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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 6066 | . . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1st2nd2 6066 | . . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
3 | 1, 2 | breqan12d 3940 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ~Q 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
4 | xp1st 6056 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
5 | xp2nd 6057 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
6 | 4, 5 | jca 304 | . . 3 ⊢ (𝐴 ∈ (N × N) → ((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N)) |
7 | xp1st 6056 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N) | |
8 | xp2nd 6057 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N) | |
9 | 7, 8 | jca 304 | . . 3 ⊢ (𝐵 ∈ (N × N) → ((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N)) |
10 | enqbreq 7157 | . . 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 7132 | . . . 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 2149 | . 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 3525 class class class wbr 3924 × cxp 4532 ‘cfv 5118 (class class class)co 5767 1st c1st 6029 2nd c2nd 6030 Ncnpi 7073 ·N cmi 7075 ~Q ceq 7080 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-omul 6311 df-ni 7105 df-mi 7107 df-enq 7148 |
This theorem is referenced by: (None) |
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