Theorem mptcfsupp 42161

Description: A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)

Hypotheses

Ref	Expression
suppmptcfin.b	|- B = ( Base ` M )
suppmptcfin.r	|- R = (Scalar ` M )
suppmptcfin.0	|- .0. = ( 0g ` R )
suppmptcfin.1	|- .1. = ( 1r ` R )
suppmptcfin.f	|- F = ( x e. V |-> if ( x = X , .1. , .0. ) )

Assertion

Ref	Expression
mptcfsupp	|- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> F finSupp .0. )

Distinct variable groups:   x, B   x, F   x, M   x, V   x, X   x, .1.   x, .0.

Allowed substitution hint:   R( x)

Proof of Theorem mptcfsupp

Step	Hyp	Ref	Expression
1		suppmptcfin.f	. . . 4 |- F = ( x e. V |-> if ( x = X , .1. , .0. ) )
2	1	funmpt2 5927	. . 3 |- Fun F
3	2	a1i 11	. 2 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> Fun F )
4		suppmptcfin.b	. . 3 |- B = ( Base ` M )
5		suppmptcfin.r	. . 3 |- R = (Scalar ` M )
6		suppmptcfin.0	. . 3 |- .0. = ( 0g ` R )
7		suppmptcfin.1	. . 3 |- .1. = ( 1r ` R )
8	4, 5, 6, 7, 1	suppmptcfin 42160	. 2 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> ( F supp .0. ) e. Fin )
9		mptexg 6484	. . . . 5 |- ( V e. ~P B -> ( x e. V |-> if ( x = X , .1. , .0. ) ) e. _V )
10	1, 9	syl5eqel 2705	. . . 4 |- ( V e. ~P B -> F e. _V )
11	10	3ad2ant2 1083	. . 3 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> F e. _V )
12		fvex 6201	. . . 4 |- ( 0g ` R ) e. _V
13	6, 12	eqeltri 2697	. . 3 |- .0. e. _V
14		isfsupp 8279	. . 3 |- ( ( F e. _V /\ .0. e. _V ) -> ( F finSupp .0. <-> ( Fun F /\ ( F supp .0. ) e. Fin ) ) )
15	11, 13, 14	sylancl 694	. 2 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> ( F finSupp .0. <-> ( Fun F /\ ( F supp .0. ) e. Fin ) ) )
16	3, 8, 15	mpbir2and 957	1 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> F finSupp .0. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   1rcur 18501   LModclmod 18863

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

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-ral 2917  df-rex 2918  df-reu 2919  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-supp 7296  df-1o 7560  df-er 7742  df-en 7956  df-fin 7959  df-fsupp 8276

This theorem is referenced by:  lcoss  42225  el0ldep  42255