Theorem slmdvs0 29778

Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 27881 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
slmdvs0.f	|- F = (Scalar ` W )
slmdvs0.s	|- .x. = ( .s ` W )
slmdvs0.k	|- K = ( Base ` F )
slmdvs0.z	|- .0. = ( 0g ` W )

Assertion

Ref	Expression
slmdvs0	|- ( ( W e. SLMod /\ X e. K ) -> ( X .x. .0. ) = .0. )

Proof of Theorem slmdvs0

Step	Hyp	Ref	Expression
1		slmdvs0.f	. . . . 5 |- F = (Scalar ` W )
2	1	slmdsrg 29760	. . . 4 |- ( W e. SLMod -> F e. SRing )
3		slmdvs0.k	. . . . 5 |- K = ( Base ` F )
4		eqid 2622	. . . . 5 |- ( .r ` F ) = ( .r ` F )
5		eqid 2622	. . . . 5 |- ( 0g ` F ) = ( 0g ` F )
6	3, 4, 5	srgrz 18526	. . . 4 |- ( ( F e. SRing /\ X e. K ) -> ( X ( .r ` F ) ( 0g ` F ) ) = ( 0g ` F ) )
7	2, 6	sylan 488	. . 3 |- ( ( W e. SLMod /\ X e. K ) -> ( X ( .r ` F ) ( 0g ` F ) ) = ( 0g ` F ) )
8	7	oveq1d 6665	. 2 |- ( ( W e. SLMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( ( 0g ` F ) .x. .0. ) )
9		simpl 473	. . . 4 |- ( ( W e. SLMod /\ X e. K ) -> W e. SLMod )
10		simpr 477	. . . 4 |- ( ( W e. SLMod /\ X e. K ) -> X e. K )
11	2	adantr 481	. . . . 5 |- ( ( W e. SLMod /\ X e. K ) -> F e. SRing )
12	3, 5	srg0cl 18519	. . . . 5 |- ( F e. SRing -> ( 0g ` F ) e. K )
13	11, 12	syl 17	. . . 4 |- ( ( W e. SLMod /\ X e. K ) -> ( 0g ` F ) e. K )
14		eqid 2622	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
15		slmdvs0.z	. . . . . 6 |- .0. = ( 0g ` W )
16	14, 15	slmd0vcl 29774	. . . . 5 |- ( W e. SLMod -> .0. e. ( Base ` W ) )
17	16	adantr 481	. . . 4 |- ( ( W e. SLMod /\ X e. K ) -> .0. e. ( Base ` W ) )
18		slmdvs0.s	. . . . 5 |- .x. = ( .s ` W )
19	14, 1, 18, 3, 4	slmdvsass 29770	. . . 4 |- ( ( W e. SLMod /\ ( X e. K /\ ( 0g ` F ) e. K /\ .0. e. ( Base ` W ) ) ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. ( ( 0g ` F ) .x. .0. ) ) )
20	9, 10, 13, 17, 19	syl13anc 1328	. . 3 |- ( ( W e. SLMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. ( ( 0g ` F ) .x. .0. ) ) )
21	14, 1, 18, 5, 15	slmd0vs 29777	. . . . 5 |- ( ( W e. SLMod /\ .0. e. ( Base ` W ) ) -> ( ( 0g ` F ) .x. .0. ) = .0. )
22	17, 21	syldan 487	. . . 4 |- ( ( W e. SLMod /\ X e. K ) -> ( ( 0g ` F ) .x. .0. ) = .0. )
23	22	oveq2d 6666	. . 3 |- ( ( W e. SLMod /\ X e. K ) -> ( X .x. ( ( 0g ` F ) .x. .0. ) ) = ( X .x. .0. ) )
24	20, 23	eqtrd 2656	. 2 |- ( ( W e. SLMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. .0. ) )
25	8, 24, 22	3eqtr3d 2664	1 |- ( ( W e. SLMod /\ X e. K ) -> ( X .x. .0. ) = .0. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100  SRingcsrg 18505  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-8 1992  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-pow 4843  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-opab 4713  df-mpt 4730  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-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cmn 18195  df-srg 18506  df-slmd 29754

This theorem is referenced by:  gsumvsca1  29782