Theorem slmdvscl 29767

Description: Closure of scalar product for a semiring left module. (hvmulcl 27870 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
slmdvscl.v	|- V = ( Base ` W )
slmdvscl.f	|- F = (Scalar ` W )
slmdvscl.s	|- .x. = ( .s ` W )
slmdvscl.k	|- K = ( Base ` F )

Assertion

Ref	Expression
slmdvscl	|- ( ( W e. SLMod /\ R e. K /\ X e. V ) -> ( R .x. X ) e. V )

Proof of Theorem slmdvscl

Step	Hyp	Ref	Expression
1		biid 251	. 2 |- ( W e. SLMod <-> W e. SLMod )
2		pm4.24 675	. 2 |- ( R e. K <-> ( R e. K /\ R e. K ) )
3		pm4.24 675	. 2 |- ( X e. V <-> ( X e. V /\ X e. V ) )
4		slmdvscl.v	. . . . 5 |- V = ( Base ` W )
5		eqid 2622	. . . . 5 |- ( +g ` W ) = ( +g ` W )
6		slmdvscl.s	. . . . 5 |- .x. = ( .s ` W )
7		eqid 2622	. . . . 5 |- ( 0g ` W ) = ( 0g ` W )
8		slmdvscl.f	. . . . 5 |- F = (Scalar ` W )
9		slmdvscl.k	. . . . 5 |- K = ( Base ` F )
10		eqid 2622	. . . . 5 |- ( +g ` F ) = ( +g ` F )
11		eqid 2622	. . . . 5 |- ( .r ` F ) = ( .r ` F )
12		eqid 2622	. . . . 5 |- ( 1r ` F ) = ( 1r ` F )
13		eqid 2622	. . . . 5 |- ( 0g ` F ) = ( 0g ` F )
14	4, 5, 6, 7, 8, 9, 10, 11, 12, 13	slmdlema 29756	. . . 4 |- ( ( W e. SLMod /\ ( R e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( ( R .x. X ) e. V /\ ( R .x. ( X ( +g ` W ) X ) ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) /\ ( ( R ( +g ` F ) R ) .x. X ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) ) /\ ( ( ( R ( .r ` F ) R ) .x. X ) = ( R .x. ( R .x. X ) ) /\ ( ( 1r ` F ) .x. X ) = X /\ ( ( 0g ` F ) .x. X ) = ( 0g ` W ) ) ) )
15	14	simpld 475	. . 3 |- ( ( W e. SLMod /\ ( R e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( R .x. X ) e. V /\ ( R .x. ( X ( +g ` W ) X ) ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) /\ ( ( R ( +g ` F ) R ) .x. X ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) ) )
16	15	simp1d 1073	. 2 |- ( ( W e. SLMod /\ ( R e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( R .x. X ) e. V )
17	1, 2, 3, 16	syl3anb 1369	1 |- ( ( W e. SLMod /\ R e. K /\ X e. V ) -> ( R .x. X ) e. V )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   1rcur 18501  SLModcslmd 29753

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-nul 4789

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-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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-slmd 29754

This theorem is referenced by:  gsumvsca1  29782  gsumvsca2  29783  sitgaddlemb  30410