Theorem vc0 27429

Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vc0.1	|- G = ( 1st ` W )
vc0.2	|- S = ( 2nd ` W )
vc0.3	|- X = ran G
vc0.4	|- Z = (GId ` G )

Assertion

Ref	Expression
vc0	|- ( ( W e. CVecOLD /\ A e. X ) -> ( 0 S A ) = Z )

Proof of Theorem vc0

Step	Hyp	Ref	Expression
1		vc0.1	. . . 4 |- G = ( 1st ` W )
2		vc0.3	. . . 4 |- X = ran G
3		vc0.4	. . . 4 |- Z = (GId ` G )
4	1, 2, 3	vc0rid 27428	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> ( A G Z ) = A )
5		1p0e1 11133	. . . . 5 |- ( 1 + 0 ) = 1
6	5	oveq1i 6660	. . . 4 |- ( ( 1 + 0 ) S A ) = ( 1 S A )
7		0cn 10032	. . . . 5 |- 0 e. CC
8		ax-1cn 9994	. . . . . 6 |- 1 e. CC
9		vc0.2	. . . . . . 7 |- S = ( 2nd ` W )
10	1, 9, 2	vcdir 27421	. . . . . 6 |- ( ( W e. CVecOLD /\ ( 1 e. CC /\ 0 e. CC /\ A e. X ) ) -> ( ( 1 + 0 ) S A ) = ( ( 1 S A ) G ( 0 S A ) ) )
11	8, 10	mp3anr1 1421	. . . . 5 |- ( ( W e. CVecOLD /\ ( 0 e. CC /\ A e. X ) ) -> ( ( 1 + 0 ) S A ) = ( ( 1 S A ) G ( 0 S A ) ) )
12	7, 11	mpanr1 719	. . . 4 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( 1 + 0 ) S A ) = ( ( 1 S A ) G ( 0 S A ) ) )
13	1, 9, 2	vcidOLD 27419	. . . 4 |- ( ( W e. CVecOLD /\ A e. X ) -> ( 1 S A ) = A )
14	6, 12, 13	3eqtr3a 2680	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( 1 S A ) G ( 0 S A ) ) = A )
15	13	oveq1d 6665	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( 1 S A ) G ( 0 S A ) ) = ( A G ( 0 S A ) ) )
16	4, 14, 15	3eqtr2rd 2663	. 2 |- ( ( W e. CVecOLD /\ A e. X ) -> ( A G ( 0 S A ) ) = ( A G Z ) )
17	1, 9, 2	vccl 27418	. . . . 5 |- ( ( W e. CVecOLD /\ 0 e. CC /\ A e. X ) -> ( 0 S A ) e. X )
18	7, 17	mp3an2 1412	. . . 4 |- ( ( W e. CVecOLD /\ A e. X ) -> ( 0 S A ) e. X )
19	1, 2, 3	vczcl 27427	. . . . 5 |- ( W e. CVecOLD -> Z e. X )
20	19	adantr 481	. . . 4 |- ( ( W e. CVecOLD /\ A e. X ) -> Z e. X )
21		simpr 477	. . . 4 |- ( ( W e. CVecOLD /\ A e. X ) -> A e. X )
22	18, 20, 21	3jca 1242	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( 0 S A ) e. X /\ Z e. X /\ A e. X ) )
23	1, 2	vclcan 27426	. . 3 |- ( ( W e. CVecOLD /\ ( ( 0 S A ) e. X /\ Z e. X /\ A e. X ) ) -> ( ( A G ( 0 S A ) ) = ( A G Z ) <-> ( 0 S A ) = Z ) )
24	22, 23	syldan 487	. 2 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( A G ( 0 S A ) ) = ( A G Z ) <-> ( 0 S A ) = Z ) )
25	16, 24	mpbid 222	1 |- ( ( W e. CVecOLD /\ A e. X ) -> ( 0 S A ) = Z )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939  GIdcgi 27344   CVecOLDcvc 27413

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  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-vc 27414

This theorem is referenced by:  vcz  27430  vcm  27431  nv0  27492