Theorem ecopoveq 6385

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 5661	. . . 4 |- ( ( z = A /\ u = D ) -> ( z .+ u ) = ( A .+ D ) )
2		oveq12 5661	. . . 4 |- ( ( w = B /\ v = C ) -> ( w .+ v ) = ( B .+ C ) )
3	1, 2	eqeqan12d 2103	. . 3 |- ( ( ( z = A /\ u = D ) /\ ( w = B /\ v = C ) ) -> ( ( z .+ u ) = ( w .+ v ) <-> ( A .+ D ) = ( B .+ C ) ) )
4	3	an42s 556	. 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 4517	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 102   <-> wb 103   = wceq 1289   E.wex 1426   e. wcel 1438   <.cop 3449   class class class wbr 3845   {copab 3898   X. cxp 4436  (class class class)co 5652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-iota 4980  df-fv 5023  df-ov 5655

This theorem is referenced by:  ecopovsym  6386  ecopovtrn  6387  ecopover  6388  ecopovsymg  6389  ecopovtrng  6390  ecopoverg  6391  enqbreq  6913  enrbreq  7278  prsrlem1  7286