Theorem 0idsr 11020

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	⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)

Proof of Theorem 0idsr

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 10979	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 7375	. . 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)
4	2, 3	eqeq12d 2753	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 +R 0R) = 𝐴))
5		df-0r 10983	. . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R
6	5	oveq2i 7379	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R +R 0R) = ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R )
7		1pr 10938	. . . . 5 ⊢ 1P ∈ P
8		addsrpr 10998	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (1P ∈ P ∧ 1P ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
9	7, 7, 8	mpanr12 706	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
10		addclpr 10941	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P)
11	7, 10	mpan2 692	. . . . . 6 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P)
12		addclpr 10941	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 +P 1P) ∈ P)
13	7, 12	mpan2 692	. . . . . 6 ⊢ (𝑦 ∈ P → (𝑦 +P 1P) ∈ P)
14	11, 13	anim12i 614	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P))
15		vex 3446	. . . . . . 7 ⊢ 𝑥 ∈ V
16		vex 3446	. . . . . . 7 ⊢ 𝑦 ∈ V
17	7	elexi 3465	. . . . . . 7 ⊢ 1P ∈ V
18		addcompr 10944	. . . . . . 7 ⊢ (𝑧 +P 𝑤) = (𝑤 +P 𝑧)
19		addasspr 10945	. . . . . . 7 ⊢ ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))
20	15, 16, 17, 18, 19	caov12 7596	. . . . . 6 ⊢ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P))
21		enreceq 10989	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ((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))))
22	20, 21	mpbiri 258	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
23	14, 22	mpdan 688	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
24	9, 23	eqtr4d 2775	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
25	6, 24	eqtrid 2784	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R )
26	1, 4, 25	ecoptocl 8756	1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4588  (class class class)co 7368  [cec 8643  Pcnp 10782  1_Pc1p 10783   +_P cpp 10784   ~_R cer 10787  Rcnr 10788  0_Rc0r 10789   +_R cplr 10792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-omul 8412  df-er 8645  df-ec 8647  df-qs 8651  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-mpq 10832  df-ltpq 10833  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-mq 10838  df-1nq 10839  df-rq 10840  df-ltnq 10841  df-np 10904  df-1p 10905  df-plp 10906  df-ltp 10908  df-enr 10978  df-nr 10979  df-plr 10980  df-0r 10983

This theorem is referenced by:  addgt0sr  11027  sqgt0sr  11029  map2psrpr  11033  supsrlem  11034  addresr  11061  mulresr  11062  axi2m1  11082  axcnre  11087