Theorem enq0breq 6418

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 5464	. . . . . 6 ⊢ ((z = A ∧ u = 𝐷) → (z ·𝑜 u) = (A ·𝑜 𝐷))
2		oveq12 5464	. . . . . 6 ⊢ ((w = B ∧ v = 𝐶) → (w ·𝑜 v) = (B ·𝑜 𝐶))
3	1, 2	eqeqan12d 2052	. . . . 5 ⊢ (((z = A ∧ u = 𝐷) ∧ (w = B ∧ v = 𝐶)) → ((z ·𝑜 u) = (w ·𝑜 v) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
4	3	an42s 523	. . . 4 ⊢ (((z = A ∧ w = B) ∧ (v = 𝐶 ∧ u = 𝐷)) → ((z ·𝑜 u) = (w ·𝑜 v) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
5	4	copsex4g 3975	. . 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 437	. 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 3955	. . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ N) → ⟨A, B⟩ ∈ V)
8		opexg 3955	. . 3 ⊢ ((𝐶 ∈ 𝜔 ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ V)
9		eleq1 2097	. . . . . 6 ⊢ (x = ⟨A, B⟩ → (x ∈ (𝜔 × N) ↔ ⟨A, B⟩ ∈ (𝜔 × N)))
10	9	anbi1d 438	. . . . 5 ⊢ (x = ⟨A, B⟩ → ((x ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N)) ↔ (⟨A, B⟩ ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N))))
11		eqeq1 2043	. . . . . . . 8 ⊢ (x = ⟨A, B⟩ → (x = ⟨z, w⟩ ↔ ⟨A, B⟩ = ⟨z, w⟩))
12	11	anbi1d 438	. . . . . . 7 ⊢ (x = ⟨A, B⟩ → ((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩)))
13	12	anbi1d 438	. . . . . 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 1747	. . . . 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 442	. . . 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 2097	. . . . . 6 ⊢ (y = ⟨𝐶, 𝐷⟩ → (y ∈ (𝜔 × N) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N)))
17	16	anbi2d 437	. . . . 5 ⊢ (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B⟩ ∈ (𝜔 × N) ∧ y ∈ (𝜔 × N)) ↔ (⟨A, B⟩ ∈ (𝜔 × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝜔 × N))))
18		eqeq1 2043	. . . . . . . 8 ⊢ (y = ⟨𝐶, 𝐷⟩ → (y = ⟨v, u⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩))
19	18	anbi2d 437	. . . . . . 7 ⊢ (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B⟩ = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩)))
20	19	anbi1d 438	. . . . . 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 1747	. . . . 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 442	. . . 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 6406	. . . 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 3997	. . 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 4319	. . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ N) → ⟨A, B⟩ ∈ (𝜔 × N))
27		opelxpi 4319	. . . 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 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   class class class wbr 3755  𝜔com 4256   × cxp 4286  (class class class)co 5455   ·_𝑜 comu 5938  Ncnpi 6256   ~_Q0 ceq0 6270

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-iota 4810  df-fv 4853  df-ov 5458  df-enq0 6406

This theorem is referenced by:  enq0eceq  6419  nqnq0pi  6420  addcmpblnq0  6425  mulcmpblnq0  6426  mulcanenq0ec  6427  nnnq0lem1  6428