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Theorem 0idsr 9903
 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 9863 . 2 R = ((P × P) / ~R )
2 oveq1 6642 . . 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 2635 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 +R 0R) = 𝐴))
5 df-0r 9867 . . . 4 0R = [⟨1P, 1P⟩] ~R
65oveq2i 6646 . . 3 ([⟨𝑥, 𝑦⟩] ~R +R 0R) = ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R )
7 1pr 9822 . . . . 5 1PP
8 addsrpr 9881 . . . . 5 (((𝑥P𝑦P) ∧ (1PP ∧ 1PP)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
97, 7, 8mpanr12 720 . . . 4 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
10 addclpr 9825 . . . . . . 7 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) ∈ P)
117, 10mpan2 706 . . . . . 6 (𝑥P → (𝑥 +P 1P) ∈ P)
12 addclpr 9825 . . . . . . 7 ((𝑦P ∧ 1PP) → (𝑦 +P 1P) ∈ P)
137, 12mpan2 706 . . . . . 6 (𝑦P → (𝑦 +P 1P) ∈ P)
1411, 13anim12i 589 . . . . 5 ((𝑥P𝑦P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P))
15 vex 3198 . . . . . . 7 𝑥 ∈ V
16 vex 3198 . . . . . . 7 𝑦 ∈ V
177elexi 3208 . . . . . . 7 1P ∈ V
18 addcompr 9828 . . . . . . 7 (𝑧 +P 𝑤) = (𝑤 +P 𝑧)
19 addasspr 9829 . . . . . . 7 ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))
2015, 16, 17, 18, 19caov12 6847 . . . . . 6 (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P))
21 enreceq 9872 . . . . . 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 248 . . . . 5 (((𝑥P𝑦P) ∧ ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
2314, 22mpdan 701 . . . 4 ((𝑥P𝑦P) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
249, 23eqtr4d 2657 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
256, 24syl5eq 2666 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R )
261, 4, 25ecoptocl 7822 1 (𝐴R → (𝐴 +R 0R) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ⟨cop 4174  (class class class)co 6635  [cec 7725  Pcnp 9666  1Pc1p 9667   +P cpp 9668   ~R cer 9671  Rcnr 9672  0Rc0r 9673   +R cplr 9676 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-omul 7550  df-er 7727  df-ec 7729  df-qs 7733  df-ni 9679  df-pli 9680  df-mi 9681  df-lti 9682  df-plpq 9715  df-mpq 9716  df-ltpq 9717  df-enq 9718  df-nq 9719  df-erq 9720  df-plq 9721  df-mq 9722  df-1nq 9723  df-rq 9724  df-ltnq 9725  df-np 9788  df-1p 9789  df-plp 9790  df-ltp 9792  df-enr 9862  df-nr 9863  df-plr 9864  df-0r 9867 This theorem is referenced by:  addgt0sr  9910  sqgt0sr  9912  map2psrpr  9916  supsrlem  9917  addresr  9944  mulresr  9945  axi2m1  9965  axcnre  9970
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