . . . 7  |-  ( R : I --> Mnd  ->  R  Fn  I )
116, 10syl 15 . . . . . 6  |-  ( ph  ->  R  Fn  I )
121, 8, 2, 9, 5, 11dsmmelbas 27308 . . . . 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 27308 . . . . 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 14419 . 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 13389 . . . . 5  |-  ( (
ph  /\  a  e.  I )  ->  (
( J  .+  K
) `  a )  =  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) ) )
2726neeq1d 2472 . . . 4  |-  ( (
ph  /\  a  e.  I )  ->  (
( ( J  .+  K ) `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  ( ( J `  a )
( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) ) )
2827rabbidva 2792 . . 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 7140 . . . . 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 2546 . . . . . . . . . 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 5679 . . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( Base `  ( R `  a
) )  =  (
Base `  ( R `  a ) )
39 eqid 2296 . . . . . . . . . . . 12  |-  ( 0g
`  ( R `  a ) )  =  ( 0g `  ( R `  a )
)
4038, 39mndidcl 14407 . . . . . . . . . . 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 2296 . . . . . . . . . . 11  |-  ( +g  `  ( R `  a
) )  =  ( +g  `  ( R `
 a ) )
4338, 42, 39mndlid 14409 . . . . . . . . . 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 5883 . . . . . . . . . 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 2304 . . . . . . . . 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 2526 . . . . . 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 3267 . . . . 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 3452 . . . . 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 3238 . . . 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 7099 . . . 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 2370 . 2  |-  ( ph  ->  { a  e.  I  |  ( ( J 
.+  K ) `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
561, 8, 2, 9, 5, 11dsmmelbas 27308 . 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 1632    e. wcel 1696    =/= wne 2459   {crab 2560    u. cun 3163    C_ wss 3165    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   +g cplusg 13224   X_scprds 13362   0gc0g 13416   Mndcmnd 14377    (+)m cdsmm 27300
This theorem is referenced by:  dsmmsubg  27312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-0g 13420  df-mnd 14383  df-dsmm 27301
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