Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwslnmlem2 Structured version   Unicode version

Theorem pwslnmlem2 31222
Description: A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypotheses
Ref Expression
pwslnmlem2.a  |-  A  e. 
_V
pwslnmlem2.b  |-  B  e. 
_V
pwslnmlem2.x  |-  X  =  ( W  ^s  A )
pwslnmlem2.y  |-  Y  =  ( W  ^s  B )
pwslnmlem2.z  |-  Z  =  ( W  ^s  ( A  u.  B ) )
pwslnmlem2.w  |-  ( ph  ->  W  e.  LMod )
pwslnmlem2.dj  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
pwslnmlem2.xn  |-  ( ph  ->  X  e. LNoeM )
pwslnmlem2.yn  |-  ( ph  ->  Y  e. LNoeM )
Assertion
Ref Expression
pwslnmlem2  |-  ( ph  ->  Z  e. LNoeM )

Proof of Theorem pwslnmlem2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwslnmlem2.w . . 3  |-  ( ph  ->  W  e.  LMod )
2 pwslnmlem2.a . . . . 5  |-  A  e. 
_V
3 pwslnmlem2.b . . . . 5  |-  B  e. 
_V
42, 3unex 6597 . . . 4  |-  ( A  u.  B )  e. 
_V
54a1i 11 . . 3  |-  ( ph  ->  ( A  u.  B
)  e.  _V )
6 ssun1 3663 . . . 4  |-  A  C_  ( A  u.  B
)
76a1i 11 . . 3  |-  ( ph  ->  A  C_  ( A  u.  B ) )
8 pwslnmlem2.z . . . 4  |-  Z  =  ( W  ^s  ( A  u.  B ) )
9 pwslnmlem2.x . . . 4  |-  X  =  ( W  ^s  A )
10 eqid 2457 . . . 4  |-  ( Base `  Z )  =  (
Base `  Z )
11 eqid 2457 . . . 4  |-  ( Base `  X )  =  (
Base `  X )
12 eqid 2457 . . . 4  |-  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) )  =  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )
138, 9, 10, 11, 12pwssplit3 17834 . . 3  |-  ( ( W  e.  LMod  /\  ( A  u.  B )  e.  _V  /\  A  C_  ( A  u.  B
) )  ->  (
x  e.  ( Base `  Z )  |->  ( x  |`  A ) )  e.  ( Z LMHom  X ) )
141, 5, 7, 13syl3anc 1228 . 2  |-  ( ph  ->  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )  e.  ( Z LMHom 
X ) )
15 fvex 5882 . . . . . 6  |-  ( 0g
`  X )  e. 
_V
1612mptiniseg 5507 . . . . . 6  |-  ( ( 0g `  X )  e.  _V  ->  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( 0g `  X ) } )
1715, 16ax-mp 5 . . . . 5  |-  ( `' ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  {
x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( 0g `  X ) }
18 lmodgrp 17646 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Grp )
19 grpmnd 16189 . . . . . . . . . 10  |-  ( W  e.  Grp  ->  W  e.  Mnd )
201, 18, 193syl 20 . . . . . . . . 9  |-  ( ph  ->  W  e.  Mnd )
21 eqid 2457 . . . . . . . . . 10  |-  ( 0g
`  W )  =  ( 0g `  W
)
229, 21pws0g 16083 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  A  e.  _V )  ->  ( A  X.  {
( 0g `  W
) } )  =  ( 0g `  X
) )
2320, 2, 22sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( A  X.  {
( 0g `  W
) } )  =  ( 0g `  X
) )
2423eqcomd 2465 . . . . . . 7  |-  ( ph  ->  ( 0g `  X
)  =  ( A  X.  { ( 0g
`  W ) } ) )
2524eqeq2d 2471 . . . . . 6  |-  ( ph  ->  ( ( x  |`  A )  =  ( 0g `  X )  <-> 
( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) ) )
2625rabbidv 3101 . . . . 5  |-  ( ph  ->  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( 0g `  X ) }  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2717, 26syl5eq 2510 . . . 4  |-  ( ph  ->  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2827oveq2d 6312 . . 3  |-  ( ph  ->  ( Zs  ( `' ( x  e.  ( Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  =  ( Zs  { x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } ) )
29 pwslnmlem2.yn . . . 4  |-  ( ph  ->  Y  e. LNoeM )
30 pwslnmlem2.dj . . . . . 6  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
31 eqid 2457 . . . . . . 7  |-  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  =  { x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }
32 eqid 2457 . . . . . . 7  |-  ( y  e.  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  |->  ( y  |`  B )
)  =  ( y  e.  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  |->  ( y  |`  B )
)
33 pwslnmlem2.y . . . . . . 7  |-  Y  =  ( W  ^s  B )
34 eqid 2457 . . . . . . 7  |-  ( Zs  { x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )  =  ( Zs  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
358, 10, 21, 31, 32, 9, 33, 34pwssplit4 31218 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( A  u.  B )  e.  _V  /\  ( A  i^i  B )  =  (/) )  ->  ( y  e.  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  |->  ( y  |`  B )
)  e.  ( ( Zs  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } ) LMIso 
Y ) )
361, 5, 30, 35syl3anc 1228 . . . . 5  |-  ( ph  ->  ( y  e.  {
x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  |->  ( y  |`  B ) )  e.  ( ( Zs  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } ) LMIso 
Y ) )
37 brlmici 17842 . . . . 5  |-  ( ( y  e.  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  |->  ( y  |`  B )
)  e.  ( ( Zs  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } ) LMIso 
Y )  ->  ( Zs  { x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )  ~=ph𝑚 
Y )
38 lnmlmic 31217 . . . . 5  |-  ( ( Zs  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } ) 
~=ph𝑚  Y  ->  ( ( Zs  { x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )  e. LNoeM  <->  Y  e. LNoeM ) )
3936, 37, 383syl 20 . . . 4  |-  ( ph  ->  ( ( Zs  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )  e. LNoeM 
<->  Y  e. LNoeM ) )
4029, 39mpbird 232 . . 3  |-  ( ph  ->  ( Zs  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )  e. LNoeM )
4128, 40eqeltrd 2545 . 2  |-  ( ph  ->  ( Zs  ( `' ( x  e.  ( Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  e. LNoeM )
428, 9, 10, 11, 12pwssplit1 17832 . . . . . . 7  |-  ( ( W  e.  Mnd  /\  ( A  u.  B
)  e.  _V  /\  A  C_  ( A  u.  B ) )  -> 
( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) : ( Base `  Z ) -onto-> ( Base `  X ) )
4320, 5, 7, 42syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) : ( Base `  Z ) -onto-> ( Base `  X ) )
44 forn 5804 . . . . . 6  |-  ( ( x  e.  ( Base `  Z )  |->  ( x  |`  A ) ) : ( Base `  Z
) -onto-> ( Base `  X
)  ->  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) )  =  ( Base `  X
) )
4543, 44syl 16 . . . . 5  |-  ( ph  ->  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) )  =  ( Base `  X
) )
4645oveq2d 6312 . . . 4  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  ( Xs  ( Base `  X ) ) )
47 pwslnmlem2.xn . . . . 5  |-  ( ph  ->  X  e. LNoeM )
4811ressid 14706 . . . . 5  |-  ( X  e. LNoeM  ->  ( Xs  ( Base `  X ) )  =  X )
4947, 48syl 16 . . . 4  |-  ( ph  ->  ( Xs  ( Base `  X
) )  =  X )
5046, 49eqtrd 2498 . . 3  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  X )
5150, 47eqeltrd 2545 . 2  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  e. LNoeM )
52 eqid 2457 . . 3  |-  ( 0g
`  X )  =  ( 0g `  X
)
53 eqid 2457 . . 3  |-  ( `' ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )
54 eqid 2457 . . 3  |-  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  =  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )
55 eqid 2457 . . 3  |-  ( Xs  ran  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) )  =  ( Xs 
ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )
5652, 53, 54, 55lmhmlnmsplit 31216 . 2  |-  ( ( ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )  e.  ( Z LMHom 
X )  /\  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  e. LNoeM  /\  ( Xs  ran  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) )  e. LNoeM )  ->  Z  e. LNoeM )
5714, 41, 51, 56syl3anc 1228 1  |-  ( ph  ->  Z  e. LNoeM )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   ran crn 5009    |` cres 5010   "cima 5011   -onto->wfo 5592   ` cfv 5594  (class class class)co 6296   Basecbs 14644   ↾s cress 14645   0gc0g 14857    ^s cpws 14864   Mndcmnd 16046   Grpcgrp 16180   LModclmod 17639   LMHom clmhm 17792   LMIso clmim 17793    ~=ph𝑚 clmic 17794  LNoeMclnm 31204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-hom 14736  df-cco 14737  df-0g 14859  df-prds 14865  df-pws 14867  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-subg 16325  df-ghm 16392  df-cntz 16482  df-lsm 16783  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-lmod 17641  df-lss 17706  df-lsp 17745  df-lmhm 17795  df-lmim 17796  df-lmic 17797  df-lfig 31197  df-lnm 31205
This theorem is referenced by:  pwslnm  31223
  Copyright terms: Public domain W3C validator