Theorem 0idsr 11186

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 11145	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 7428	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = (𝐴 +R 0R))
3		id 23	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → [⟨𝑥, 𝑦⟩] ~R = 𝐴)
4	2, 3	eqeq12d 2776	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 +R 0R) = 𝐴))
5		df-0r 11149	. . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R
6	5	oveq2i 7432	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R +R 0R) = ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R )
7		1pr 11104	. . . . 5 ⊢ 1P ∈ P
8		addsrpr 11164	. . . . 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 718	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
10		addclpr 11107	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P)
11	7, 10	mpan2 704	. . . . . 6 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P)
12		addclpr 11107	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 +P 1P) ∈ P)
13	7, 12	mpan2 704	. . . . . 6 ⊢ (𝑦 ∈ P → (𝑦 +P 1P) ∈ P)
14	11, 13	anim12i 625	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P))
15		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
16		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
17	7	elexi 3472	. . . . . . 7 ⊢ 1P ∈ V
18		addcompr 11110	. . . . . . 7 ⊢ (𝑧 +P 𝑤) = (𝑤 +P 𝑧)
19		addasspr 11111	. . . . . . 7 ⊢ ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))
20	15, 16, 17, 18, 19	caov12 7650	. . . . . 6 ⊢ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P))
21		enreceq 11155	. . . . . 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 261	. . . . 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 700	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R )
24	9, 23	eqtr4d 2798	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
25	6, 24	eqtrid 2807	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R )
26	1, 4, 25	ecoptocl 8832	1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ⟨cop 4834  (class class class)co 7421  [cec 8719  Pcnp 10948  1_Pc1p 10949   +_P cpp 10950   ~_R cer 10953  Rcnr 10954  0_Rc0r 10955   +_R cplr 10958

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-inf2 9646

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8721  df-ec 8723  df-qs 8727  df-ni 10961  df-pli 10962  df-mi 10963  df-lti 10964  df-plpq 10997  df-mpq 10998  df-ltpq 10999  df-enq 11000  df-nq 11001  df-erq 11002  df-plq 11003  df-mq 11004  df-1nq 11005  df-rq 11006  df-ltnq 11007  df-np 11070  df-1p 11071  df-plp 11072  df-ltp 11074  df-enr 11144  df-nr 11145  df-plr 11146  df-0r 11149

This theorem is used by:  addgt0sr  11193  sqgt0sr  11195  map2psrpr  11199  supsrlem  11200  addresr  11227  mulresr  11228  axi2m1  11248  axcnre  11253