Theorem lcoval 42201

Description: The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)

Hypotheses

Ref	Expression
lcoop.b	|- B = ( Base ` M )
lcoop.s	|- S = (Scalar ` M )
lcoop.r	|- R = ( Base ` S )

Assertion

Ref	Expression
lcoval	|- ( ( M e. X /\ V e. ~P B ) -> ( C e. ( M LinCo V ) <-> ( C e. B /\ E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ C = ( s ( linC ` M ) V ) ) ) ) )

Distinct variable groups:   M, s   R, s   V, s   C, s

Allowed substitution hints:   B( s)   S( s)   X( s)

Proof of Theorem lcoval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcoop.b	. . . 4 |- B = ( Base ` M )
2		lcoop.s	. . . 4 |- S = (Scalar ` M )
3		lcoop.r	. . . 4 |- R = ( Base ` S )
4	1, 2, 3	lcoop 42200	. . 3 |- ( ( M e. X /\ V e. ~P B ) -> ( M LinCo V ) = { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } )
5	4	eleq2d 2687	. 2 |- ( ( M e. X /\ V e. ~P B ) -> ( C e. ( M LinCo V ) <-> C e. { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } ) )
6		eqeq1 2626	. . . . 5 |- ( c = C -> ( c = ( s ( linC ` M ) V ) <-> C = ( s ( linC ` M ) V ) ) )
7	6	anbi2d 740	. . . 4 |- ( c = C -> ( ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) <-> ( s finSupp ( 0g ` S ) /\ C = ( s ( linC ` M ) V ) ) ) )
8	7	rexbidv 3052	. . 3 |- ( c = C -> ( E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) <-> E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ C = ( s ( linC ` M ) V ) ) ) )
9	8	elrab 3363	. 2 |- ( C e. { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } <-> ( C e. B /\ E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ C = ( s ( linC ` M ) V ) ) ) )
10	5, 9	syl6bb 276	1 |- ( ( M e. X /\ V e. ~P B ) -> ( C e. ( M LinCo V ) <-> ( C e. B /\ E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ C = ( s ( linC ` M ) V ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   ~Pcpw 4158   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   linC clinc 42193   LinCo clinco 42194

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-lco 42196

This theorem is referenced by:  lcoel0  42217  lincsumcl  42220  lincscmcl  42221  lincolss  42223  ellcoellss  42224  lcoss  42225  lindslinindsimp1  42246  lindslinindsimp2  42252