Theorem ecopoveq 6864

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 e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }

Assertion

Ref	Expression
ecopoveq	|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. 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 6059	. . . 4 |- ( ( z = A /\ u = D ) -> ( z .+ u ) = ( A .+ D ) )
2		oveq12 6059	. . . 4 |- ( ( w = B /\ v = C ) -> ( w .+ v ) = ( B .+ C ) )
3	1, 2	eqeqan12d 2248	. . 3 |- ( ( ( z = A /\ u = D ) /\ ( w = B /\ v = C ) ) -> ( ( z .+ u ) = ( w .+ v ) <-> ( A .+ D ) = ( B .+ C ) ) )
4	3	an42s 593	. 2 |- ( ( ( z = A /\ w = B ) /\ ( v = C /\ u = D ) ) -> ( ( z .+ u ) = ( w .+ v ) <-> ( A .+ D ) = ( B .+ C ) ) )
5		ecopopr.1	. 2 |- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }
6	4, 5	opbrop 4829	1 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. .~ <. C , D >. <-> ( A .+ D ) = ( B .+ C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2203   <.cop 3692   class class class wbr 4109   {copab 4170   X. cxp 4747  (class class class)co 6050

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  ecopovsym  6865  ecopovtrn  6866  ecopover  6867  ecopovsymg  6868  ecopovtrng  6869  ecopoverg  6870  enqbreq  7671  enrbreq  8049  prsrlem1  8057