Theorem 0idsr 7568

Description: The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)

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 7528	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 5774	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = (𝐴 +R 0R))
3		id 19	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → [⟨𝑥, 𝑦⟩] ~R = 𝐴)
4	2, 3	eqeq12d 2152	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 +R 0R) = 𝐴))
5		df-0r 7532	. . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R
6	5	oveq2i 5778	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R +R 0R) = ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R )
7		1pr 7355	. . . . 5 ⊢ 1P ∈ P
8		addsrpr 7546	. . . . 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 435	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
10		simpl 108	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑥 ∈ P)
11		simpr 109	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑦 ∈ P)
12	7	a1i 9	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 1P ∈ P)
13		addcomprg 7379	. . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
14	13	adantl 275	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
15		addassprg 7380	. . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P ∧ 𝑣 ∈ P) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
16	15	adantl 275	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P ∧ 𝑣 ∈ P)) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
17	10, 11, 12, 14, 16	caov12d 5945	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)))
18		addclpr 7338	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P)
19	7, 18	mpan2 421	. . . . . . 7 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P)
20		addclpr 7338	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 +P 1P) ∈ P)
21	7, 20	mpan2 421	. . . . . . 7 ⊢ (𝑦 ∈ P → (𝑦 +P 1P) ∈ P)
22	19, 21	anim12i 336	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P))
23		enreceq 7537	. . . . . 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))))
24	22, 23	mpdan 417	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R ↔ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P))))
25	17, 24	mpbird 166	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
26	9, 25	eqtr4d 2173	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
27	6, 26	syl5eq 2182	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R )
28	1, 4, 27	ecoptocl 6509	1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ⟨cop 3525  (class class class)co 5767  [cec 6420  Pcnp 7092  1_Pc1p 7093   +_P cpp 7094   ~_R cer 7097  Rcnr 7098  0_Rc0r 7099   +_R cplr 7102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-enr 7527  df-nr 7528  df-plr 7529  df-0r 7532

This theorem is referenced by:  addgt0sr  7576  ltadd1sr  7577  ltm1sr  7578  caucvgsrlemoffval  7597  caucvgsrlemoffres  7601  caucvgsr  7603  map2psrprg  7606  suplocsrlempr  7608  addresr  7638  mulresr  7639  axi2m1  7676  ax0id  7679  axcnre  7682