Theorem 00sr 8142

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 8100	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 6095	. . 3 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R 0R ) = ( A .R 0R ) )
3	2	eqeq1d 2247	. 2 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R 0R ) = 0R <-> ( A .R 0R ) = 0R ) )
4		1pr 7927	. . . . 5 |- 1P e. P.
5		mulsrpr 8119	. . . . 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 443	. . . 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 7945	. . . . . . . . . 10 |- ( ( x e. P. /\ 1P e. P. ) -> ( x .P. 1P ) e. P. )
8	4, 7	mpan2 429	. . . . . . . . 9 |- ( x e. P. -> ( x .P. 1P ) e. P. )
9		mulclpr 7945	. . . . . . . . . 10 |- ( ( y e. P. /\ 1P e. P. ) -> ( y .P. 1P ) e. P. )
10	4, 9	mpan2 429	. . . . . . . . 9 |- ( y e. P. -> ( y .P. 1P ) e. P. )
11		addclpr 7910	. . . . . . . . 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 2238	. . . . . . . 8 |- ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P ) = ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P )
15		enreceq 8109	. . . . . . . 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 443	. . . . 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 401	. . . 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 2271	. . 3 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R ) = [ <. 1P , 1P >. ] ~R )
21		df-0r 8104	. . . 4 |- 0R = [ <. 1P , 1P >. ] ~R
22	21	oveq2i 6099	. . 3 |- ( [ <. x , y >. ] ~R .R 0R ) = ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R )
23	20, 22, 21	3eqtr4g 2296	. 2 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R 0R ) = 0R )
24	1, 3, 23	ecoptocl 6900	1 |- ( A e. R. -> ( A .R 0R ) = 0R )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   <.cop 3712  (class class class)co 6088   [cec 6809   P.cnp 7664   1Pc1p 7665   +P. cpp 7666   .P. cmp 7667   ~R cer 7669   R.cnr 7670   0Rc0r 7671   .R cmr 7675

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-i1p 7840  df-iplp 7841  df-imp 7842  df-enr 8099  df-nr 8100  df-mr 8102  df-0r 8104

This theorem is used by:  pn0sr  8144  mulresr  8211  axi2m1  8248  axcnre  8254