Theorem ecopoveq 6620

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 5874	. . . 4 |- ( ( z = A /\ u = D ) -> ( z .+ u ) = ( A .+ D ) )
2		oveq12 5874	. . . 4 |- ( ( w = B /\ v = C ) -> ( w .+ v ) = ( B .+ C ) )
3	1, 2	eqeqan12d 2191	. . 3 |- ( ( ( z = A /\ u = D ) /\ ( w = B /\ v = C ) ) -> ( ( z .+ u ) = ( w .+ v ) <-> ( A .+ D ) = ( B .+ C ) ) )
4	3	an42s 589	. 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 4699	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 1353   E.wex 1490   e. wcel 2146   <.cop 3592   class class class wbr 3998   {copab 4058   X. cxp 4618  (class class class)co 5865

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-xp 4626  df-iota 5170  df-fv 5216  df-ov 5868

This theorem is referenced by:  ecopovsym  6621  ecopovtrn  6622  ecopover  6623  ecopovsymg  6624  ecopovtrng  6625  ecopoverg  6626  enqbreq  7330  enrbreq  7708  prsrlem1  7716