Theorem pwslnmlem2 37663

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.

Step	Hyp	Ref	Expression
1		pwslnmlem2.w	. . 3 |- ( ph -> W e. LMod )
2		pwslnmlem2.a	. . . . 5 |- A e. _V
3		pwslnmlem2.b	. . . . 5 |- B e. _V
4	2, 3	unex 6956	. . . 4 |- ( A u. B ) e. _V
5	4	a1i 11	. . 3 |- ( ph -> ( A u. B ) e. _V )
6		ssun1 3776	. . . 4 |- A C_ ( A u. B )
7	6	a1i 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 2622	. . . 4 |- ( Base ` Z ) = ( Base ` Z )
11		eqid 2622	. . . 4 |- ( Base ` X ) = ( Base ` X )
12		eqid 2622	. . . 4 |- ( x e. ( Base ` Z ) |-> ( x |` A ) ) = ( x e. ( Base ` Z ) |-> ( x |` A ) )
13	8, 9, 10, 11, 12	pwssplit3 19061	. . 3 |- ( ( W e. LMod /\ ( A u. B ) e. _V /\ A C_ ( A u. B ) ) -> ( x e. ( Base ` Z ) |-> ( x |` A ) ) e. ( Z LMHom X ) )
14	1, 5, 7, 13	syl3anc 1326	. 2 |- ( ph -> ( x e. ( Base ` Z ) |-> ( x |` A ) ) e. ( Z LMHom X ) )
15		fvex 6201	. . . . . 6 |- ( 0g ` X ) e. _V
16	12	mptiniseg 5629	. . . . . 6 |- ( ( 0g ` X ) e. _V -> ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } ) = { x e. ( Base ` Z ) | ( x |` A ) = ( 0g ` X ) } )
17	15, 16	ax-mp 5	. . . . 5 |- ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } ) = { x e. ( Base ` Z ) | ( x |` A ) = ( 0g ` X ) }
18		lmodgrp 18870	. . . . . . . . . 10 |- ( W e. LMod -> W e. Grp )
19		grpmnd 17429	. . . . . . . . . 10 |- ( W e. Grp -> W e. Mnd )
20	1, 18, 19	3syl 18	. . . . . . . . 9 |- ( ph -> W e. Mnd )
21		eqid 2622	. . . . . . . . . 10 |- ( 0g ` W ) = ( 0g ` W )
22	9, 21	pws0g 17326	. . . . . . . . 9 |- ( ( W e. Mnd /\ A e. _V ) -> ( A X. { ( 0g ` W ) } ) = ( 0g ` X ) )
23	20, 2, 22	sylancl 694	. . . . . . . 8 |- ( ph -> ( A X. { ( 0g ` W ) } ) = ( 0g ` X ) )
24	23	eqcomd 2628	. . . . . . 7 |- ( ph -> ( 0g ` X ) = ( A X. { ( 0g ` W ) } ) )
25	24	eqeq2d 2632	. . . . . 6 |- ( ph -> ( ( x |` A ) = ( 0g ` X ) <-> ( x |` A ) = ( A X. { ( 0g ` W ) } ) ) )
26	25	rabbidv 3189	. . . . 5 |- ( ph -> { x e. ( Base ` Z ) | ( x |` A ) = ( 0g ` X ) } = { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } )
27	17, 26	syl5eq 2668	. . . 4 |- ( ph -> ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } ) = { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } )
28	27	oveq2d 6666	. . 3 |- ( ph -> ( Z ↾s ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } ) ) = ( Z ↾s { 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 2622	. . . . . . 7 |- { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } = { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) }
32		eqid 2622	. . . . . . 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 2622	. . . . . . 7 |- ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } ) = ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } )
35	8, 10, 21, 31, 32, 9, 33, 34	pwssplit4 37659	. . . . . 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. ( ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } ) LMIso Y ) )
36	1, 5, 30, 35	syl3anc 1326	. . . . 5 |- ( ph -> ( y e. { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } |-> ( y |` B ) ) e. ( ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } ) LMIso Y ) )
37		brlmici 19069	. . . . 5 |- ( ( y e. { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } |-> ( y |` B ) ) e. ( ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } ) LMIso Y ) -> ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } ) ~=ph𝑚 Y )
38		lnmlmic 37658	. . . . 5 |- ( ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } ) ~=ph𝑚 Y -> ( ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } ) e. LNoeM <-> Y e. LNoeM ) )
39	36, 37, 38	3syl 18	. . . 4 |- ( ph -> ( ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } ) e. LNoeM <-> Y e. LNoeM ) )
40	29, 39	mpbird 247	. . 3 |- ( ph -> ( Z ↾s { x e. ( Base ` Z ) | ( x |` A ) = ( A X. { ( 0g ` W ) } ) } ) e. LNoeM )
41	28, 40	eqeltrd 2701	. 2 |- ( ph -> ( Z ↾s ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } ) ) e. LNoeM )
42	8, 9, 10, 11, 12	pwssplit1 19059	. . . . . . 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 ) )
43	20, 5, 7, 42	syl3anc 1326	. . . . . 6 |- ( ph -> ( x e. ( Base ` Z ) |-> ( x |` A ) ) : ( Base ` Z ) -onto-> ( Base ` X ) )
44		forn 6118	. . . . . 6 |- ( ( x e. ( Base ` Z ) |-> ( x |` A ) ) : ( Base ` Z ) -onto-> ( Base ` X ) -> ran ( x e. ( Base ` Z ) |-> ( x |` A ) ) = ( Base ` X ) )
45	43, 44	syl 17	. . . . 5 |- ( ph -> ran ( x e. ( Base ` Z ) |-> ( x |` A ) ) = ( Base ` X ) )
46	45	oveq2d 6666	. . . 4 |- ( ph -> ( X ↾s ran ( x e. ( Base ` Z ) |-> ( x |` A ) ) ) = ( X ↾s ( Base ` X ) ) )
47		pwslnmlem2.xn	. . . . 5 |- ( ph -> X e. LNoeM )
48	11	ressid 15935	. . . . 5 |- ( X e. LNoeM -> ( X ↾s ( Base ` X ) ) = X )
49	47, 48	syl 17	. . . 4 |- ( ph -> ( X ↾s ( Base ` X ) ) = X )
50	46, 49	eqtrd 2656	. . 3 |- ( ph -> ( X ↾s ran ( x e. ( Base ` Z ) |-> ( x |` A ) ) ) = X )
51	50, 47	eqeltrd 2701	. 2 |- ( ph -> ( X ↾s ran ( x e. ( Base ` Z ) |-> ( x |` A ) ) ) e. LNoeM )
52		eqid 2622	. . 3 |- ( 0g ` X ) = ( 0g ` X )
53		eqid 2622	. . 3 |- ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } ) = ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } )
54		eqid 2622	. . 3 |- ( Z ↾s ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } ) ) = ( Z ↾s ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } ) )
55		eqid 2622	. . 3 |- ( X ↾s ran ( x e. ( Base ` Z ) |-> ( x |` A ) ) ) = ( X ↾s ran ( x e. ( Base ` Z ) |-> ( x |` A ) ) )
56	52, 53, 54, 55	lmhmlnmsplit 37657	. 2 |- ( ( ( x e. ( Base ` Z ) |-> ( x |` A ) ) e. ( Z LMHom X ) /\ ( Z ↾s ( `' ( x e. ( Base ` Z ) |-> ( x |` A ) ) " { ( 0g ` X ) } ) ) e. LNoeM /\ ( X ↾s ran ( x e. ( Base ` Z ) |-> ( x |` A ) ) ) e. LNoeM ) -> Z e. LNoeM )
57	14, 41, 51, 56	syl3anc 1326	1 |- ( ph -> Z e. LNoeM )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   ^_s cpws 16107   Mndcmnd 17294   Grpcgrp 17422   LModclmod 18863   LMHom clmhm 19019   LMIso clmim 19020   ~=ph_𝑚 clmic 19021  LNoeMclnm 37645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lmhm 19022  df-lmim 19023  df-lmic 19024  df-lfig 37638  df-lnm 37646

This theorem is referenced by:  pwslnm  37664