Theorem 00sr 7979

Description: A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)

Assertion

Ref	Expression
00sr	|- ( A e. R. -> ( A .R 0R ) = 0R )

Proof of Theorem 00sr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 7937	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 6020	. . 3 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R 0R ) = ( A .R 0R ) )
3	2	eqeq1d 2238	. 2 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R 0R ) = 0R <-> ( A .R 0R ) = 0R ) )
4		1pr 7764	. . . . 5 |- 1P e. P.
5		mulsrpr 7956	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( 1P e. P. /\ 1P e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R ) = [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R )
6	4, 4, 5	mpanr12 439	. . . 4 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R ) = [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R )
7		mulclpr 7782	. . . . . . . . . 10 |- ( ( x e. P. /\ 1P e. P. ) -> ( x .P. 1P ) e. P. )
8	4, 7	mpan2 425	. . . . . . . . 9 |- ( x e. P. -> ( x .P. 1P ) e. P. )
9		mulclpr 7782	. . . . . . . . . 10 |- ( ( y e. P. /\ 1P e. P. ) -> ( y .P. 1P ) e. P. )
10	4, 9	mpan2 425	. . . . . . . . 9 |- ( y e. P. -> ( y .P. 1P ) e. P. )
11		addclpr 7747	. . . . . . . . 9 |- ( ( ( x .P. 1P ) e. P. /\ ( y .P. 1P ) e. P. ) -> ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. )
12	8, 10, 11	syl2an 289	. . . . . . . 8 |- ( ( x e. P. /\ y e. P. ) -> ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. )
13	12, 12	anim12i 338	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( x e. P. /\ y e. P. ) ) -> ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. /\ ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. ) )
14		eqid 2229	. . . . . . . 8 |- ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P ) = ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P )
15		enreceq 7946	. . . . . . . 8 |- ( ( ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. /\ ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. ) /\ ( 1P e. P. /\ 1P e. P. ) ) -> ( [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R <-> ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P ) = ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P ) ) )
16	14, 15	mpbiri 168	. . . . . . 7 |- ( ( ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. /\ ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. ) /\ ( 1P e. P. /\ 1P e. P. ) ) -> [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R )
17	13, 16	sylan 283	. . . . . 6 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( x e. P. /\ y e. P. ) ) /\ ( 1P e. P. /\ 1P e. P. ) ) -> [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R )
18	4, 4, 17	mpanr12 439	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( x e. P. /\ y e. P. ) ) -> [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R )
19	18	anidms 397	. . . 4 |- ( ( x e. P. /\ y e. P. ) -> [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R )
20	6, 19	eqtrd 2262	. . 3 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R ) = [ <. 1P , 1P >. ] ~R )
21		df-0r 7941	. . . 4 |- 0R = [ <. 1P , 1P >. ] ~R
22	21	oveq2i 6024	. . 3 |- ( [ <. x , y >. ] ~R .R 0R ) = ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R )
23	20, 22, 21	3eqtr4g 2287	. 2 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R 0R ) = 0R )
24	1, 3, 23	ecoptocl 6786	1 |- ( A e. R. -> ( A .R 0R ) = 0R )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   <.cop 3670  (class class class)co 6013   [cec 6695   P.cnp 7501   1Pc1p 7502   +P. cpp 7503   .P. cmp 7504   ~R cer 7506   R.cnr 7507   0Rc0r 7508   .R cmr 7512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-i1p 7677  df-iplp 7678  df-imp 7679  df-enr 7936  df-nr 7937  df-mr 7939  df-0r 7941

This theorem is referenced by:  pn0sr  7981  mulresr  8048  axi2m1  8085  axcnre  8091