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Theorem dsmmacl 19235
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 2429 . . 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 2429 . . . . . 6  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
9 dsmmcl.h . . . . . 6  |-  H  =  ( Base `  ( S  (+)m  R ) )
10 ffn 5746 . . . . . . 7  |-  ( R : I --> Mnd  ->  R  Fn  I )
116, 10syl 17 . . . . . 6  |-  ( ph  ->  R  Fn  I )
121, 8, 2, 9, 5, 11dsmmelbas 19233 . . . . 5  |-  ( ph  ->  ( J  e.  H  <->  ( J  e.  ( Base `  P )  /\  {
a  e.  I  |  ( J `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
137, 12mpbid 213 . . . 4  |-  ( ph  ->  ( J  e.  (
Base `  P )  /\  { a  e.  I  |  ( J `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) )
1413simpld 460 . . 3  |-  ( ph  ->  J  e.  ( Base `  P ) )
15 dsmmacl.k . . . . 5  |-  ( ph  ->  K  e.  H )
161, 8, 2, 9, 5, 11dsmmelbas 19233 . . . . 5  |-  ( ph  ->  ( K  e.  H  <->  ( K  e.  ( Base `  P )  /\  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
1715, 16mpbid 213 . . . 4  |-  ( ph  ->  ( K  e.  (
Base `  P )  /\  { a  e.  I  |  ( K `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin ) )
1817simpld 460 . . 3  |-  ( ph  ->  K  e.  ( Base `  P ) )
191, 2, 3, 4, 5, 6, 14, 18prdsplusgcl 16518 . 2  |-  ( ph  ->  ( J  .+  K
)  e.  ( Base `  P ) )
204adantr 466 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  S  e.  V )
215adantr 466 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  I  e.  W )
2211adantr 466 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  R  Fn  I )
2314adantr 466 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  J  e.  ( Base `  P
) )
2418adantr 466 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  K  e.  ( Base `  P
) )
25 simpr 462 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  a  e.  I )
261, 2, 20, 21, 22, 23, 24, 3, 25prdsplusgfval 15331 . . . . 5  |-  ( (
ph  /\  a  e.  I )  ->  (
( J  .+  K
) `  a )  =  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) ) )
2726neeq1d 2708 . . . 4  |-  ( (
ph  /\  a  e.  I )  ->  (
( ( J  .+  K ) `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  ( ( J `  a )
( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) ) )
2827rabbidva 3078 . . 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 464 . . . . 5  |-  ( ph  ->  { a  e.  I  |  ( J `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
3017simprd 464 . . . . 5  |-  ( ph  ->  { a  e.  I  |  ( K `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
31 unfi 7844 . . . . 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 665 . . . 4  |-  ( ph  ->  ( { a  e.  I  |  ( J `
 a )  =/=  ( 0g `  ( R `  a )
) }  u.  {
a  e.  I  |  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) } )  e.  Fin )
33 neorian 2758 . . . . . . . . . 10  |-  ( ( ( J `  a
)  =/=  ( 0g
`  ( R `  a ) )  \/  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) )  <->  -.  ( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) ) )
3433bicomi 205 . . . . . . . . 9  |-  ( -.  ( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) )  <->  ( ( J `
 a )  =/=  ( 0g `  ( R `  a )
)  \/  ( K `
 a )  =/=  ( 0g `  ( R `  a )
) ) )
3534con1bii 332 . . . . . . . 8  |-  ( -.  ( ( J `  a )  =/=  ( 0g `  ( R `  a ) )  \/  ( K `  a
)  =/=  ( 0g
`  ( R `  a ) ) )  <-> 
( ( J `  a )  =  ( 0g `  ( R `
 a ) )  /\  ( K `  a )  =  ( 0g `  ( R `
 a ) ) ) )
366ffvelrnda 6037 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  ( R `  a )  e.  Mnd )
37 eqid 2429 . . . . . . . . . . . 12  |-  ( Base `  ( R `  a
) )  =  (
Base `  ( R `  a ) )
38 eqid 2429 . . . . . . . . . . . 12  |-  ( 0g
`  ( R `  a ) )  =  ( 0g `  ( R `  a )
)
3937, 38mndidcl 16505 . . . . . . . . . . 11  |-  ( ( R `  a )  e.  Mnd  ->  ( 0g `  ( R `  a ) )  e.  ( Base `  ( R `  a )
) )
4036, 39syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  ( 0g `  ( R `  a ) )  e.  ( Base `  ( R `  a )
) )
41 eqid 2429 . . . . . . . . . . 11  |-  ( +g  `  ( R `  a
) )  =  ( +g  `  ( R `
 a ) )
4237, 41, 38mndlid 16508 . . . . . . . . . 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 665 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g `  ( R `  a )
) ( +g  `  ( R `  a )
) ( 0g `  ( R `  a ) ) )  =  ( 0g `  ( R `
 a ) ) )
44 oveq12 6314 . . . . . . . . . 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 2431 . . . . . . . . 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 225 . . . . . . . 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 220 . . . . . . 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 2647 . . . . . 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 3548 . . . . 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 3750 . . . . 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 3517 . . . 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 7798 . . . 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 665 . . 3  |-  ( ph  ->  { a  e.  I  |  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
5428, 53eqeltrd 2517 . 2  |-  ( ph  ->  { a  e.  I  |  ( ( J 
.+  K ) `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
551, 8, 2, 9, 5, 11dsmmelbas 19233 . 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 930 1  |-  ( ph  ->  ( J  .+  K
)  e.  H )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   {crab 2786    u. cun 3440    C_ wss 3442    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   Fincfn 7577   Basecbs 15084   +g cplusg 15152   0gc0g 15297   X_scprds 15303   Mndcmnd 16486    (+)m cdsmm 19225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-hom 15176  df-cco 15177  df-0g 15299  df-prds 15305  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-dsmm 19226
This theorem is referenced by:  dsmmsubg  19237
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