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Theorem addclsr 9849
Description: Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addclsr ((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ R)

Proof of Theorem addclsr
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 9823 . . 3 R = ((P × P) / ~R )
2 oveq1 6612 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 +R [⟨𝑧, 𝑤⟩] ~R ))
32eleq1d 2688 . . 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 6613 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 +R 𝐵))
54eleq1d 2688 . . 3 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 +R [⟨𝑧, 𝑤⟩] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 +R 𝐵) ∈ ((P × P) / ~R )))
6 addsrpr 9841 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R )
7 addclpr 9785 . . . . . . 7 ((𝑥P𝑧P) → (𝑥 +P 𝑧) ∈ P)
8 addclpr 9785 . . . . . . 7 ((𝑦P𝑤P) → (𝑦 +P 𝑤) ∈ P)
97, 8anim12i 589 . . . . . 6 (((𝑥P𝑧P) ∧ (𝑦P𝑤P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P))
109an4s 868 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P))
11 opelxpi 5113 . . . . 5 (((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P) → ⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩ ∈ (P × P))
12 enrex 9833 . . . . . 6 ~R ∈ V
1312ecelqsi 7749 . . . . 5 (⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩ ∈ (P × P) → [⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R ∈ ((P × P) / ~R ))
1410, 11, 133syl 18 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → [⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R ∈ ((P × P) / ~R ))
156, 14eqeltrd 2704 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) ∈ ((P × P) / ~R ))
161, 3, 5, 152ecoptocl 7784 . 2 ((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ ((P × P) / ~R ))
1716, 1syl6eleqr 2715 1 ((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ R)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  cop 4159   × cxp 5077  (class class class)co 6605  [cec 7686   / cqs 7687  Pcnp 9626   +P cpp 9628   ~R cer 9631  Rcnr 9632   +R cplr 9636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-omul 7511  df-er 7688  df-ec 7690  df-qs 7694  df-ni 9639  df-pli 9640  df-mi 9641  df-lti 9642  df-plpq 9675  df-mpq 9676  df-ltpq 9677  df-enq 9678  df-nq 9679  df-erq 9680  df-plq 9681  df-mq 9682  df-1nq 9683  df-rq 9684  df-ltnq 9685  df-np 9748  df-plp 9750  df-ltp 9752  df-enr 9822  df-nr 9823  df-plr 9824
This theorem is referenced by:  dmaddsr  9851  map2psrpr  9876  axaddf  9911  axmulf  9912  axaddrcl  9918  axaddass  9922  axmulass  9923  axdistr  9924
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