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Theorem 0idsr 9769
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 9729 . 2 R = ((P × P) / ~R )
2 oveq1 6529 . . 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 2619 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 +R 0R) = 𝐴))
5 df-0r 9733 . . . 4 0R = [⟨1P, 1P⟩] ~R
65oveq2i 6533 . . 3 ([⟨𝑥, 𝑦⟩] ~R +R 0R) = ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R )
7 1pr 9688 . . . . 5 1PP
8 addsrpr 9747 . . . . 5 (((𝑥P𝑦P) ∧ (1PP ∧ 1PP)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
97, 7, 8mpanr12 716 . . . 4 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
10 addclpr 9691 . . . . . . 7 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) ∈ P)
117, 10mpan2 702 . . . . . 6 (𝑥P → (𝑥 +P 1P) ∈ P)
12 addclpr 9691 . . . . . . 7 ((𝑦P ∧ 1PP) → (𝑦 +P 1P) ∈ P)
137, 12mpan2 702 . . . . . 6 (𝑦P → (𝑦 +P 1P) ∈ P)
1411, 13anim12i 587 . . . . 5 ((𝑥P𝑦P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P))
15 vex 3170 . . . . . . 7 𝑥 ∈ V
16 vex 3170 . . . . . . 7 𝑦 ∈ V
177elexi 3180 . . . . . . 7 1P ∈ V
18 addcompr 9694 . . . . . . 7 (𝑧 +P 𝑤) = (𝑤 +P 𝑧)
19 addasspr 9695 . . . . . . 7 ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))
2015, 16, 17, 18, 19caov12 6732 . . . . . 6 (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P))
21 enreceq 9738 . . . . . 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 246 . . . . 5 (((𝑥P𝑦P) ∧ ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
2314, 22mpdan 698 . . . 4 ((𝑥P𝑦P) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
249, 23eqtr4d 2641 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
256, 24syl5eq 2650 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R )
261, 4, 25ecoptocl 7696 1 (𝐴R → (𝐴 +R 0R) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  cop 4125  (class class class)co 6522  [cec 7599  Pcnp 9532  1Pc1p 9533   +P cpp 9534   ~R cer 9537  Rcnr 9538  0Rc0r 9539   +R cplr 9542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423  df-omul 7424  df-er 7601  df-ec 7603  df-qs 7607  df-ni 9545  df-pli 9546  df-mi 9547  df-lti 9548  df-plpq 9581  df-mpq 9582  df-ltpq 9583  df-enq 9584  df-nq 9585  df-erq 9586  df-plq 9587  df-mq 9588  df-1nq 9589  df-rq 9590  df-ltnq 9591  df-np 9654  df-1p 9655  df-plp 9656  df-ltp 9658  df-enr 9728  df-nr 9729  df-plr 9730  df-0r 9733
This theorem is referenced by:  addgt0sr  9776  sqgt0sr  9778  map2psrpr  9782  supsrlem  9783  addresr  9810  mulresr  9811  axi2m1  9831  axcnre  9836
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