enqbreq2 - Metamath Proof Explorer

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Theorem enqbreq2 10838

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 7974	. . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2nd2 7974	. . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	1, 2	breqan12d 5091	. 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 7967	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1^st ‘𝐴) ∈ N)
5		xp2nd 7968	. . . 4 ⊢ (𝐴 ∈ (N × N) → (2^nd ‘𝐴) ∈ N)
6	4, 5	jca 517	. . 3 ⊢ (𝐴 ∈ (N × N) → ((1^st ‘𝐴) ∈ N ∧ (2^nd ‘𝐴) ∈ N))
7		xp1st 7967	. . . 4 ⊢ (𝐵 ∈ (N × N) → (1^st ‘𝐵) ∈ N)
8		xp2nd 7968	. . . 4 ⊢ (𝐵 ∈ (N × N) → (2^nd ‘𝐵) ∈ N)
9	7, 8	jca 517	. . 3 ⊢ (𝐵 ∈ (N × N) → ((1^st ‘𝐵) ∈ N ∧ (2^nd ‘𝐵) ∈ N))
10		enqbreq 10837	. . 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 603	. 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 10814	. . . 4 ⊢ ((2^nd ‘𝐴) ·_N (1^st ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴))
13	12	eqeq2i 2754	. . 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 307	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 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ⟨cop 4564 class class class wbr 5075 × cxp 5619 ‘cfv 6489 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 Ncnpi 10762 ·_N cmi 10764 ~_Q ceq 10769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-oadd 8403 df-omul 8404 df-ni 10790 df-mi 10792 df-enq 10829

This theorem is referenced by: adderpqlem 10872 mulerpqlem 10873 ltsonq 10887 lterpq 10888

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