Theorem mndpsuppfi 42156

Description: The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)

Hypothesis

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

Assertion

Ref	Expression
mndpsuppfi	|- ( ( ( M e. Mnd /\ V e. X ) /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) /\ ( ( A supp ( 0g ` M ) ) e. Fin /\ ( B supp ( 0g ` M ) ) e. Fin ) ) -> ( ( A oF ( +g ` M ) B ) supp ( 0g ` M ) ) e. Fin )

Proof of Theorem mndpsuppfi

Step	Hyp	Ref	Expression
1		unfi 8227	. . 3 |- ( ( ( A supp ( 0g ` M ) ) e. Fin /\ ( B supp ( 0g ` M ) ) e. Fin ) -> ( ( A supp ( 0g ` M ) ) u. ( B supp ( 0g ` M ) ) ) e. Fin )
2	1	3ad2ant3 1084	. 2 |- ( ( ( M e. Mnd /\ V e. X ) /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) /\ ( ( A supp ( 0g ` M ) ) e. Fin /\ ( B supp ( 0g ` M ) ) e. Fin ) ) -> ( ( A supp ( 0g ` M ) ) u. ( B supp ( 0g ` M ) ) ) e. Fin )
3		mndpsuppfi.r	. . . 4 |- R = ( Base ` M )
4	3	mndpsuppss 42152	. . 3 |- ( ( ( M e. Mnd /\ V e. X ) /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) -> ( ( A oF ( +g ` M ) B ) supp ( 0g ` M ) ) C_ ( ( A supp ( 0g ` M ) ) u. ( B supp ( 0g ` M ) ) ) )
5	4	3adant3 1081	. 2 |- ( ( ( M e. Mnd /\ V e. X ) /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) /\ ( ( A supp ( 0g ` M ) ) e. Fin /\ ( B supp ( 0g ` M ) ) e. Fin ) ) -> ( ( A oF ( +g ` M ) B ) supp ( 0g ` M ) ) C_ ( ( A supp ( 0g ` M ) ) u. ( B supp ( 0g ` M ) ) ) )
6		ssfi 8180	. 2 |- ( ( ( ( A supp ( 0g ` M ) ) u. ( B supp ( 0g ` M ) ) ) e. Fin /\ ( ( A oF ( +g ` M ) B ) supp ( 0g ` M ) ) C_ ( ( A supp ( 0g ` M ) ) u. ( B supp ( 0g ` M ) ) ) ) -> ( ( A oF ( +g ` M ) B ) supp ( 0g ` M ) ) e. Fin )
7	2, 5, 6	syl2anc 693	1 |- ( ( ( M e. Mnd /\ V e. X ) /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) /\ ( ( A supp ( 0g ` M ) ) e. Fin /\ ( B supp ( 0g ` M ) ) e. Fin ) ) -> ( ( A oF ( +g ` M ) B ) supp ( 0g ` M ) ) e. Fin )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   ` cfv 5888  (class class class)co 6650   oFcof 6895   supp csupp 7295   ^m cmap 7857   Fincfn 7955   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294

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-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-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295

This theorem is referenced by:  mndpfsupp  42157