Theorem axaddrcl 11040

Description: Closure law for addition in the real subfield of complex numbers. Axiom 5 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 11064 be used later. Instead, in most cases use readdcl 11086. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
axaddrcl	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)

Proof of Theorem axaddrcl

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

Step	Hyp	Ref	Expression
1		elreal 11019	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 11019	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		oveq1 7353	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ + ⟨𝑦, 0R⟩) = (𝐴 + ⟨𝑦, 0R⟩))
4	3	eleq1d 2816	. 2 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ + ⟨𝑦, 0R⟩) ∈ ℝ ↔ (𝐴 + ⟨𝑦, 0R⟩) ∈ ℝ))
5		oveq2 7354	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 + ⟨𝑦, 0R⟩) = (𝐴 + 𝐵))
6	5	eleq1d 2816	. 2 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 + ⟨𝑦, 0R⟩) ∈ ℝ ↔ (𝐴 + 𝐵) ∈ ℝ))
7		addresr 11026	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑥 +R 𝑦), 0R⟩)
8		addclsr 10971	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 𝑦) ∈ R)
9		opelreal 11018	. . . 4 ⊢ (⟨(𝑥 +R 𝑦), 0R⟩ ∈ ℝ ↔ (𝑥 +R 𝑦) ∈ R)
10	8, 9	sylibr 234	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → ⟨(𝑥 +R 𝑦), 0R⟩ ∈ ℝ)
11	7, 10	eqeltrd 2831	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 0R⟩ + ⟨𝑦, 0R⟩) ∈ ℝ)
12	1, 2, 4, 6, 11	2gencl 3479	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4582  (class class class)co 7346  Rcnr 10753  0_Rc0r 10754   +_R cplr 10757  ℝcr 11002   + caddc 11006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-omul 8390  df-er 8622  df-ec 8624  df-qs 8628  df-ni 10760  df-pli 10761  df-mi 10762  df-lti 10763  df-plpq 10796  df-mpq 10797  df-ltpq 10798  df-enq 10799  df-nq 10800  df-erq 10801  df-plq 10802  df-mq 10803  df-1nq 10804  df-rq 10805  df-ltnq 10806  df-np 10869  df-1p 10870  df-plp 10871  df-ltp 10873  df-enr 10943  df-nr 10944  df-plr 10945  df-0r 10948  df-c 11009  df-r 11013  df-add 11014

This theorem is referenced by: (None)