Theorem enrbreq 7784

Description: Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)

Assertion

Ref	Expression
enrbreq	⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (⟨𝐴, 𝐵⟩ ~R ⟨𝐶, 𝐷⟩ ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))

Proof of Theorem enrbreq

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

Step	Hyp	Ref	Expression
1		df-enr 7776	. 2 ⊢ ~R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))}
2	1	ecopoveq 6675	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (⟨𝐴, 𝐵⟩ ~R ⟨𝐶, 𝐷⟩ ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ⟨cop 3621   class class class wbr 4029  (class class class)co 5910  Pcnp 7341   +_P cpp 7343   ~_R cer 7346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4661  df-iota 5207  df-fv 5254  df-ov 5913  df-enr 7776

This theorem is referenced by:  enreceq  7786  addcmpblnr  7789  mulcmpblnr  7791