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Theorem enqbreq2 10843

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 7981	. . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2nd2 7981	. . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	1, 2	breqan12d 5101	. 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 7974	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1^st ‘𝐴) ∈ N)
5		xp2nd 7975	. . . 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 7974	. . . 4 ⊢ (𝐵 ∈ (N × N) → (1^st ‘𝐵) ∈ N)
8		xp2nd 7975	. . . 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 10842	. . 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 597	. 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 10819	. . . 4 ⊢ ((2^nd ‘𝐴) ·_N (1^st ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴))
13	12	eqeq2i 2749	. . 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 1542 ∈ wcel 2114 ⟨cop 4573 class class class wbr 5085 × cxp 5629 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 Ncnpi 10767 ·_N cmi 10769 ~_Q ceq 10774

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-oadd 8409 df-omul 8410 df-ni 10795 df-mi 10797 df-enq 10834

This theorem is referenced by: adderpqlem 10877 mulerpqlem 10878 ltsonq 10892 lterpq 10893

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