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Theorem pwslnmlem2 29446
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 6378 . . . 4  |-  ( A  u.  B )  e. 
_V
54a1i 11 . . 3  |-  ( ph  ->  ( A  u.  B
)  e.  _V )
6 ssun1 3519 . . . 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 2443 . . . 4  |-  ( Base `  Z )  =  (
Base `  Z )
11 eqid 2443 . . . 4  |-  ( Base `  X )  =  (
Base `  X )
12 eqid 2443 . . . 4  |-  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) )  =  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )
138, 9, 10, 11, 12pwssplit3 17142 . . 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 1218 . 2  |-  ( ph  ->  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )  e.  ( Z LMHom 
X ) )
15 fvex 5701 . . . . . 6  |-  ( 0g
`  X )  e. 
_V
1612mptiniseg 5332 . . . . . 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 16955 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Grp )
19 grpmnd 15550 . . . . . . . . . 10  |-  ( W  e.  Grp  ->  W  e.  Mnd )
201, 18, 193syl 20 . . . . . . . . 9  |-  ( ph  ->  W  e.  Mnd )
21 eqid 2443 . . . . . . . . . 10  |-  ( 0g
`  W )  =  ( 0g `  W
)
229, 21pws0g 15457 . . . . . . . . 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 2448 . . . . . . 7  |-  ( ph  ->  ( 0g `  X
)  =  ( A  X.  { ( 0g
`  W ) } ) )
2524eqeq2d 2454 . . . . . 6  |-  ( ph  ->  ( ( x  |`  A )  =  ( 0g `  X )  <-> 
( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) ) )
2625rabbidv 2964 . . . . 5  |-  ( ph  ->  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( 0g `  X ) }  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2717, 26syl5eq 2487 . . . 4  |-  ( ph  ->  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2827oveq2d 6107 . . 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 2443 . . . . . . 7  |-  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  =  { x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }
32 eqid 2443 . . . . . . 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 2443 . . . . . . 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 29442 . . . . . 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 1218 . . . . 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 17150 . . . . 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 29441 . . . . 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 2517 . 2  |-  ( ph  ->  ( Zs  ( `' ( x  e.  ( Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  e. LNoeM )
428, 9, 10, 11, 12pwssplit1 17140 . . . . . . 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 1218 . . . . . 6  |-  ( ph  ->  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) : ( Base `  Z ) -onto-> ( Base `  X ) )
44 forn 5623 . . . . . 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 6107 . . . 4  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  ( Xs  ( Base `  X ) ) )
47 pwslnmlem2.xn . . . . 5  |-  ( ph  ->  X  e. LNoeM )
4811ressid 14233 . . . . 5  |-  ( X  e. LNoeM  ->  ( Xs  ( Base `  X ) )  =  X )
4947, 48syl 16 . . . 4  |-  ( ph  ->  ( Xs  ( Base `  X
) )  =  X )
5046, 49eqtrd 2475 . . 3  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  X )
5150, 47eqeltrd 2517 . 2  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  e. LNoeM )
52 eqid 2443 . . 3  |-  ( 0g
`  X )  =  ( 0g `  X
)
53 eqid 2443 . . 3  |-  ( `' ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )
54 eqid 2443 . . 3  |-  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  =  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )
55 eqid 2443 . . 3  |-  ( Xs  ran  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) )  =  ( Xs 
ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )
5652, 53, 54, 55lmhmlnmsplit 29440 . 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 1218 1  |-  ( ph  ->  Z  e. LNoeM )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    u. cun 3326    i^i cin 3327    C_ wss 3328   (/)c0 3637   {csn 3877   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   `'ccnv 4839   ran crn 4841    |` cres 4842   "cima 4843   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091   Basecbs 14174   ↾s cress 14175   0gc0g 14378    ^s cpws 14385   Mndcmnd 15409   Grpcgrp 15410   LModclmod 16948   LMHom clmhm 17100   LMIso clmim 17101    ~=ph𝑚 clmic 17102  LNoeMclnm 29428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-0g 14380  df-prds 14386  df-pws 14388  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-ghm 15745  df-cntz 15835  df-lsm 16135  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-lmod 16950  df-lss 17014  df-lsp 17053  df-lmhm 17103  df-lmim 17104  df-lmic 17105  df-lfig 29421  df-lnm 29429
This theorem is referenced by:  pwslnm  29447
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