Theorem enq0breq 6291

Description: Equivalence relation for non-negative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.)

Assertion

Ref	Expression
enq0breq	⊢ (((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) → (⟨A, B⟩ ~Q0 ⟨𝐶, 𝐷⟩ ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))

Proof of Theorem enq0breq

Dummy variables x y z w v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq12 5445	. . . . . 6 ⊢ ((z = A ∧ u = 𝐷) → (z ·𝑜 u) = (A ·𝑜 𝐷))
2		oveq12 5445	. . . . . 6 ⊢ ((w = B ∧ v = 𝐶) → (w ·𝑜 v) = (B ·𝑜 𝐶))
3	1, 2	eqeqan12d 2037	. . . . 5 ⊢ (((z = A ∧ u = 𝐷) ∧ (w = B ∧ v = 𝐶)) → ((z ·𝑜 u) = (w ·𝑜 v) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
4	3	an42s 510	. . . 4 ⊢ (((z = A ∧ w = B) ∧ (v = 𝐶 ∧ u = 𝐷)) → ((z ·𝑜 u) = (w ·𝑜 v) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
5	4	copsex4g 3958	. . 3 ⊢ (((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) → (∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
6	5	anbi2d 440	. 2 ⊢ (((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) → (((⟨A, B⟩ ∈ (𝜔 × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v))) ↔ ((⟨A, B⟩ ∈ (𝜔 × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N)) ∧ (A ·𝑜 𝐷) = (B ·𝑜 𝐶))))
7		opexg 3938	. . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ N) → ⟨A, B⟩ ∈ V)
8		opexg 3938	. . 3 ⊢ ((𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ V)
9		eleq1 2082	. . . . . 6 ⊢ (x = ⟨A, B⟩ → (x ∈ (𝜔 × N) ↔ ⟨A, B⟩ ∈ (𝜔 × N)))
10	9	anbi1d 441	. . . . 5 ⊢ (x = ⟨A, B⟩ → ((x ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N)) ↔ (⟨A, B⟩ ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N))))
11		eqeq1 2028	. . . . . . . 8 ⊢ (x = ⟨A, B⟩ → (x = ⟨z, w⟩ ↔ ⟨A, B⟩ = ⟨z, w⟩))
12	11	anbi1d 441	. . . . . . 7 ⊢ (x = ⟨A, B⟩ → ((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩)))
13	12	anbi1d 441	. . . . . 6 ⊢ (x = ⟨A, B⟩ → (((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)) ↔ ((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v))))
14	13	4exbidv 1732	. . . . 5 ⊢ (x = ⟨A, B⟩ → (∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)) ↔ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v))))
15	10, 14	anbi12d 445	. . . 4 ⊢ (x = ⟨A, B⟩ → (((x ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v))) ↔ ((⟨A, B⟩ ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)))))
16		eleq1 2082	. . . . . 6 ⊢ (y = ⟨𝐶, 𝐷⟩ → (y ∈ (𝜔 × N) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N)))
17	16	anbi2d 440	. . . . 5 ⊢ (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B⟩ ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N)) ↔ (⟨A, B⟩ ∈ (𝜔 × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N))))
18		eqeq1 2028	. . . . . . . 8 ⊢ (y = ⟨𝐶, 𝐷⟩ → (y = ⟨v, u⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩))
19	18	anbi2d 440	. . . . . . 7 ⊢ (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩)))
20	19	anbi1d 441	. . . . . 6 ⊢ (y = ⟨𝐶, 𝐷⟩ → (((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)) ↔ ((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v))))
21	20	4exbidv 1732	. . . . 5 ⊢ (y = ⟨𝐶, 𝐷⟩ → (∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)) ↔ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v))))
22	17, 21	anbi12d 445	. . . 4 ⊢ (y = ⟨𝐶, 𝐷⟩ → (((⟨A, B⟩ ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v))) ↔ ((⟨A, B⟩ ∈ (𝜔 × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)))))
23		df-enq0 6279	. . . 4 ⊢ ~Q0 = {⟨x, y⟩ ∣ ((x ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)))}
24	15, 22, 23	brabg 3980	. . 3 ⊢ ((⟨A, B⟩ ∈ V ∧ ⟨𝐶, 𝐷⟩ ∈ V) → (⟨A, B⟩ ~Q0 ⟨𝐶, 𝐷⟩ ↔ ((⟨A, B⟩ ∈ (𝜔 × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)))))
25	7, 8, 24	syl2an 273	. 2 ⊢ (((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) → (⟨A, B⟩ ~Q0 ⟨𝐶, 𝐷⟩ ↔ ((⟨A, B⟩ ∈ (𝜔 × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N)) ∧ ∃z∃w∃v∃u((⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩) ∧ (z ·𝑜 u) = (w ·𝑜 v)))))
26		opelxpi 4303	. . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ N) → ⟨A, B⟩ ∈ (𝜔 × N))
27		opelxpi 4303	. . . 4 ⊢ ((𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N))
28	26, 27	anim12i 321	. . 3 ⊢ (((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) → (⟨A, B⟩ ∈ (𝜔 × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N)))
29	28	biantrurd 289	. 2 ⊢ (((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) → ((A ·𝑜 𝐷) = (B ·𝑜 𝐶) ↔ ((⟨A, B⟩ ∈ (𝜔 × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N)) ∧ (A ·𝑜 𝐷) = (B ·𝑜 𝐶))))
30	6, 25, 29	3bitr4d 209	1 ⊢ (((A ∈ 𝜔 ∧ B ∈ N) ∧ (𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N)) → (⟨A, B⟩ ~Q0 ⟨𝐶, 𝐷⟩ ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535  ⟨cop 3353   class class class wbr 3738  𝜔com 4240   × cxp 4270  (class class class)co 5436   ·_𝑜 comu 5914  Ncnpi 6130   ~_Q0 ceq0 6144

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-xp 4278  df-iota 4794  df-fv 4837  df-ov 5439  df-enq0 6279

This theorem is referenced by:  enq0eceq  6292  nqnq0pi  6293  addcmpblnq0  6298  mulcmpblnq0  6299  mulcanenq0ec  6300  nnnq0lem1  6301