Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > enrbreq | GIF version |
Description: Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) |
Ref | Expression |
---|---|
enrbreq | ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (〈𝐴, 𝐵〉 ~R 〈𝐶, 𝐷〉 ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enr 7527 | . 2 ⊢ ~R = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))} | |
2 | 1 | ecopoveq 6517 | 1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (〈𝐴, 𝐵〉 ~R 〈𝐶, 𝐷〉 ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 〈cop 3525 class class class wbr 3924 (class class class)co 5767 Pcnp 7092 +P cpp 7094 ~R cer 7097 |
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 df-enr 7527 |
This theorem is referenced by: enreceq 7537 addcmpblnr 7540 mulcmpblnr 7542 |
Copyright terms: Public domain | W3C validator |