Theorem 00sr 11091

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 11048	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 7409	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = (𝐴 ·R 0R))
3	2	eqeq1d 2726	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R ↔ (𝐴 ·R 0R) = 0R))
4		1pr 11007	. . . . 5 ⊢ 1P ∈ P
5		mulsrpr 11068	. . . . 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 702	. . . 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 11012	. . . . . . . . . 10 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 ·P 1P) ∈ P)
8	4, 7	mpan2 688	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) ∈ P)
9		mulclpr 11012	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 ·P 1P) ∈ P)
10	4, 9	mpan2 688	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) ∈ P)
11		addclpr 11010	. . . . . . . . 9 ⊢ (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
12	8, 10, 11	syl2an 595	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P 1P)) ∈ P)
13	12, 12	anim12i 612	. . . . . . 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 2724	. . . . . . . 8 ⊢ (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P) = (((𝑥 ·P 1P) +P (𝑦 ·P 1P)) +P 1P)
15		enreceq 11058	. . . . . . . 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 579	. . . . . 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 702	. . . . 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 2764	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R )
21		df-0r 11052	. . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R
22	21	oveq2i 7413	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨1P, 1P⟩] ~R )
23	20, 22, 21	3eqtr4g 2789	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R 0R) = 0R)
24	1, 3, 23	ecoptocl 8798	1 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ⟨cop 4627  (class class class)co 7402  [cec 8698  Pcnp 10851  1_Pc1p 10852   +_P cpp 10853   ·_P cmp 10854   ~_R cer 10856  Rcnr 10857  0_Rc0r 10858   ·_R cmr 10862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-omul 8467  df-er 8700  df-ec 8702  df-qs 8706  df-ni 10864  df-pli 10865  df-mi 10866  df-lti 10867  df-plpq 10900  df-mpq 10901  df-ltpq 10902  df-enq 10903  df-nq 10904  df-erq 10905  df-plq 10906  df-mq 10907  df-1nq 10908  df-rq 10909  df-ltnq 10910  df-np 10973  df-1p 10974  df-plp 10975  df-mp 10976  df-ltp 10977  df-enr 11047  df-nr 11048  df-mr 11050  df-0r 11052

This theorem is referenced by:  pn0sr  11093  mulresr  11131  axi2m1  11151  axcnre  11156