![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 7748 | . . . 4 ⊢ ~R Er (P × P) | |
2 | 1 | a1i 9 | . . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ~R Er (P × P)) |
3 | opelxpi 4670 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 〈𝐴, 𝐵〉 ∈ (P × P)) | |
4 | 3 | adantr 276 | . . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → 〈𝐴, 𝐵〉 ∈ (P × P)) |
5 | 2, 4 | erth 6593 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (〈𝐴, 𝐵〉 ~R 〈𝐶, 𝐷〉 ↔ [〈𝐴, 𝐵〉] ~R = [〈𝐶, 𝐷〉] ~R )) |
6 | enrbreq 7747 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (〈𝐴, 𝐵〉 ~R 〈𝐶, 𝐷〉 ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶))) | |
7 | 5, 6 | bitr3d 190 | 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 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 〈cop 3607 class class class wbr 4015 × cxp 4636 (class class class)co 5888 Er wer 6546 [cec 6547 Pcnp 7304 +P cpp 7306 ~R cer 7309 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-eprel 4301 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-irdg 6385 df-1o 6431 df-2o 6432 df-oadd 6435 df-omul 6436 df-er 6549 df-ec 6551 df-qs 6555 df-ni 7317 df-pli 7318 df-mi 7319 df-lti 7320 df-plpq 7357 df-mpq 7358 df-enq 7360 df-nqqs 7361 df-plqqs 7362 df-mqqs 7363 df-1nqqs 7364 df-rq 7365 df-ltnqqs 7366 df-enq0 7437 df-nq0 7438 df-0nq0 7439 df-plq0 7440 df-mq0 7441 df-inp 7479 df-iplp 7481 df-enr 7739 |
This theorem is referenced by: ltsrprg 7760 m1p1sr 7773 m1m1sr 7774 0idsr 7780 1idsr 7781 00sr 7782 recexgt0sr 7786 aptisr 7792 srpospr 7796 prsradd 7799 map2psrprg 7818 pitonnlem1p1 7859 recidpirq 7871 |
Copyright terms: Public domain | W3C validator |