Theorem ecopoveq 6842

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 6037	. . . 4 |- ( ( z = A /\ u = D ) -> ( z .+ u ) = ( A .+ D ) )
2		oveq12 6037	. . . 4 |- ( ( w = B /\ v = C ) -> ( w .+ v ) = ( B .+ C ) )
3	1, 2	eqeqan12d 2247	. . 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 4811	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 2202   <.cop 3676   class class class wbr 4093   {copab 4154   X. cxp 4729  (class class class)co 6028

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  ecopovsym  6843  ecopovtrn  6844  ecopover  6845  ecopovsymg  6846  ecopovtrng  6847  ecopoverg  6848  enqbreq  7636  enrbreq  8014  prsrlem1  8022