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Theorem pwslnmlem2 27195
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 4518 . . . 4  |-  ( A  u.  B )  e. 
_V
54a1i 10 . . 3  |-  ( ph  ->  ( A  u.  B
)  e.  _V )
6 ssun1 3338 . . . 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 2283 . . . 4  |-  ( Base `  Z )  =  (
Base `  Z )
11 eqid 2283 . . . 4  |-  ( Base `  X )  =  (
Base `  X )
12 eqid 2283 . . . 4  |-  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) )  =  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )
138, 9, 10, 11, 12pwssplit3 27190 . . 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 5539 . . . . . 6  |-  ( 0g
`  X )  e. 
_V
1612mptiniseg 5167 . . . . . 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 15634 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Grp )
19 grpmnd 14494 . . . . . . . . . 10  |-  ( W  e.  Grp  ->  W  e.  Mnd )
201, 18, 193syl 18 . . . . . . . . 9  |-  ( ph  ->  W  e.  Mnd )
21 eqid 2283 . . . . . . . . . 10  |-  ( 0g
`  W )  =  ( 0g `  W
)
229, 21pws0g 14408 . . . . . . . . 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 2288 . . . . . . 7  |-  ( ph  ->  ( 0g `  X
)  =  ( A  X.  { ( 0g
`  W ) } ) )
2524eqeq2d 2294 . . . . . 6  |-  ( ph  ->  ( ( x  |`  A )  =  ( 0g `  X )  <-> 
( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) ) )
2625rabbidv 2780 . . . . 5  |-  ( ph  ->  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( 0g `  X ) }  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2717, 26syl5eq 2327 . . . 4  |-  ( ph  ->  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2827oveq2d 5874 . . 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 2283 . . . . . . 7  |-  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  =  { x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }
32 eqid 2283 . . . . . . 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 2283 . . . . . . 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 27191 . . . . . 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 15822 . . . . 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 27186 . . . . 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 2357 . 2  |-  ( ph  ->  ( Zs  ( `' ( x  e.  ( Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  e. LNoeM )
428, 9, 10, 11, 12pwssplit1 27188 . . . . . . 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 5454 . . . . . 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 5874 . . . 4  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  ( Xs  ( Base `  X ) ) )
47 pwslnmlem2.xn . . . . 5  |-  ( ph  ->  X  e. LNoeM )
4811ressid 13203 . . . . 5  |-  ( X  e. LNoeM  ->  ( Xs  ( Base `  X ) )  =  X )
4947, 48syl 15 . . . 4  |-  ( ph  ->  ( Xs  ( Base `  X
) )  =  X )
5046, 49eqtrd 2315 . . 3  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  X )
5150, 47eqeltrd 2357 . 2  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  e. LNoeM )
52 eqid 2283 . . 3  |-  ( 0g
`  X )  =  ( 0g `  X
)
53 eqid 2283 . . 3  |-  ( `' ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )
54 eqid 2283 . . 3  |-  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  =  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )
55 eqid 2283 . . 3  |-  ( Xs  ran  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) )  =  ( Xs 
ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )
5652, 53, 54, 55lmhmlnmsplit 27185 . 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 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149    ^s cpws 13347   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362   LModclmod 15627   LMHom clmhm 15776   LMIso clmim 15777    ~=ph𝑚 clmic 15778  LNoeMclnm 27173
This theorem is referenced by:  pwslnm  27196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lmhm 15779  df-lmim 15780  df-lmic 15781  df-lfig 27166  df-lnm 27174
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