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Theorem ecopoveq 6137
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 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (⟨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 5464 . . . 4 ((z = A u = 𝐷) → (z + u) = (A + 𝐷))
2 oveq12 5464 . . . 4 ((w = B v = 𝐶) → (w + v) = (B + 𝐶))
31, 2eqeqan12d 2052 . . 3 (((z = A u = 𝐷) (w = B v = 𝐶)) → ((z + u) = (w + v) ↔ (A + 𝐷) = (B + 𝐶)))
43an42s 523 . 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)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))}
64, 5opbrop 4362 1 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (⟨A, B𝐶, 𝐷⟩ ↔ (A + 𝐷) = (B + 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  cop 3370   class class class wbr 3755  {copab 3808   × cxp 4286  (class class class)co 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  ecopovsym  6138  ecopovtrn  6139  ecopover  6140  ecopovsymg  6141  ecopovtrng  6142  ecopoverg  6143  enqbreq  6340  enrbreq  6662  prsrlem1  6670
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