enqbreq2 - Metamath Proof Explorer

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Theorem enqbreq2 10989

Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
enqbreq2	⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~_Q 𝐵 ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴))))

**Proof of Theorem enqbreq2**
Step	Hyp	Ref	Expression
1		1st2nd2 8069	. . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2nd2 8069	. . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	1, 2	breqan12d 5182	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~_Q 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ~_Q ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
4		xp1st 8062	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1^st ‘𝐴) ∈ N)
5		xp2nd 8063	. . . 4 ⊢ (𝐴 ∈ (N × N) → (2^nd ‘𝐴) ∈ N)
6	4, 5	jca 511	. . 3 ⊢ (𝐴 ∈ (N × N) → ((1^st ‘𝐴) ∈ N ∧ (2^nd ‘𝐴) ∈ N))
7		xp1st 8062	. . . 4 ⊢ (𝐵 ∈ (N × N) → (1^st ‘𝐵) ∈ N)
8		xp2nd 8063	. . . 4 ⊢ (𝐵 ∈ (N × N) → (2^nd ‘𝐵) ∈ N)
9	7, 8	jca 511	. . 3 ⊢ (𝐵 ∈ (N × N) → ((1^st ‘𝐵) ∈ N ∧ (2^nd ‘𝐵) ∈ N))
10		enqbreq 10988	. . 3 ⊢ ((((1^st ‘𝐴) ∈ N ∧ (2^nd ‘𝐴) ∈ N) ∧ ((1^st ‘𝐵) ∈ N ∧ (2^nd ‘𝐵) ∈ N)) → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ~_Q ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) = ((2^nd ‘𝐴) ·_N (1^st ‘𝐵))))
11	6, 9, 10	syl2an 595	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ~_Q ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) = ((2^nd ‘𝐴) ·_N (1^st ‘𝐵))))
12		mulcompi 10965	. . . 4 ⊢ ((2^nd ‘𝐴) ·_N (1^st ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴))
13	12	eqeq2i 2753	. . 3 ⊢ (((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) = ((2^nd ‘𝐴) ·_N (1^st ‘𝐵)) ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴)))
14	13	a1i 11	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) = ((2^nd ‘𝐴) ·_N (1^st ‘𝐵)) ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴))))
15	3, 11, 14	3bitrd 305	1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~_Q 𝐵 ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⟨cop 4654 class class class wbr 5166 × cxp 5698 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 Ncnpi 10913 ·_N cmi 10915 ~_Q ceq 10920

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-oadd 8526 df-omul 8527 df-ni 10941 df-mi 10943 df-enq 10980

This theorem is referenced by: adderpqlem 11023 mulerpqlem 11024 ltsonq 11038 lterpq 11039

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