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Theorem pwslnmlem2 27298
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 4534 . . . 4  |-  ( A  u.  B )  e. 
_V
54a1i 10 . . 3  |-  ( ph  ->  ( A  u.  B
)  e.  _V )
6 ssun1 3351 . . . 4  |-  A  C_  ( A  u.  B
)
76a1i 10 . . 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 2296 . . . 4  |-  ( Base `  Z )  =  (
Base `  Z )
11 eqid 2296 . . . 4  |-  ( Base `  X )  =  (
Base `  X )
12 eqid 2296 . . . 4  |-  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) )  =  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )
138, 9, 10, 11, 12pwssplit3 27293 . . 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 1182 . 2  |-  ( ph  ->  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )  e.  ( Z LMHom 
X ) )
15 fvex 5555 . . . . . 6  |-  ( 0g
`  X )  e. 
_V
1612mptiniseg 5183 . . . . . 6  |-  ( ( 0g `  X )  e.  _V  ->  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( 0g `  X ) } )
1715, 16ax-mp 8 . . . . 5  |-  ( `' ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  {
x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( 0g `  X ) }
18 lmodgrp 15650 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Grp )
19 grpmnd 14510 . . . . . . . . . 10  |-  ( W  e.  Grp  ->  W  e.  Mnd )
201, 18, 193syl 18 . . . . . . . . 9  |-  ( ph  ->  W  e.  Mnd )
21 eqid 2296 . . . . . . . . . 10  |-  ( 0g
`  W )  =  ( 0g `  W
)
229, 21pws0g 14424 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  A  e.  _V )  ->  ( A  X.  {
( 0g `  W
) } )  =  ( 0g `  X
) )
2320, 2, 22sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( A  X.  {
( 0g `  W
) } )  =  ( 0g `  X
) )
2423eqcomd 2301 . . . . . . 7  |-  ( ph  ->  ( 0g `  X
)  =  ( A  X.  { ( 0g
`  W ) } ) )
2524eqeq2d 2307 . . . . . 6  |-  ( ph  ->  ( ( x  |`  A )  =  ( 0g `  X )  <-> 
( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) ) )
2625rabbidv 2793 . . . . 5  |-  ( ph  ->  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( 0g `  X ) }  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2717, 26syl5eq 2340 . . . 4  |-  ( ph  ->  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2827oveq2d 5890 . . 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 2296 . . . . . . 7  |-  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  =  { x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }
32 eqid 2296 . . . . . . 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 2296 . . . . . . 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 27294 . . . . . 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 1182 . . . . 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 15838 . . . . 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 27289 . . . . 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 18 . . . 4  |-  ( ph  ->  ( ( Zs  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )  e. LNoeM 
<->  Y  e. LNoeM ) )
4029, 39mpbird 223 . . 3  |-  ( ph  ->  ( Zs  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )  e. LNoeM )
4128, 40eqeltrd 2370 . 2  |-  ( ph  ->  ( Zs  ( `' ( x  e.  ( Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  e. LNoeM )
428, 9, 10, 11, 12pwssplit1 27291 . . . . . . 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 1182 . . . . . 6  |-  ( ph  ->  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) : ( Base `  Z ) -onto-> ( Base `  X ) )
44 forn 5470 . . . . . 6  |-  ( ( x  e.  ( Base `  Z )  |->  ( x  |`  A ) ) : ( Base `  Z
) -onto-> ( Base `  X
)  ->  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) )  =  ( Base `  X
) )
4543, 44syl 15 . . . . 5  |-  ( ph  ->  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) )  =  ( Base `  X
) )
4645oveq2d 5890 . . . 4  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  ( Xs  ( Base `  X ) ) )
47 pwslnmlem2.xn . . . . 5  |-  ( ph  ->  X  e. LNoeM )
4811ressid 13219 . . . . 5  |-  ( X  e. LNoeM  ->  ( Xs  ( Base `  X ) )  =  X )
4947, 48syl 15 . . . 4  |-  ( ph  ->  ( Xs  ( Base `  X
) )  =  X )
5046, 49eqtrd 2328 . . 3  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  X )
5150, 47eqeltrd 2370 . 2  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  e. LNoeM )
52 eqid 2296 . . 3  |-  ( 0g
`  X )  =  ( 0g `  X
)
53 eqid 2296 . . 3  |-  ( `' ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )
54 eqid 2296 . . 3  |-  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  =  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )
55 eqid 2296 . . 3  |-  ( Xs  ran  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) )  =  ( Xs 
ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )
5652, 53, 54, 55lmhmlnmsplit 27288 . 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 1182 1  |-  ( ph  ->  Z  e. LNoeM )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   ran crn 4706    |` cres 4707   "cima 4708   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165    ^s cpws 13363   0gc0g 13416   Mndcmnd 14377   Grpcgrp 14378   LModclmod 15643   LMHom clmhm 15792   LMIso clmim 15793    ~=ph𝑚 clmic 15794  LNoeMclnm 27276
This theorem is referenced by:  pwslnm  27299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lmhm 15795  df-lmim 15796  df-lmic 15797  df-lfig 27269  df-lnm 27277
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