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Mirrors > Home > ILE Home > Th. List > enqbreq2 | GIF version |
Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) |
Ref | Expression |
---|---|
enqbreq2 | ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 6073 | . . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1st2nd2 6073 | . . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
3 | 1, 2 | breqan12d 3945 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ~Q 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
4 | xp1st 6063 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
5 | xp2nd 6064 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
6 | 4, 5 | jca 304 | . . 3 ⊢ (𝐴 ∈ (N × N) → ((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N)) |
7 | xp1st 6063 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N) | |
8 | xp2nd 6064 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N) | |
9 | 7, 8 | jca 304 | . . 3 ⊢ (𝐵 ∈ (N × N) → ((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N)) |
10 | enqbreq 7167 | . . 3 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ ((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N)) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ~Q 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐴) ·N (1st ‘𝐵)))) | |
11 | 6, 9, 10 | syl2an 287 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ~Q 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐴) ·N (1st ‘𝐵)))) |
12 | mulcompig 7142 | . . . 4 ⊢ (((2nd ‘𝐴) ∈ N ∧ (1st ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))) | |
13 | 5, 7, 12 | syl2an 287 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))) |
14 | 13 | eqeq2d 2151 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐴) ·N (1st ‘𝐵)) ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))) |
15 | 3, 11, 14 | 3bitrd 213 | 1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))) |
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 ‘cfv 5123 (class class class)co 5774 1st c1st 6036 2nd c2nd 6037 Ncnpi 7083 ·N cmi 7085 ~Q ceq 7090 |
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-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-id 4215 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-oadd 6317 df-omul 6318 df-ni 7115 df-mi 7117 df-enq 7158 |
This theorem is referenced by: (None) |
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