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| Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) | 
| 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 8054 | . . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
| 2 | 1st2nd2 8054 | . . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
| 3 | 1, 2 | breqan12d 5158 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ~Q 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) | 
| 4 | xp1st 8047 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
| 5 | xp2nd 8048 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
| 6 | 4, 5 | jca 511 | . . 3 ⊢ (𝐴 ∈ (N × N) → ((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N)) | 
| 7 | xp1st 8047 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N) | |
| 8 | xp2nd 8048 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N) | |
| 9 | 7, 8 | jca 511 | . . 3 ⊢ (𝐵 ∈ (N × N) → ((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N)) | 
| 10 | enqbreq 10960 | . . 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 596 | . 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 | mulcompi 10937 | . . . 4 ⊢ ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) | |
| 13 | 12 | eqeq2i 2749 | . . 3 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐴) ·N (1st ‘𝐵)) ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))) | 
| 14 | 13 | a1i 11 | . 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 305 | 1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 〈cop 4631 class class class wbr 5142 × cxp 5682 ‘cfv 6560 (class class class)co 7432 1st c1st 8013 2nd c2nd 8014 Ncnpi 10885 ·N cmi 10887 ~Q ceq 10892 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-oadd 8511 df-omul 8512 df-ni 10913 df-mi 10915 df-enq 10952 | 
| This theorem is referenced by: adderpqlem 10995 mulerpqlem 10996 ltsonq 11010 lterpq 11011 | 
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