MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ecopoveq Structured version   Visualization version   GIF version

Theorem ecopoveq 7833
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)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
Assertion
Ref Expression
ecopoveq (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝐶, 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
StepHypRef Expression
1 oveq12 6644 . . . 4 ((𝑧 = 𝐴𝑢 = 𝐷) → (𝑧 + 𝑢) = (𝐴 + 𝐷))
2 oveq12 6644 . . . 4 ((𝑤 = 𝐵𝑣 = 𝐶) → (𝑤 + 𝑣) = (𝐵 + 𝐶))
31, 2eqeqan12d 2636 . . 3 (((𝑧 = 𝐴𝑢 = 𝐷) ∧ (𝑤 = 𝐵𝑣 = 𝐶)) → ((𝑧 + 𝑢) = (𝑤 + 𝑣) ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
43an42s 869 . 2 (((𝑧 = 𝐴𝑤 = 𝐵) ∧ (𝑣 = 𝐶𝑢 = 𝐷)) → ((𝑧 + 𝑢) = (𝑤 + 𝑣) ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
5 ecopopr.1 . 2 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
64, 5opbrop 5188 1 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  cop 4174   class class class wbr 4644  {copab 4703   × cxp 5102  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-iota 5839  df-fv 5884  df-ov 6638
This theorem is referenced by:  ecopovsym  7834  ecopovtrn  7835  ecopover  7836  ecopoverOLD  7837  enqbreq  9726  enrbreq  9870  prsrlem1  9878
  Copyright terms: Public domain W3C validator