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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 5959 | . . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1st2nd2 5959 | . . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
3 | 1, 2 | breqan12d 3866 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ~Q 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
4 | xp1st 5950 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
5 | xp2nd 5951 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
6 | 4, 5 | jca 301 | . . 3 ⊢ (𝐴 ∈ (N × N) → ((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N)) |
7 | xp1st 5950 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N) | |
8 | xp2nd 5951 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N) | |
9 | 7, 8 | jca 301 | . . 3 ⊢ (𝐵 ∈ (N × N) → ((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N)) |
10 | enqbreq 6976 | . . 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 284 | . 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 6951 | . . . 4 ⊢ (((2nd ‘𝐴) ∈ N ∧ (1st ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))) | |
13 | 5, 7, 12 | syl2an 284 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))) |
14 | 13 | eqeq2d 2100 | . 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 1290 ∈ wcel 1439 〈cop 3453 class class class wbr 3851 × cxp 4450 ‘cfv 5028 (class class class)co 5666 1st c1st 5923 2nd c2nd 5924 Ncnpi 6892 ·N cmi 6894 ~Q ceq 6899 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-oadd 6199 df-omul 6200 df-ni 6924 df-mi 6926 df-enq 6967 |
This theorem is referenced by: (None) |
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