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Theorem slmd0vs 29777
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 27867 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmd0vs.v |- V = ( Base ` W )
slmd0vs.f |- F = (Scalar ` W )
slmd0vs.s |- .x. = ( .s ` W )
slmd0vs.o |- O = ( 0g ` F )
slmd0vs.z |- .0. = ( 0g ` W )
Assertion
Ref Expression
slmd0vs |- ( ( W e. SLMod /\ X e. V ) -> ( O .x. X ) = .0. )
Proof of Theorem slmd0vs
StepHypRef Expression
1 simpl 473 . . . 4 |- ( ( W e. SLMod /\ X e. V ) -> W e. SLMod )
2 slmd0vs.f . . . . . 6 |- F = (Scalar ` W )
3 eqid 2622 . . . . . 6 |- ( Base ` F ) = ( Base ` F )
4 slmd0vs.o . . . . . 6 |- O = ( 0g ` F )
52, 3, 4slmd0cl 29771 . . . . 5 |- ( W e. SLMod -> O e. ( Base ` F ) )
65adantr 481 . . . 4 |- ( ( W e. SLMod /\ X e. V ) -> O e. ( Base ` F ) )
7 simpr 477 . . . 4 |- ( ( W e. SLMod /\ X e. V ) -> X e. V )
8 slmd0vs.v . . . . 5 |- V = ( Base ` W )
9 eqid 2622 . . . . 5 |- ( +g ` W ) = ( +g ` W )
10 slmd0vs.s . . . . 5 |- .x. = ( .s ` W )
11 slmd0vs.z . . . . 5 |- .0. = ( 0g ` W )
12 eqid 2622 . . . . 5 |- ( +g ` F ) = ( +g ` F )
13 eqid 2622 . . . . 5 |- ( .r ` F ) = ( .r ` F )
14 eqid 2622 . . . . 5 |- ( 1r ` F ) = ( 1r ` F )
158, 9, 10, 11, 2, 3, 12, 13, 14, 4slmdlema 29756 . . . 4 |- ( ( W e. SLMod /\ ( O e. ( Base ` F ) /\ O e. ( Base ` F ) ) /\ ( X e. V /\ X e. V ) ) -> ( ( ( O .x. X ) e. V /\ ( O .x. ( X ( +g ` W ) X ) ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) /\ ( ( O ( +g ` F ) O ) .x. X ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) ) /\ ( ( ( O ( .r ` F ) O ) .x. X ) = ( O .x. ( O .x. X ) ) /\ ( ( 1r ` F ) .x. X ) = X /\ ( O .x. X ) = .0. ) ) )
161, 6, 6, 7, 7, 15syl122anc 1335 . . 3 |- ( ( W e. SLMod /\ X e. V ) -> ( ( ( O .x. X ) e. V /\ ( O .x. ( X ( +g ` W ) X ) ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) /\ ( ( O ( +g ` F ) O ) .x. X ) = ( ( O .x. X ) ( +g ` W ) ( O .x. X ) ) ) /\ ( ( ( O ( .r ` F ) O ) .x. X ) = ( O .x. ( O .x. X ) ) /\ ( ( 1r ` F ) .x. X ) = X /\ ( O .x. X ) = .0. ) ) )
1716simprd 479 . 2 |- ( ( W e. SLMod /\ X e. V ) -> ( ( ( O ( .r ` F ) O ) .x. X ) = ( O .x. ( O .x. X ) ) /\ ( ( 1r ` F ) .x. X ) = X /\ ( O .x. X ) = .0. ) )
1817simp3d 1075 1 |- ( ( W e. SLMod /\ X e. V ) -> ( O .x. X ) = .0. )
Colors of variables: wff setvar class
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-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:  slmdvs0  29778  gsumvsca2  29783
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