Theorem mpllsslem 19435

Description: If A is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set D of finite bags (the primary applications being A = Fin and A = ~P B for some B), then the set of all power series whose coefficient functions are supported on an element of A is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 16-Jul-2019.)

Hypotheses

Ref	Expression
mplsubglem.s	|- S = ( I mPwSer R )
mplsubglem.b	|- B = ( Base ` S )
mplsubglem.z	|- .0. = ( 0g ` R )
mplsubglem.d	|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
mplsubglem.i	|- ( ph -> I e. W )
mplsubglem.0	|- ( ph -> (/) e. A )
mplsubglem.a	|- ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x u. y ) e. A )
mplsubglem.y	|- ( ( ph /\ ( x e. A /\ y C_ x ) ) -> y e. A )
mplsubglem.u	|- ( ph -> U = { g e. B | ( g supp .0. ) e. A } )
mpllsslem.r	|- ( ph -> R e. Ring )

Assertion

Ref	Expression
mpllsslem	|- ( ph -> U e. ( LSubSp ` S ) )

Distinct variable groups:   f, g, x, y, .0.   A, f, g, x, y   B, f, g   D, g   f, I   ph, x, y   S, f, g, y

Allowed substitution hints:   ph( f, g)   B( x, y)   D( x, y, f)   R( x, y, f, g)   S( x)   U( x, y, f, g)   I( x, y, g)   W( x, y, f, g)

Proof of Theorem mpllsslem

Dummy variables k u v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplsubglem.s	. . 3 |- S = ( I mPwSer R )
2		mplsubglem.i	. . 3 |- ( ph -> I e. W )
3		mpllsslem.r	. . 3 |- ( ph -> R e. Ring )
4	1, 2, 3	psrsca 19389	. 2 |- ( ph -> R = (Scalar ` S ) )
5		eqidd 2623	. 2 |- ( ph -> ( Base ` R ) = ( Base ` R ) )
6		mplsubglem.b	. . 3 |- B = ( Base ` S )
7	6	a1i 11	. 2 |- ( ph -> B = ( Base ` S ) )
8		eqidd 2623	. 2 |- ( ph -> ( +g ` S ) = ( +g ` S ) )
9		eqidd 2623	. 2 |- ( ph -> ( .s ` S ) = ( .s ` S ) )
10		eqidd 2623	. 2 |- ( ph -> ( LSubSp ` S ) = ( LSubSp ` S ) )
11		mplsubglem.z	. . . 4 |- .0. = ( 0g ` R )
12		mplsubglem.d	. . . 4 |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
13		mplsubglem.0	. . . 4 |- ( ph -> (/) e. A )
14		mplsubglem.a	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x u. y ) e. A )
15		mplsubglem.y	. . . 4 |- ( ( ph /\ ( x e. A /\ y C_ x ) ) -> y e. A )
16		mplsubglem.u	. . . 4 |- ( ph -> U = { g e. B | ( g supp .0. ) e. A } )
17		ringgrp 18552	. . . . 5 |- ( R e. Ring -> R e. Grp )
18	3, 17	syl 17	. . . 4 |- ( ph -> R e. Grp )
19	1, 6, 11, 12, 2, 13, 14, 15, 16, 18	mplsubglem 19434	. . 3 |- ( ph -> U e. (SubGrp ` S ) )
20	6	subgss 17595	. . 3 |- ( U e. (SubGrp ` S ) -> U C_ B )
21	19, 20	syl 17	. 2 |- ( ph -> U C_ B )
22		eqid 2622	. . . 4 |- ( 0g ` S ) = ( 0g ` S )
23	22	subg0cl 17602	. . 3 |- ( U e. (SubGrp ` S ) -> ( 0g ` S ) e. U )
24		ne0i 3921	. . 3 |- ( ( 0g ` S ) e. U -> U =/= (/) )
25	19, 23, 24	3syl 18	. 2 |- ( ph -> U =/= (/) )
26	19	adantr 481	. . 3 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> U e. (SubGrp ` S ) )
27		eqid 2622	. . . . . 6 |- ( .s ` S ) = ( .s ` S )
28		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
29	3	adantr 481	. . . . . 6 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> R e. Ring )
30		simprl 794	. . . . . 6 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> u e. ( Base ` R ) )
31		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> v e. U )
32	16	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> U = { g e. B | ( g supp .0. ) e. A } )
33	32	eleq2d 2687	. . . . . . . . 9 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v e. U <-> v e. { g e. B | ( g supp .0. ) e. A } ) )
34		oveq1 6657	. . . . . . . . . . 11 |- ( g = v -> ( g supp .0. ) = ( v supp .0. ) )
35	34	eleq1d 2686	. . . . . . . . . 10 |- ( g = v -> ( ( g supp .0. ) e. A <-> ( v supp .0. ) e. A ) )
36	35	elrab 3363	. . . . . . . . 9 |- ( v e. { g e. B | ( g supp .0. ) e. A } <-> ( v e. B /\ ( v supp .0. ) e. A ) )
37	33, 36	syl6bb 276	. . . . . . . 8 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v e. U <-> ( v e. B /\ ( v supp .0. ) e. A ) ) )
38	31, 37	mpbid 222	. . . . . . 7 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v e. B /\ ( v supp .0. ) e. A ) )
39	38	simpld 475	. . . . . 6 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> v e. B )
40	1, 27, 28, 6, 29, 30, 39	psrvscacl 19393	. . . . 5 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .s ` S ) v ) e. B )
41		ovex 6678	. . . . . . 7 |- ( ( u ( .s ` S ) v ) supp .0. ) e. _V
42	41	a1i 11	. . . . . 6 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. _V )
43	38	simprd 479	. . . . . . 7 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v supp .0. ) e. A )
44	15	expr 643	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( y C_ x -> y e. A ) )
45	44	alrimiv 1855	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> A. y ( y C_ x -> y e. A ) )
46	45	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. x e. A A. y ( y C_ x -> y e. A ) )
47	46	adantr 481	. . . . . . 7 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> A. x e. A A. y ( y C_ x -> y e. A ) )
48		sseq2 3627	. . . . . . . . . 10 |- ( x = ( v supp .0. ) -> ( y C_ x <-> y C_ ( v supp .0. ) ) )
49	48	imbi1d 331	. . . . . . . . 9 |- ( x = ( v supp .0. ) -> ( ( y C_ x -> y e. A ) <-> ( y C_ ( v supp .0. ) -> y e. A ) ) )
50	49	albidv 1849	. . . . . . . 8 |- ( x = ( v supp .0. ) -> ( A. y ( y C_ x -> y e. A ) <-> A. y ( y C_ ( v supp .0. ) -> y e. A ) ) )
51	50	rspcv 3305	. . . . . . 7 |- ( ( v supp .0. ) e. A -> ( A. x e. A A. y ( y C_ x -> y e. A ) -> A. y ( y C_ ( v supp .0. ) -> y e. A ) ) )
52	43, 47, 51	sylc 65	. . . . . 6 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> A. y ( y C_ ( v supp .0. ) -> y e. A ) )
53	1, 28, 12, 6, 40	psrelbas 19379	. . . . . . 7 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .s ` S ) v ) : D --> ( Base ` R ) )
54		eqid 2622	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
55	30	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> u e. ( Base ` R ) )
56	39	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> v e. B )
57		eldifi 3732	. . . . . . . . . 10 |- ( k e. ( D \ ( v supp .0. ) ) -> k e. D )
58	57	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> k e. D )
59	1, 27, 28, 6, 54, 12, 55, 56, 58	psrvscaval 19392	. . . . . . . 8 |- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( ( u ( .s ` S ) v ) ` k ) = ( u ( .r ` R ) ( v ` k ) ) )
60	1, 28, 12, 6, 39	psrelbas 19379	. . . . . . . . . 10 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> v : D --> ( Base ` R ) )
61		ssid 3624	. . . . . . . . . . 11 |- ( v supp .0. ) C_ ( v supp .0. )
62	61	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v supp .0. ) C_ ( v supp .0. ) )
63		ovex 6678	. . . . . . . . . . . 12 |- ( NN0 ^m I ) e. _V
64	12, 63	rabex2 4815	. . . . . . . . . . 11 |- D e. _V
65	64	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> D e. _V )
66		fvex 6201	. . . . . . . . . . . 12 |- ( 0g ` R ) e. _V
67	11, 66	eqeltri 2697	. . . . . . . . . . 11 |- .0. e. _V
68	67	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> .0. e. _V )
69	60, 62, 65, 68	suppssr 7326	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( v ` k ) = .0. )
70	69	oveq2d 6666	. . . . . . . 8 |- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( u ( .r ` R ) ( v ` k ) ) = ( u ( .r ` R ) .0. ) )
71	28, 54, 11	ringrz 18588	. . . . . . . . . 10 |- ( ( R e. Ring /\ u e. ( Base ` R ) ) -> ( u ( .r ` R ) .0. ) = .0. )
72	3, 30, 71	syl2an2r 876	. . . . . . . . 9 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .r ` R ) .0. ) = .0. )
73	72	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( u ( .r ` R ) .0. ) = .0. )
74	59, 70, 73	3eqtrd 2660	. . . . . . 7 |- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( ( u ( .s ` S ) v ) ` k ) = .0. )
75	53, 74	suppss 7325	. . . . . 6 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) )
76		sseq1 3626	. . . . . . . 8 |- ( y = ( ( u ( .s ` S ) v ) supp .0. ) -> ( y C_ ( v supp .0. ) <-> ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) ) )
77		eleq1 2689	. . . . . . . 8 |- ( y = ( ( u ( .s ` S ) v ) supp .0. ) -> ( y e. A <-> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) )
78	76, 77	imbi12d 334	. . . . . . 7 |- ( y = ( ( u ( .s ` S ) v ) supp .0. ) -> ( ( y C_ ( v supp .0. ) -> y e. A ) <-> ( ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) )
79	78	spcgv 3293	. . . . . 6 |- ( ( ( u ( .s ` S ) v ) supp .0. ) e. _V -> ( A. y ( y C_ ( v supp .0. ) -> y e. A ) -> ( ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) )
80	42, 52, 75, 79	syl3c 66	. . . . 5 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. A )
81	32	eleq2d 2687	. . . . . 6 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) e. U <-> ( u ( .s ` S ) v ) e. { g e. B | ( g supp .0. ) e. A } ) )
82		oveq1 6657	. . . . . . . 8 |- ( g = ( u ( .s ` S ) v ) -> ( g supp .0. ) = ( ( u ( .s ` S ) v ) supp .0. ) )
83	82	eleq1d 2686	. . . . . . 7 |- ( g = ( u ( .s ` S ) v ) -> ( ( g supp .0. ) e. A <-> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) )
84	83	elrab 3363	. . . . . 6 |- ( ( u ( .s ` S ) v ) e. { g e. B | ( g supp .0. ) e. A } <-> ( ( u ( .s ` S ) v ) e. B /\ ( ( u ( .s ` S ) v ) supp .0. ) e. A ) )
85	81, 84	syl6bb 276	. . . . 5 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) e. U <-> ( ( u ( .s ` S ) v ) e. B /\ ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) )
86	40, 80, 85	mpbir2and 957	. . . 4 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .s ` S ) v ) e. U )
87	86	3adantr3 1222	. . 3 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> ( u ( .s ` S ) v ) e. U )
88		simpr3 1069	. . 3 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> w e. U )
89		eqid 2622	. . . 4 |- ( +g ` S ) = ( +g ` S )
90	89	subgcl 17604	. . 3 |- ( ( U e. (SubGrp ` S ) /\ ( u ( .s ` S ) v ) e. U /\ w e. U ) -> ( ( u ( .s ` S ) v ) ( +g ` S ) w ) e. U )
91	26, 87, 88, 90	syl3anc 1326	. 2 |- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> ( ( u ( .s ` S ) v ) ( +g ` S ) w ) e. U )
92	4, 5, 7, 8, 9, 10, 21, 25, 91	islssd 18936	1 |- ( ph -> U e. ( LSubSp ` S ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   supp csupp 7295   ^m cmap 7857   Fincfn 7955   NNcn 11020   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   .scvsca 15945   0gc0g 16100   Grpcgrp 17422  SubGrpcsubg 17588   Ringcrg 18547   LSubSpclss 18932   mPwSer cmps 19351

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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  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-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-tset 15960  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-mgp 18490  df-ring 18549  df-lss 18933  df-psr 19356

This theorem is referenced by:  mpllss  19438