Theorem ecopoveq 6794

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 6022	. . . 4 |- ( ( z = A /\ u = D ) -> ( z .+ u ) = ( A .+ D ) )
2		oveq12 6022	. . . 4 |- ( ( w = B /\ v = C ) -> ( w .+ v ) = ( B .+ C ) )
3	1, 2	eqeqan12d 2245	. . 3 |- ( ( ( z = A /\ u = D ) /\ ( w = B /\ v = C ) ) -> ( ( z .+ u ) = ( w .+ v ) <-> ( A .+ D ) = ( B .+ C ) ) )
4	3	an42s 591	. 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 4803	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 1395   E.wex 1538   e. wcel 2200   <.cop 3670   class class class wbr 4086   {copab 4147   X. cxp 4721  (class class class)co 6013

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-xp 4729  df-iota 5284  df-fv 5332  df-ov 6016

This theorem is referenced by:  ecopovsym  6795  ecopovtrn  6796  ecopover  6797  ecopovsymg  6798  ecopovtrng  6799  ecopoverg  6800  enqbreq  7566  enrbreq  7944  prsrlem1  7952