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Theorem pwslnmlem2 27172
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 4707 . . . 4  |-  ( A  u.  B )  e. 
_V
54a1i 11 . . 3  |-  ( ph  ->  ( A  u.  B
)  e.  _V )
6 ssun1 3510 . . . 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 2436 . . . 4  |-  ( Base `  Z )  =  (
Base `  Z )
11 eqid 2436 . . . 4  |-  ( Base `  X )  =  (
Base `  X )
12 eqid 2436 . . . 4  |-  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) )  =  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )
138, 9, 10, 11, 12pwssplit3 27167 . . 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 1184 . 2  |-  ( ph  ->  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) )  e.  ( Z LMHom 
X ) )
15 fvex 5742 . . . . . 6  |-  ( 0g
`  X )  e. 
_V
1612mptiniseg 5364 . . . . . 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 15957 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Grp )
19 grpmnd 14817 . . . . . . . . . 10  |-  ( W  e.  Grp  ->  W  e.  Mnd )
201, 18, 193syl 19 . . . . . . . . 9  |-  ( ph  ->  W  e.  Mnd )
21 eqid 2436 . . . . . . . . . 10  |-  ( 0g
`  W )  =  ( 0g `  W
)
229, 21pws0g 14731 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  A  e.  _V )  ->  ( A  X.  {
( 0g `  W
) } )  =  ( 0g `  X
) )
2320, 2, 22sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( A  X.  {
( 0g `  W
) } )  =  ( 0g `  X
) )
2423eqcomd 2441 . . . . . . 7  |-  ( ph  ->  ( 0g `  X
)  =  ( A  X.  { ( 0g
`  W ) } ) )
2524eqeq2d 2447 . . . . . 6  |-  ( ph  ->  ( ( x  |`  A )  =  ( 0g `  X )  <-> 
( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) ) )
2625rabbidv 2948 . . . . 5  |-  ( ph  ->  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( 0g `  X ) }  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2717, 26syl5eq 2480 . . . 4  |-  ( ph  ->  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )
2827oveq2d 6097 . . 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 2436 . . . . . . 7  |-  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }  =  { x  e.  ( Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) }
32 eqid 2436 . . . . . . 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 2436 . . . . . . 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 27168 . . . . . 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 1184 . . . . 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 16141 . . . . 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 27163 . . . . 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 19 . . . 4  |-  ( ph  ->  ( ( Zs  { x  e.  ( Base `  Z
)  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )  e. LNoeM 
<->  Y  e. LNoeM ) )
4029, 39mpbird 224 . . 3  |-  ( ph  ->  ( Zs  { x  e.  (
Base `  Z )  |  ( x  |`  A )  =  ( A  X.  { ( 0g `  W ) } ) } )  e. LNoeM )
4128, 40eqeltrd 2510 . 2  |-  ( ph  ->  ( Zs  ( `' ( x  e.  ( Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  e. LNoeM )
428, 9, 10, 11, 12pwssplit1 27165 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) : ( Base `  Z ) -onto-> ( Base `  X ) )
44 forn 5656 . . . . . 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 6097 . . . 4  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  ( Xs  ( Base `  X ) ) )
47 pwslnmlem2.xn . . . . 5  |-  ( ph  ->  X  e. LNoeM )
4811ressid 13524 . . . . 5  |-  ( X  e. LNoeM  ->  ( Xs  ( Base `  X ) )  =  X )
4947, 48syl 16 . . . 4  |-  ( ph  ->  ( Xs  ( Base `  X
) )  =  X )
5046, 49eqtrd 2468 . . 3  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  =  X )
5150, 47eqeltrd 2510 . 2  |-  ( ph  ->  ( Xs  ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )  e. LNoeM )
52 eqid 2436 . . 3  |-  ( 0g
`  X )  =  ( 0g `  X
)
53 eqid 2436 . . 3  |-  ( `' ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )  =  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } )
54 eqid 2436 . . 3  |-  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )  =  ( Zs  ( `' ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) " { ( 0g `  X ) } ) )
55 eqid 2436 . . 3  |-  ( Xs  ran  ( x  e.  (
Base `  Z )  |->  ( x  |`  A ) ) )  =  ( Xs 
ran  ( x  e.  ( Base `  Z
)  |->  ( x  |`  A ) ) )
5652, 53, 54, 55lmhmlnmsplit 27162 . 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 1184 1  |-  ( ph  ->  Z  e. LNoeM )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814   class class class wbr 4212    e. cmpt 4266    X. cxp 4876   `'ccnv 4877   ran crn 4879    |` cres 4880   "cima 4881   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470    ^s cpws 13670   0gc0g 13723   Mndcmnd 14684   Grpcgrp 14685   LModclmod 15950   LMHom clmhm 16095   LMIso clmim 16096    ~=ph𝑚 clmic 16097  LNoeMclnm 27150
This theorem is referenced by:  pwslnm  27173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-hom 13553  df-cco 13554  df-prds 13671  df-pws 13673  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-ghm 15004  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lmhm 16098  df-lmim 16099  df-lmic 16100  df-lfig 27143  df-lnm 27151
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