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Mirrors > Home > MPE Home > Th. List > enqbreq2 | Structured version Visualization version GIF version |
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 7890 | . . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1st2nd2 7890 | . . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
3 | 1, 2 | breqan12d 5093 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ~Q 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
4 | xp1st 7883 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
5 | xp2nd 7884 | . . . 4 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
6 | 4, 5 | jca 511 | . . 3 ⊢ (𝐴 ∈ (N × N) → ((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N)) |
7 | xp1st 7883 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N) | |
8 | xp2nd 7884 | . . . 4 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N) | |
9 | 7, 8 | jca 511 | . . 3 ⊢ (𝐵 ∈ (N × N) → ((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N)) |
10 | enqbreq 10703 | . . 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 595 | . 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 10680 | . . . 4 ⊢ ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) | |
13 | 12 | eqeq2i 2746 | . . 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 304 | 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 205 ∧ wa 395 = wceq 1537 ∈ wcel 2101 〈cop 4570 class class class wbr 5077 × cxp 5589 ‘cfv 6447 (class class class)co 7295 1st c1st 7849 2nd c2nd 7850 Ncnpi 10628 ·N cmi 10630 ~Q ceq 10635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7608 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-oadd 8321 df-omul 8322 df-ni 10656 df-mi 10658 df-enq 10695 |
This theorem is referenced by: adderpqlem 10738 mulerpqlem 10739 ltsonq 10753 lterpq 10754 |
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