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Theorem rmsupp0 42149
Description: The support of a mapping of a multiplication of zero with a function into a ring is empty. (Contributed by AV, 10-Apr-2019.)
Hypothesis
Ref Expression
rmsuppss.r |- R = ( Base ` M )
Assertion
Ref Expression
rmsupp0 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) = (/) )
Distinct variable groups:   v, A   v, C   v, M   v, R   v, X   v, V
Proof of Theorem rmsupp0
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 fveq2 6191 . . . . 5 |- ( v = w -> ( A ` v ) = ( A ` w ) )
21oveq2d 6666 . . . 4 |- ( v = w -> ( C ( .r ` M ) ( A ` v ) ) = ( C ( .r ` M ) ( A ` w ) ) )
32cbvmptv 4750 . . 3 |- ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) = ( w e. V |-> ( C ( .r ` M ) ( A ` w ) ) )
4 simpl2 1065 . . 3 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> V e. X )
5 fvexd 6203 . . 3 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> ( 0g ` M ) e. _V )
6 ovexd 6680 . . 3 |- ( ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( C ( .r ` M ) ( A ` w ) ) e. _V )
73, 4, 5, 6mptsuppd 7318 . 2 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ 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 ) } )
8 simpll3 1102 . . . . . 6 |- ( ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> C = ( 0g ` M ) )
98oveq1d 6665 . . . . 5 |- ( ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( C ( .r ` M ) ( A ` w ) ) = ( ( 0g ` M ) ( .r ` M ) ( A ` w ) ) )
10 simpll1 1100 . . . . . 6 |- ( ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> M e. Ring )
11 elmapi 7879 . . . . . . . . 9 |- ( A e. ( R ^m V ) -> A : V --> R )
12 ffvelrn 6357 . . . . . . . . . . 11 |- ( ( A : V --> R /\ w e. V ) -> ( A ` w ) e. R )
13 rmsuppss.r . . . . . . . . . . 11 |- R = ( Base ` M )
1412, 13syl6eleq 2711 . . . . . . . . . 10 |- ( ( A : V --> R /\ w e. V ) -> ( A ` w ) e. ( Base ` M ) )
1514ex 450 . . . . . . . . 9 |- ( A : V --> R -> ( w e. V -> ( A ` w ) e. ( Base ` M ) ) )
1611, 15syl 17 . . . . . . . 8 |- ( A e. ( R ^m V ) -> ( w e. V -> ( A ` w ) e. ( Base ` M ) ) )
1716adantl 482 . . . . . . 7 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> ( w e. V -> ( A ` w ) e. ( Base ` M ) ) )
1817imp 445 . . . . . 6 |- ( ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( A ` w ) e. ( Base ` M ) )
19 eqid 2622 . . . . . . 7 |- ( Base ` M ) = ( Base ` M )
20 eqid 2622 . . . . . . 7 |- ( .r ` M ) = ( .r ` M )
21 eqid 2622 . . . . . . 7 |- ( 0g ` M ) = ( 0g ` M )
2219, 20, 21ringlz 18587 . . . . . 6 |- ( ( M e. Ring /\ ( A ` w ) e. ( Base ` M ) ) -> ( ( 0g ` M ) ( .r ` M ) ( A ` w ) ) = ( 0g ` M ) )
2310, 18, 22syl2anc 693 . . . . 5 |- ( ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( ( 0g ` M ) ( .r ` M ) ( A ` w ) ) = ( 0g ` M ) )
249, 23eqtrd 2656 . . . 4 |- ( ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( C ( .r ` M ) ( A ` w ) ) = ( 0g ` M ) )
2524neeq1d 2853 . . 3 |- ( ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) /\ w e. V ) -> ( ( C ( .r ` M ) ( A ` w ) ) =/= ( 0g ` M ) <-> ( 0g ` M ) =/= ( 0g ` M ) ) )
2625rabbidva 3188 . 2 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> { w e. V | ( C ( .r ` M ) ( A ` w ) ) =/= ( 0g ` M ) } = { w e. V | ( 0g ` M ) =/= ( 0g ` M ) } )
27 neirr 2803 . . . . 5 |- -. ( 0g ` M ) =/= ( 0g ` M )
2827a1i 11 . . . 4 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> -. ( 0g ` M ) =/= ( 0g ` M ) )
2928ralrimivw 2967 . . 3 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> A. w e. V -. ( 0g ` M ) =/= ( 0g ` M ) )
30 rabeq0 3957 . . 3 |- ( { w e. V | ( 0g ` M ) =/= ( 0g ` M ) } = (/) <-> A. w e. V -. ( 0g ` M ) =/= ( 0g ` M ) )
3129, 30sylibr 224 . 2 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> { w e. V | ( 0g ` M ) =/= ( 0g ` M ) } = (/) )
327, 26, 313eqtrd 2660 1 |- ( ( ( M e. Ring /\ V e. X /\ C = ( 0g ` M ) ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) = (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   -->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-minusg 17426  df-mgp 18490  df-ring 18549
This theorem is referenced by: (None)
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