Theorem mptscmfsuppd 18929

Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 19666. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.) (Proof shortened by AV, 18-Oct-2019.)

Hypotheses

Ref	Expression
mptscmfsuppd.b	|- B = ( Base ` P )
mptscmfsuppd.s	|- S = (Scalar ` P )
mptscmfsuppd.n	|- .x. = ( .s ` P )
mptscmfsuppd.p	|- ( ph -> P e. LMod )
mptscmfsuppd.x	|- ( ph -> X e. V )
mptscmfsuppd.z	|- ( ( ph /\ k e. X ) -> Z e. B )
mptscmfsuppd.a	|- ( ph -> A : X --> Y )
mptscmfsuppd.f	|- ( ph -> A finSupp ( 0g ` S ) )

Assertion

Ref	Expression
mptscmfsuppd	|- ( ph -> ( k e. X |-> ( ( A ` k ) .x. Z ) ) finSupp ( 0g ` P ) )

Distinct variable groups:   A, k   B, k   P, k   S, k   k, X   .x. , k   ph, k

Allowed substitution hints:   V( k)   Y( k)   Z( k)

Proof of Theorem mptscmfsuppd

Step	Hyp	Ref	Expression
1		mptscmfsuppd.x	. 2 |- ( ph -> X e. V )
2		mptscmfsuppd.p	. 2 |- ( ph -> P e. LMod )
3		mptscmfsuppd.s	. . 3 |- S = (Scalar ` P )
4	3	a1i 11	. 2 |- ( ph -> S = (Scalar ` P ) )
5		mptscmfsuppd.b	. 2 |- B = ( Base ` P )
6		fvexd 6203	. 2 |- ( ( ph /\ k e. X ) -> ( A ` k ) e. _V )
7		mptscmfsuppd.z	. 2 |- ( ( ph /\ k e. X ) -> Z e. B )
8		eqid 2622	. 2 |- ( 0g ` P ) = ( 0g ` P )
9		eqid 2622	. 2 |- ( 0g ` S ) = ( 0g ` S )
10		mptscmfsuppd.n	. 2 |- .x. = ( .s ` P )
11		mptscmfsuppd.a	. . . 4 |- ( ph -> A : X --> Y )
12	11	feqmptd 6249	. . 3 |- ( ph -> A = ( k e. X |-> ( A ` k ) ) )
13		mptscmfsuppd.f	. . 3 |- ( ph -> A finSupp ( 0g ` S ) )
14	12, 13	eqbrtrrd 4677	. 2 |- ( ph -> ( k e. X |-> ( A ` k ) ) finSupp ( 0g ` S ) )
15	1, 2, 4, 5, 6, 7, 8, 9, 10, 14	mptscmfsupp0 18928	1 |- ( ph -> ( k e. X |-> ( ( A ` k ) .x. Z ) ) finSupp ( 0g ` P ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   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-fal 1489  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-supp 7296  df-er 7742  df-en 7956  df-fin 7959  df-fsupp 8276  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ring 18549  df-lmod 18865

This theorem is referenced by:  ply1coefsupp  19665