Theorem 00sr 6697

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

Assertion

Ref	Expression
00sr	⊢ (A ∈ 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 6655	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 5462	. . 3 ⊢ ([⟨x, y⟩] ~R = A → ([⟨x, y⟩] ~R ·R 0R) = (A ·R 0R))
3	2	eqeq1d 2045	. 2 ⊢ ([⟨x, y⟩] ~R = A → (([⟨x, y⟩] ~R ·R 0R) = 0R ↔ (A ·R 0R) = 0R))
4		1pr 6535	. . . . 5 ⊢ 1P ∈ P
5		mulsrpr 6674	. . . . 5 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (1P ∈ P ∧ 1P ∈ 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 415	. . . 4 ⊢ ((x ∈ P ∧ y ∈ 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 6553	. . . . . . . . . 10 ⊢ ((x ∈ P ∧ 1P ∈ P) → (x ·P 1P) ∈ P)
8	4, 7	mpan2 401	. . . . . . . . 9 ⊢ (x ∈ P → (x ·P 1P) ∈ P)
9		mulclpr 6553	. . . . . . . . . 10 ⊢ ((y ∈ P ∧ 1P ∈ P) → (y ·P 1P) ∈ P)
10	4, 9	mpan2 401	. . . . . . . . 9 ⊢ (y ∈ P → (y ·P 1P) ∈ P)
11		addclpr 6520	. . . . . . . . 9 ⊢ (((x ·P 1P) ∈ P ∧ (y ·P 1P) ∈ P) → ((x ·P 1P) +P (y ·P 1P)) ∈ P)
12	8, 10, 11	syl2an 273	. . . . . . . 8 ⊢ ((x ∈ P ∧ y ∈ P) → ((x ·P 1P) +P (y ·P 1P)) ∈ P)
13	12, 12	anim12i 321	. . . . . . 7 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (x ∈ P ∧ y ∈ P)) → (((x ·P 1P) +P (y ·P 1P)) ∈ P ∧ ((x ·P 1P) +P (y ·P 1P)) ∈ P))
14		eqid 2037	. . . . . . . 8 ⊢ (((x ·P 1P) +P (y ·P 1P)) +P 1P) = (((x ·P 1P) +P (y ·P 1P)) +P 1P)
15		enreceq 6664	. . . . . . . 8 ⊢ (((((x ·P 1P) +P (y ·P 1P)) ∈ P ∧ ((x ·P 1P) +P (y ·P 1P)) ∈ P) ∧ (1P ∈ P ∧ 1P ∈ 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 157	. . . . . . 7 ⊢ (((((x ·P 1P) +P (y ·P 1P)) ∈ P ∧ ((x ·P 1P) +P (y ·P 1P)) ∈ P) ∧ (1P ∈ P ∧ 1P ∈ P)) → [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
17	13, 16	sylan 267	. . . . . 6 ⊢ ((((x ∈ P ∧ y ∈ P) ∧ (x ∈ P ∧ y ∈ P)) ∧ (1P ∈ P ∧ 1P ∈ P)) → [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
18	4, 4, 17	mpanr12 415	. . . . 5 ⊢ (((x ∈ P ∧ y ∈ P) ∧ (x ∈ P ∧ y ∈ P)) → [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
19	18	anidms 377	. . . 4 ⊢ ((x ∈ P ∧ y ∈ P) → [⟨((x ·P 1P) +P (y ·P 1P)), ((x ·P 1P) +P (y ·P 1P))⟩] ~R = [⟨1P, 1P⟩] ~R )
20	6, 19	eqtrd 2069	. . 3 ⊢ ((x ∈ P ∧ y ∈ P) → ([⟨x, y⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R )
21		df-0r 6659	. . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R
22	21	oveq2i 5466	. . 3 ⊢ ([⟨x, y⟩] ~R ·R 0R) = ([⟨x, y⟩] ~R ·R [⟨1P, 1P⟩] ~R )
23	20, 22, 21	3eqtr4g 2094	. 2 ⊢ ((x ∈ P ∧ y ∈ P) → ([⟨x, y⟩] ~R ·R 0R) = 0R)
24	1, 3, 23	ecoptocl 6129	1 ⊢ (A ∈ R → (A ·R 0R) = 0R)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ⟨cop 3370  (class class class)co 5455  [cec 6040  Pcnp 6275  1_Pc1p 6276   +_P cpp 6277   ·_P cmp 6278   ~_R cer 6280  Rcnr 6281  0_Rc0r 6282   ·_R cmr 6286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-imp 6452  df-enr 6654  df-nr 6655  df-mr 6657  df-0r 6659

This theorem is referenced by:  pn0sr  6699  mulresr  6735  axi2m1  6759  axcnre  6765