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Theorem addclsr 9864
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 9838 . . 3 R = ((P × P) / ~R )
2 oveq1 6622 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 +R [⟨𝑧, 𝑤⟩] ~R ))
32eleq1d 2683 . . 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 6623 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 +R 𝐵))
54eleq1d 2683 . . 3 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 +R [⟨𝑧, 𝑤⟩] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 +R 𝐵) ∈ ((P × P) / ~R )))
6 addsrpr 9856 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R )
7 addclpr 9800 . . . . . . 7 ((𝑥P𝑧P) → (𝑥 +P 𝑧) ∈ P)
8 addclpr 9800 . . . . . . 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 5118 . . . . 5 (((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P) → ⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩ ∈ (P × P))
12 enrex 9848 . . . . . 6 ~R ∈ V
1312ecelqsi 7763 . . . . 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 2698 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) ∈ ((P × P) / ~R ))
161, 3, 5, 152ecoptocl 7798 . 2 ((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ ((P × P) / ~R ))
1716, 1syl6eleqr 2709 1 ((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ R)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cop 4161   × cxp 5082  (class class class)co 6615  [cec 7700   / cqs 7701  Pcnp 9641   +P cpp 9643   ~R cer 9646  Rcnr 9647   +R cplr 9651
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-ec 7704  df-qs 7708  df-ni 9654  df-pli 9655  df-mi 9656  df-lti 9657  df-plpq 9690  df-mpq 9691  df-ltpq 9692  df-enq 9693  df-nq 9694  df-erq 9695  df-plq 9696  df-mq 9697  df-1nq 9698  df-rq 9699  df-ltnq 9700  df-np 9763  df-plp 9765  df-ltp 9767  df-enr 9837  df-nr 9838  df-plr 9839
This theorem is referenced by:  dmaddsr  9866  map2psrpr  9891  axaddf  9926  axmulf  9927  axaddrcl  9933  axaddass  9937  axmulass  9938  axdistr  9939
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