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Theorem 0idsr 10497
 Description: The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
0idsr (𝐴R → (𝐴 +R 0R) = 𝐴)

Proof of Theorem 0idsr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 10456 . 2 R = ((P × P) / ~R )
2 oveq1 7140 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = (𝐴 +R 0R))
3 id 22 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → [⟨𝑥, 𝑦⟩] ~R = 𝐴)
42, 3eqeq12d 2836 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 +R 0R) = 𝐴))
5 df-0r 10460 . . . 4 0R = [⟨1P, 1P⟩] ~R
65oveq2i 7144 . . 3 ([⟨𝑥, 𝑦⟩] ~R +R 0R) = ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R )
7 1pr 10415 . . . . 5 1PP
8 addsrpr 10475 . . . . 5 (((𝑥P𝑦P) ∧ (1PP ∧ 1PP)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
97, 7, 8mpanr12 703 . . . 4 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
10 addclpr 10418 . . . . . . 7 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) ∈ P)
117, 10mpan2 689 . . . . . 6 (𝑥P → (𝑥 +P 1P) ∈ P)
12 addclpr 10418 . . . . . . 7 ((𝑦P ∧ 1PP) → (𝑦 +P 1P) ∈ P)
137, 12mpan2 689 . . . . . 6 (𝑦P → (𝑦 +P 1P) ∈ P)
1411, 13anim12i 614 . . . . 5 ((𝑥P𝑦P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P))
15 vex 3476 . . . . . . 7 𝑥 ∈ V
16 vex 3476 . . . . . . 7 𝑦 ∈ V
177elexi 3492 . . . . . . 7 1P ∈ V
18 addcompr 10421 . . . . . . 7 (𝑧 +P 𝑤) = (𝑤 +P 𝑧)
19 addasspr 10422 . . . . . . 7 ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))
2015, 16, 17, 18, 19caov12 7354 . . . . . 6 (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P))
21 enreceq 10466 . . . . . 6 (((𝑥P𝑦P) ∧ ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R ↔ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P))))
2220, 21mpbiri 260 . . . . 5 (((𝑥P𝑦P) ∧ ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
2314, 22mpdan 685 . . . 4 ((𝑥P𝑦P) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
249, 23eqtr4d 2858 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
256, 24syl5eq 2867 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R )
261, 4, 25ecoptocl 8365 1 (𝐴R → (𝐴 +R 0R) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ⟨cop 4549  (class class class)co 7133  [cec 8265  Pcnp 10259  1Pc1p 10260   +P cpp 10261   ~R cer 10264  Rcnr 10265  0Rc0r 10266   +R cplr 10269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-omul 8085  df-er 8267  df-ec 8269  df-qs 8273  df-ni 10272  df-pli 10273  df-mi 10274  df-lti 10275  df-plpq 10308  df-mpq 10309  df-ltpq 10310  df-enq 10311  df-nq 10312  df-erq 10313  df-plq 10314  df-mq 10315  df-1nq 10316  df-rq 10317  df-ltnq 10318  df-np 10381  df-1p 10382  df-plp 10383  df-ltp 10385  df-enr 10455  df-nr 10456  df-plr 10457  df-0r 10460 This theorem is referenced by:  addgt0sr  10504  sqgt0sr  10506  map2psrpr  10510  supsrlem  10511  addresr  10538  mulresr  10539  axi2m1  10559  axcnre  10564
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