Theorem addclsr 9783

Description: Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
addclsr	⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R)

Proof of Theorem addclsr

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 9757	. . 3 ⊢ R = ((P × P) / ~R )
2		oveq1 6556	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 +R [⟨𝑧, 𝑤⟩] ~R ))
3	2	eleq1d 2672	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 +R [⟨𝑧, 𝑤⟩] ~R ) ∈ ((P × P) / ~R )))
4		oveq2 6557	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 +R 𝐵))
5	4	eleq1d 2672	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 +R [⟨𝑧, 𝑤⟩] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 +R 𝐵) ∈ ((P × P) / ~R )))
6		addsrpr 9775	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R )
7		addclpr 9719	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 +P 𝑧) ∈ P)
8		addclpr 9719	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 +P 𝑤) ∈ P)
9	7, 8	anim12i 588	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P))
10	9	an4s 865	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P))
11		opelxpi 5072	. . . . 5 ⊢ (((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P) → ⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩ ∈ (P × P))
12		enrex 9767	. . . . . 6 ⊢ ~R ∈ V
13	12	ecelqsi 7690	. . . . 5 ⊢ (⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩ ∈ (P × P) → [⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R ∈ ((P × P) / ~R ))
14	10, 11, 13	3syl 18	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → [⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R ∈ ((P × P) / ~R ))
15	6, 14	eqeltrd 2688	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) ∈ ((P × P) / ~R ))
16	1, 3, 5, 15	2ecoptocl 7725	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ ((P × P) / ~R ))
17	16, 1	syl6eleqr 2699	1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036  (class class class)co 6549  [cec 7627   / cqs 7628  Pcnp 9560   +_P cpp 9562   ~_R cer 9565  Rcnr 9566   +_R cplr 9570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-plp 9684  df-ltp 9686  df-enr 9756  df-nr 9757  df-plr 9758

This theorem is referenced by:  dmaddsr  9785  map2psrpr  9810  axaddf  9845  axmulf  9846  axaddrcl  9852  axaddass  9856  axmulass  9857  axdistr  9858