Theorem rmsuppss 42151

Description: The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.)

Hypothesis

Ref	Expression
rmsuppss.r	|- R = ( Base ` M )

Assertion

Ref	Expression
rmsuppss	|- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) C_ ( A supp ( 0g ` M ) ) )

Distinct variable groups:   v, A   v, C   v, M   v, R   v, X   v, V

Proof of Theorem rmsuppss

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . 7 |- ( ( A ` w ) = ( 0g ` M ) -> ( C ( .r ` M ) ( A ` w ) ) = ( C ( .r ` M ) ( 0g ` M ) ) )
2		simpll1 1100	. . . . . . . 8 |- ( ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> M e. Ring )
3		simpll3 1102	. . . . . . . 8 |- ( ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> C e. R )
4		rmsuppss.r	. . . . . . . . 9 |- R = ( Base ` M )
5		eqid 2622	. . . . . . . . 9 |- ( .r ` M ) = ( .r ` M )
6		eqid 2622	. . . . . . . . 9 |- ( 0g ` M ) = ( 0g ` M )
7	4, 5, 6	ringrz 18588	. . . . . . . 8 |- ( ( M e. Ring /\ C e. R ) -> ( C ( .r ` M ) ( 0g ` M ) ) = ( 0g ` M ) )
8	2, 3, 7	syl2anc 693	. . . . . . 7 |- ( ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( C ( .r ` M ) ( 0g ` M ) ) = ( 0g ` M ) )
9	1, 8	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) /\ w e. V ) /\ ( A ` w ) = ( 0g ` M ) ) -> ( C ( .r ` M ) ( A ` w ) ) = ( 0g ` M ) )
10	9	ex 450	. . . . 5 |- ( ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( ( A ` w ) = ( 0g ` M ) -> ( C ( .r ` M ) ( A ` w ) ) = ( 0g ` M ) ) )
11	10	necon3d 2815	. . . 4 |- ( ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( ( C ( .r ` M ) ( A ` w ) ) =/= ( 0g ` M ) -> ( A ` w ) =/= ( 0g ` M ) ) )
12	11	ss2rabdv 3683	. . 3 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> { w e. V | ( C ( .r ` M ) ( A ` w ) ) =/= ( 0g ` M ) } C_ { w e. V | ( A ` w ) =/= ( 0g ` M ) } )
13		elmapi 7879	. . . . . 6 |- ( A e. ( R ^m V ) -> A : V --> R )
14		fdm 6051	. . . . . 6 |- ( A : V --> R -> dom A = V )
15	13, 14	syl 17	. . . . 5 |- ( A e. ( R ^m V ) -> dom A = V )
16	15	adantl 482	. . . 4 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> dom A = V )
17		rabeq 3192	. . . 4 |- ( dom A = V -> { w e. dom A | ( A ` w ) =/= ( 0g ` M ) } = { w e. V | ( A ` w ) =/= ( 0g ` M ) } )
18	16, 17	syl 17	. . 3 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> { w e. dom A | ( A ` w ) =/= ( 0g ` M ) } = { w e. V | ( A ` w ) =/= ( 0g ` M ) } )
19	12, 18	sseqtr4d 3642	. 2 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> { w e. V | ( C ( .r ` M ) ( A ` w ) ) =/= ( 0g ` M ) } C_ { w e. dom A | ( A ` w ) =/= ( 0g ` M ) } )
20		fveq2 6191	. . . . 5 |- ( v = w -> ( A ` v ) = ( A ` w ) )
21	20	oveq2d 6666	. . . 4 |- ( v = w -> ( C ( .r ` M ) ( A ` v ) ) = ( C ( .r ` M ) ( A ` w ) ) )
22	21	cbvmptv 4750	. . 3 |- ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) = ( w e. V |-> ( C ( .r ` M ) ( A ` w ) ) )
23		simpl2 1065	. . 3 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> V e. X )
24		fvexd 6203	. . 3 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> ( 0g ` M ) e. _V )
25		ovexd 6680	. . 3 |- ( ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( C ( .r ` M ) ( A ` w ) ) e. _V )
26	22, 23, 24, 25	mptsuppd 7318	. 2 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) = { w e. V | ( C ( .r ` M ) ( A ` w ) ) =/= ( 0g ` M ) } )
27		elmapfun 7881	. . . 4 |- ( A e. ( R ^m V ) -> Fun A )
28	27	adantl 482	. . 3 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> Fun A )
29		simpr 477	. . 3 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> A e. ( R ^m V ) )
30		suppval1 7301	. . 3 |- ( ( Fun A /\ A e. ( R ^m V ) /\ ( 0g ` M ) e. _V ) -> ( A supp ( 0g ` M ) ) = { w e. dom A | ( A ` w ) =/= ( 0g ` M ) } )
31	28, 29, 24, 30	syl3anc 1326	. 2 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> ( A supp ( 0g ` M ) ) = { w e. dom A | ( A ` w ) =/= ( 0g ` M ) } )
32	19, 26, 31	3sstr4d 3648	1 |- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) C_ ( A supp ( 0g ` M ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   ^m cmap 7857   Basecbs 15857   .rcmulr 15942   0gc0g 16100   Ringcrg 18547

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-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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-mgp 18490  df-ring 18549

This theorem is referenced by:  rmsuppfi  42154