Theorem 00sr 7731

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 7689	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 5860	. . 3 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R 0R ) = ( A .R 0R ) )
3	2	eqeq1d 2179	. 2 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R 0R ) = 0R <-> ( A .R 0R ) = 0R ) )
4		1pr 7516	. . . . 5 |- 1P e. P.
5		mulsrpr 7708	. . . . 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 437	. . . 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 7534	. . . . . . . . . 10 |- ( ( x e. P. /\ 1P e. P. ) -> ( x .P. 1P ) e. P. )
8	4, 7	mpan2 423	. . . . . . . . 9 |- ( x e. P. -> ( x .P. 1P ) e. P. )
9		mulclpr 7534	. . . . . . . . . 10 |- ( ( y e. P. /\ 1P e. P. ) -> ( y .P. 1P ) e. P. )
10	4, 9	mpan2 423	. . . . . . . . 9 |- ( y e. P. -> ( y .P. 1P ) e. P. )
11		addclpr 7499	. . . . . . . . 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 287	. . . . . . . 8 |- ( ( x e. P. /\ y e. P. ) -> ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. )
13	12, 12	anim12i 336	. . . . . . 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 2170	. . . . . . . 8 |- ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P ) = ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P )
15		enreceq 7698	. . . . . . . 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 167	. . . . . . 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 281	. . . . . 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 437	. . . . 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 395	. . . 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 2203	. . 3 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R ) = [ <. 1P , 1P >. ] ~R )
21		df-0r 7693	. . . 4 |- 0R = [ <. 1P , 1P >. ] ~R
22	21	oveq2i 5864	. . 3 |- ( [ <. x , y >. ] ~R .R 0R ) = ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R )
23	20, 22, 21	3eqtr4g 2228	. 2 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R 0R ) = 0R )
24	1, 3, 23	ecoptocl 6600	1 |- ( A e. R. -> ( A .R 0R ) = 0R )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   <.cop 3586  (class class class)co 5853   [cec 6511   P.cnp 7253   1Pc1p 7254   +P. cpp 7255   .P. cmp 7256   ~R cer 7258   R.cnr 7259   0Rc0r 7260   .R cmr 7264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-imp 7431  df-enr 7688  df-nr 7689  df-mr 7691  df-0r 7693

This theorem is referenced by:  pn0sr  7733  mulresr  7800  axi2m1  7837  axcnre  7843