Theorem ecopoveq 6596

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 5851	. . . 4 ⊢ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐷) → (𝑧 + 𝑢) = (𝐴 + 𝐷))
2		oveq12 5851	. . . 4 ⊢ ((𝑤 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑤 + 𝑣) = (𝐵 + 𝐶))
3	1, 2	eqeqan12d 2181	. . 3 ⊢ (((𝑧 = 𝐴 ∧ 𝑢 = 𝐷) ∧ (𝑤 = 𝐵 ∧ 𝑣 = 𝐶)) → ((𝑧 + 𝑢) = (𝑤 + 𝑣) ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
4	3	an42s 579	. 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 4683	1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ ∼ ⟨𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ⟨cop 3579   class class class wbr 3982  {copab 4042   × cxp 4602  (class class class)co 5842

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-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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  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-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  ecopovsym  6597  ecopovtrn  6598  ecopover  6599  ecopovsymg  6600  ecopovtrng  6601  ecopoverg  6602  enqbreq  7297  enrbreq  7675  prsrlem1  7683