Theorem 0idsr 7708

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 7668	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 5849	. . 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 2180	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 +R 0R) = 𝐴))
5		df-0r 7672	. . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R
6	5	oveq2i 5853	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R +R 0R) = ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R )
7		1pr 7495	. . . . 5 ⊢ 1P ∈ P
8		addsrpr 7686	. . . . 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 436	. . . 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 7519	. . . . . . 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 7520	. . . . . . 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 6023	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)))
18		addclpr 7478	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P)
19	7, 18	mpan2 422	. . . . . . 7 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P)
20		addclpr 7478	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 +P 1P) ∈ P)
21	7, 20	mpan2 422	. . . . . . 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 7677	. . . . . 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 418	. . . . 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 2201	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
27	6, 26	syl5eq 2211	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R )
28	1, 4, 27	ecoptocl 6588	1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ⟨cop 3579  (class class class)co 5842  [cec 6499  Pcnp 7232  1_Pc1p 7233   +_P cpp 7234   ~_R cer 7237  Rcnr 7238  0_Rc0r 7239   +_R cplr 7242

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-i1p 7408  df-iplp 7409  df-enr 7667  df-nr 7668  df-plr 7669  df-0r 7672

This theorem is referenced by:  addgt0sr  7716  ltadd1sr  7717  ltm1sr  7718  caucvgsrlemoffval  7737  caucvgsrlemoffres  7741  caucvgsr  7743  map2psrprg  7746  suplocsrlempr  7748  addresr  7778  mulresr  7779  axi2m1  7816  ax0id  7819  axcnre  7822