Theorem ecopoveq 6689

Description: This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation ∼ (specified by the hypothesis) in terms of its operation 𝐹. (Contributed by NM, 16-Aug-1995.)

Hypothesis

Ref	Expression
ecopopr.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}

Assertion

Ref	Expression
ecopoveq	⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ ∼ ⟨𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢, +   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐴,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐷,𝑦,𝑧,𝑤,𝑣,𝑢

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecopoveq

Step	Hyp	Ref	Expression
1		oveq12 5931	. . . 4 ⊢ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐷) → (𝑧 + 𝑢) = (𝐴 + 𝐷))
2		oveq12 5931	. . . 4 ⊢ ((𝑤 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑤 + 𝑣) = (𝐵 + 𝐶))
3	1, 2	eqeqan12d 2212	. . 3 ⊢ (((𝑧 = 𝐴 ∧ 𝑢 = 𝐷) ∧ (𝑤 = 𝐵 ∧ 𝑣 = 𝐶)) → ((𝑧 + 𝑢) = (𝑤 + 𝑣) ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
4	3	an42s 589	. 2 ⊢ (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑢 = 𝐷)) → ((𝑧 + 𝑢) = (𝑤 + 𝑣) ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
5		ecopopr.1	. 2 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
6	4, 5	opbrop 4742	1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ ∼ ⟨𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ⟨cop 3625   class class class wbr 4033  {copab 4093   × cxp 4661  (class class class)co 5922

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  ecopovsym  6690  ecopovtrn  6691  ecopover  6692  ecopovsymg  6693  ecopovtrng  6694  ecopoverg  6695  enqbreq  7423  enrbreq  7801  prsrlem1  7809