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Theorem dsmmacl 18286
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 2452 . . 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 2452 . . . . . 6  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
9 dsmmcl.h . . . . . 6  |-  H  =  ( Base `  ( S  (+)m  R ) )
10 ffn 5662 . . . . . . 7  |-  ( R : I --> Mnd  ->  R  Fn  I )
116, 10syl 16 . . . . . 6  |-  ( ph  ->  R  Fn  I )
121, 8, 2, 9, 5, 11dsmmelbas 18284 . . . . 5  |-  ( ph  ->  ( J  e.  H  <->  ( J  e.  ( Base `  P )  /\  {
a  e.  I  |  ( J `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
137, 12mpbid 210 . . . 4  |-  ( ph  ->  ( J  e.  (
Base `  P )  /\  { a  e.  I  |  ( J `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) )
1413simpld 459 . . 3  |-  ( ph  ->  J  e.  ( Base `  P ) )
15 dsmmacl.k . . . . 5  |-  ( ph  ->  K  e.  H )
161, 8, 2, 9, 5, 11dsmmelbas 18284 . . . . 5  |-  ( ph  ->  ( K  e.  H  <->  ( K  e.  ( Base `  P )  /\  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
1715, 16mpbid 210 . . . 4  |-  ( ph  ->  ( K  e.  (
Base `  P )  /\  { a  e.  I  |  ( K `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) )
1817simpld 459 . . 3  |-  ( ph  ->  K  e.  ( Base `  P ) )
191, 2, 3, 4, 5, 6, 14, 18prdsplusgcl 15566 . 2  |-  ( ph  ->  ( J  .+  K
)  e.  ( Base `  P ) )
204adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  S  e.  V )
215adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  I  e.  W )
2211adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  R  Fn  I )
2314adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  J  e.  ( Base `  P
) )
2418adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  K  e.  ( Base `  P
) )
25 simpr 461 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  a  e.  I )
261, 2, 20, 21, 22, 23, 24, 3, 25prdsplusgfval 14526 . . . . 5  |-  ( (
ph  /\  a  e.  I )  ->  (
( J  .+  K
) `  a )  =  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) ) )
2726neeq1d 2726 . . . 4  |-  ( (
ph  /\  a  e.  I )  ->  (
( ( J  .+  K ) `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  ( ( J `  a )
( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) ) )
2827rabbidva 3063 . . 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 463 . . . . 5  |-  ( ph  ->  { a  e.  I  |  ( J `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
3017simprd 463 . . . . 5  |-  ( ph  ->  { a  e.  I  |  ( K `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
31 unfi 7685 . . . . 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 661 . . . 4  |-  ( ph  ->  ( { a  e.  I  |  ( J `
 a )  =/=  ( 0g `  ( R `  a )
) }  u.  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) } )  e.  Fin )
33 neorian 2776 . . . . . . . . . 10  |-  ( ( ( J `  a
)  =/=  ( 0g
`  ( R `  a ) )  \/  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) )  <->  -.  ( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) ) )
3433bicomi 202 . . . . . . . . 9  |-  ( -.  ( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) )  <->  ( ( J `
 a )  =/=  ( 0g `  ( R `  a )
)  \/  ( K `
 a )  =/=  ( 0g `  ( R `  a )
) ) )
3534con1bii 331 . . . . . . . 8  |-  ( -.  ( ( J `  a )  =/=  ( 0g `  ( R `  a ) )  \/  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) )  <-> 
( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) ) )
366ffvelrnda 5947 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  ( R `  a )  e.  Mnd )
37 eqid 2452 . . . . . . . . . . . 12  |-  ( Base `  ( R `  a
) )  =  (
Base `  ( R `  a ) )
38 eqid 2452 . . . . . . . . . . . 12  |-  ( 0g
`  ( R `  a ) )  =  ( 0g `  ( R `  a )
)
3937, 38mndidcl 15553 . . . . . . . . . . 11  |-  ( ( R `  a )  e.  Mnd  ->  ( 0g `  ( R `  a ) )  e.  ( Base `  ( R `  a )
) )
4036, 39syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  ( 0g `  ( R `  a ) )  e.  ( Base `  ( R `  a )
) )
41 eqid 2452 . . . . . . . . . . 11  |-  ( +g  `  ( R `  a
) )  =  ( +g  `  ( R `
 a ) )
4237, 41, 38mndlid 15555 . . . . . . . . . 10  |-  ( ( ( R `  a
)  e.  Mnd  /\  ( 0g `  ( R `
 a ) )  e.  ( Base `  ( R `  a )
) )  ->  (
( 0g `  ( R `  a )
) ( +g  `  ( R `  a )
) ( 0g `  ( R `  a ) ) )  =  ( 0g `  ( R `
 a ) ) )
4336, 40, 42syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g `  ( R `  a )
) ( +g  `  ( R `  a )
) ( 0g `  ( R `  a ) ) )  =  ( 0g `  ( R `
 a ) ) )
44 oveq12 6204 . . . . . . . . . 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 ) ) ) )
4544eqeq1d 2454 . . . . . . . . 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 ) ) ) )
4643, 45syl5ibrcom 222 . . . . . . . 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 ) ) ) )
4735, 46syl5bi 217 . . . . . . 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 ) ) ) )
4847necon1ad 2665 . . . . . 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 )
) ) ) )
4948ss2rabdv 3536 . . . . 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 )
) ) } )
50 unrab 3724 . . . . 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 )
) ) }
5149, 50syl6sseqr 3506 . . . 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 ) ) } ) )
52 ssfi 7639 . . . 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 )
5332, 51, 52syl2anc 661 . . 3  |-  ( ph  ->  { a  e.  I  |  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
5428, 53eqeltrd 2540 . 2  |-  ( ph  ->  { a  e.  I  |  ( ( J 
.+  K ) `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
551, 8, 2, 9, 5, 11dsmmelbas 18284 . 2  |-  ( ph  ->  ( ( J  .+  K )  e.  H  <->  ( ( J  .+  K
)  e.  ( Base `  P )  /\  {
a  e.  I  |  ( ( J  .+  K ) `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
5619, 54, 55mpbir2and 913 1  |-  ( ph  ->  ( J  .+  K
)  e.  H )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   {crab 2800    u. cun 3429    C_ wss 3431    Fn wfn 5516   -->wf 5517   ` cfv 5521  (class class class)co 6195   Fincfn 7415   Basecbs 14287   +g cplusg 14352   0gc0g 14492   X_scprds 14498   Mndcmnd 15523    (+)m cdsmm 18276
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-fz 11550  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-hom 14376  df-cco 14377  df-0g 14494  df-prds 14500  df-mnd 15529  df-dsmm 18277
This theorem is referenced by:  dsmmsubg  18288
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