enqbreq2 - Intuitionistic Logic Explorer

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Theorem enqbreq2 7714

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

**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 6399	. . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2nd2 6399	. . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	1, 2	breqan12d 4141	. 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 6389	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1^st ‘𝐴) ∈ N)
5		xp2nd 6390	. . . 4 ⊢ (𝐴 ∈ (N × N) → (2^nd ‘𝐴) ∈ N)
6	4, 5	jca 306	. . 3 ⊢ (𝐴 ∈ (N × N) → ((1^st ‘𝐴) ∈ N ∧ (2^nd ‘𝐴) ∈ N))
7		xp1st 6389	. . . 4 ⊢ (𝐵 ∈ (N × N) → (1^st ‘𝐵) ∈ N)
8		xp2nd 6390	. . . 4 ⊢ (𝐵 ∈ (N × N) → (2^nd ‘𝐵) ∈ N)
9	7, 8	jca 306	. . 3 ⊢ (𝐵 ∈ (N × N) → ((1^st ‘𝐵) ∈ N ∧ (2^nd ‘𝐵) ∈ N))
10		enqbreq 7713	. . 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 289	. 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		mulcompig 7688	. . . 4 ⊢ (((2^nd ‘𝐴) ∈ N ∧ (1^st ‘𝐵) ∈ N) → ((2^nd ‘𝐴) ·_N (1^st ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴)))
13	5, 7, 12	syl2an 289	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((2^nd ‘𝐴) ·_N (1^st ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴)))
14	13	eqeq2d 2250	. 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 214	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 set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ⟨cop 3708 class class class wbr 4125 × cxp 4767 ‘cfv 5372 (class class class)co 6075 1^st c1st 6362 2^nd c2nd 6363 Ncnpi 7629 ·_N cmi 7631 ~_Q ceq 7636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-iord 4506 df-on 4508 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-oadd 6681 df-omul 6682 df-ni 7661 df-mi 7663 df-enq 7704

This theorem is referenced by: (None)

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