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Mirrors > Home > ILE Home > Th. List > enreceq | GIF version |
Description: Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
Ref | Expression |
---|---|
enreceq | ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([〈𝐴, 𝐵〉] ~R = [〈𝐶, 𝐷〉] ~R ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enrer 7543 | . . . 4 ⊢ ~R Er (P × P) | |
2 | 1 | a1i 9 | . . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ~R Er (P × P)) |
3 | opelxpi 4571 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 〈𝐴, 𝐵〉 ∈ (P × P)) | |
4 | 3 | adantr 274 | . . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → 〈𝐴, 𝐵〉 ∈ (P × P)) |
5 | 2, 4 | erth 6473 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (〈𝐴, 𝐵〉 ~R 〈𝐶, 𝐷〉 ↔ [〈𝐴, 𝐵〉] ~R = [〈𝐶, 𝐷〉] ~R )) |
6 | enrbreq 7542 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (〈𝐴, 𝐵〉 ~R 〈𝐶, 𝐷〉 ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶))) | |
7 | 5, 6 | bitr3d 189 | 1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([〈𝐴, 𝐵〉] ~R = [〈𝐶, 𝐷〉] ~R ↔ (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 3530 class class class wbr 3929 × cxp 4537 (class class class)co 5774 Er wer 6426 [cec 6427 Pcnp 7099 +P cpp 7101 ~R cer 7104 |
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-in1 603 ax-in2 604 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-iplp 7276 df-enr 7534 |
This theorem is referenced by: ltsrprg 7555 m1p1sr 7568 m1m1sr 7569 0idsr 7575 1idsr 7576 00sr 7577 recexgt0sr 7581 aptisr 7587 srpospr 7591 prsradd 7594 map2psrprg 7613 pitonnlem1p1 7654 recidpirq 7666 |
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