enqbreq2 - Intuitionistic Logic Explorer

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Theorem enqbreq2 7319

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 6154	. . 3 ⊢ (𝐴 ∈ (N × N) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2nd2 6154	. . 3 ⊢ (𝐵 ∈ (N × N) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	1, 2	breqan12d 4005	. 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 6144	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1^st ‘𝐴) ∈ N)
5		xp2nd 6145	. . . 4 ⊢ (𝐴 ∈ (N × N) → (2^nd ‘𝐴) ∈ N)
6	4, 5	jca 304	. . 3 ⊢ (𝐴 ∈ (N × N) → ((1^st ‘𝐴) ∈ N ∧ (2^nd ‘𝐴) ∈ N))
7		xp1st 6144	. . . 4 ⊢ (𝐵 ∈ (N × N) → (1^st ‘𝐵) ∈ N)
8		xp2nd 6145	. . . 4 ⊢ (𝐵 ∈ (N × N) → (2^nd ‘𝐵) ∈ N)
9	7, 8	jca 304	. . 3 ⊢ (𝐵 ∈ (N × N) → ((1^st ‘𝐵) ∈ N ∧ (2^nd ‘𝐵) ∈ N))
10		enqbreq 7318	. . 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 287	. 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 7293	. . . 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 287	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((2^nd ‘𝐴) ·_N (1^st ‘𝐵)) = ((1^st ‘𝐵) ·_N (2^nd ‘𝐴)))
14	13	eqeq2d 2182	. 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 213	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 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 ⟨cop 3586 class class class wbr 3989 × cxp 4609 ‘cfv 5198 (class class class)co 5853 1^st c1st 6117 2^nd c2nd 6118 Ncnpi 7234 ·_N cmi 7236 ~_Q ceq 7241

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-omul 6400 df-ni 7266 df-mi 7268 df-enq 7309

This theorem is referenced by: (None)

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