Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ecopoveq | Unicode version |
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 . (Contributed by NM, 16-Aug-1995.) |
Ref | Expression |
---|---|
ecopopr.1 |
Ref | Expression |
---|---|
ecopoveq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 5776 | . . . 4 | |
2 | oveq12 5776 | . . . 4 | |
3 | 1, 2 | eqeqan12d 2153 | . . 3 |
4 | 3 | an42s 578 | . 2 |
5 | ecopopr.1 | . 2 | |
6 | 4, 5 | opbrop 4613 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cop 3525 class class class wbr 3924 copab 3983 cxp 4532 (class class class)co 5767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: ecopovsym 6518 ecopovtrn 6519 ecopover 6520 ecopovsymg 6521 ecopovtrng 6522 ecopoverg 6523 enqbreq 7157 enrbreq 7535 prsrlem1 7543 |
Copyright terms: Public domain | W3C validator |