MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dsmmacl Structured version   Unicode version

Theorem dsmmacl 19246
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 2428 . . 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 2428 . . . . . 6  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
9 dsmmcl.h . . . . . 6  |-  H  =  ( Base `  ( S  (+)m  R ) )
10 ffn 5689 . . . . . . 7  |-  ( R : I --> Mnd  ->  R  Fn  I )
116, 10syl 17 . . . . . 6  |-  ( ph  ->  R  Fn  I )
121, 8, 2, 9, 5, 11dsmmelbas 19244 . . . . 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 19244 . . . . 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 16510 . 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 15315 . . . . 5  |-  ( (
ph  /\  a  e.  I )  ->  (
( J  .+  K
) `  a )  =  ( ( J `
 a ) ( +g  `  ( R `
 a ) ) ( K `  a
) ) )
2726neeq1d 2660 . . . 4  |-  ( (
ph  /\  a  e.  I )  ->  (
( ( J  .+  K ) `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  ( ( J `  a )
( +g  `  ( R `
 a ) ) ( K `  a
) )  =/=  ( 0g `  ( R `  a ) ) ) )
2827rabbidva 3012 . . 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 7791 . . . . 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 2695 . . . . . . . . . 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 5981 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  ( R `  a )  e.  Mnd )
37 eqid 2428 . . . . . . . . . . . 12  |-  ( Base `  ( R `  a
) )  =  (
Base `  ( R `  a ) )
38 eqid 2428 . . . . . . . . . . . 12  |-  ( 0g
`  ( R `  a ) )  =  ( 0g `  ( R `  a )
)
3937, 38mndidcl 16497 . . . . . . . . . . 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 2428 . . . . . . . . . . 11  |-  ( +g  `  ( R `  a
) )  =  ( +g  `  ( R `
 a ) )
4237, 41, 38mndlid 16500 . . . . . . . . . 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 6258 . . . . . . . . . 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 2430 . . . . . . . . 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 2618 . . . . . 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 3485 . . . . 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 3687 . . . . 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 3454 . . . 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 7745 . . . 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 2506 . 2  |-  ( ph  ->  { a  e.  I  |  ( ( J 
.+  K ) `  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
551, 8, 2, 9, 5, 11dsmmelbas 19244 . 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 1872    =/= wne 2599   {crab 2718    u. cun 3377    C_ wss 3379    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   Fincfn 7524   Basecbs 15064   +g cplusg 15133   0gc0g 15281   X_scprds 15287   Mndcmnd 16478    (+)m cdsmm 19236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
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 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-ixp 7478  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-sup 7909  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-10 10627  df-n0 10821  df-z 10889  df-dec 11003  df-uz 11111  df-fz 11736  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-mulr 15147  df-sca 15149  df-vsca 15150  df-ip 15151  df-tset 15152  df-ple 15153  df-ds 15155  df-hom 15157  df-cco 15158  df-0g 15283  df-prds 15289  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-dsmm 19237
This theorem is referenced by:  dsmmsubg  19248
  Copyright terms: Public domain W3C validator