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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 5751 | . . . 4 | |
2 | oveq12 5751 | . . . 4 | |
3 | 1, 2 | eqeqan12d 2133 | . . 3 |
4 | 3 | an42s 563 | . 2 |
5 | ecopopr.1 | . 2 | |
6 | 4, 5 | opbrop 4588 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cop 3500 class class class wbr 3899 copab 3958 cxp 4507 (class class class)co 5742 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: ecopovsym 6493 ecopovtrn 6494 ecopover 6495 ecopovsymg 6496 ecopovtrng 6497 ecopoverg 6498 enqbreq 7132 enrbreq 7510 prsrlem1 7518 |
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