Theorem 00sr 11118

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

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 11075	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 7417	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = (𝐴 ·R 0R))
3	2	eqeq1d 2738	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R ↔ (𝐴 ·R 0R) = 0R))
4		1pr 11034	. . . . 5 ⊢ 1P ∈ P
5		mulsrpr 11095	. . . . 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 705	. . . 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 11039	. . . . . . . . . 10 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 ·P 1P) ∈ P)
8	4, 7	mpan2 691	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) ∈ P)
9		mulclpr 11039	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 ·P 1P) ∈ P)
10	4, 9	mpan2 691	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) ∈ P)
11		addclpr 11037	. . . . . . . . 9 ⊢ (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
12	8, 10, 11	syl2an 596	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
13	12, 12	anim12i 613	. . . . . . 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 2736	. . . . . . . 8 ⊢ (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P) = (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P)
15		enreceq 11085	. . . . . . . 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 258	. . . . . . 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 580	. . . . . 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 705	. . . . 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 566	. . . 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 2771	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R )
21		df-0r 11079	. . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R
22	21	oveq2i 7421	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R )
23	20, 22, 21	3eqtr4g 2796	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R)
24	1, 3, 23	ecoptocl 8826	1 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4612  (class class class)co 7410  [cec 8722  Pcnp 10878  1_Pc1p 10879   +_P cpp 10880   ·_P cmp 10881   ~_R cer 10883  Rcnr 10884  0_Rc0r 10885   ·_R cmr 10889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-omul 8490  df-er 8724  df-ec 8726  df-qs 8730  df-ni 10891  df-pli 10892  df-mi 10893  df-lti 10894  df-plpq 10927  df-mpq 10928  df-ltpq 10929  df-enq 10930  df-nq 10931  df-erq 10932  df-plq 10933  df-mq 10934  df-1nq 10935  df-rq 10936  df-ltnq 10937  df-np 11000  df-1p 11001  df-plp 11002  df-mp 11003  df-ltp 11004  df-enr 11074  df-nr 11075  df-mr 11077  df-0r 11079

This theorem is referenced by:  pn0sr  11120  mulresr  11158  axi2m1  11178  axcnre  11183