Theorem enq0breq 7584

Description: Equivalence relation for nonnegative 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 5976	. . . . . 6 ⊢ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐷) → (𝑧 ·o 𝑢) = (𝐴 ·o 𝐷))
2		oveq12 5976	. . . . . 6 ⊢ ((𝑤 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑤 ·o 𝑣) = (𝐵 ·o 𝐶))
3	1, 2	eqeqan12d 2223	. . . . 5 ⊢ (((𝑧 = 𝐴 ∧ 𝑢 = 𝐷) ∧ (𝑤 = 𝐵 ∧ 𝑣 = 𝐶)) → ((𝑧 ·o 𝑢) = (𝑤 ·o 𝑣) ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶)))
4	3	an42s 589	. . . 4 ⊢ (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑢 = 𝐷)) → ((𝑧 ·o 𝑢) = (𝑤 ·o 𝑣) ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶)))
5	4	copsex4g 4309	. . 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 464	. 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 4290	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ V)
8		opexg 4290	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ V)
9		eleq1 2270	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (ω × N) ↔ ⟨𝐴, 𝐵⟩ ∈ (ω × N)))
10	9	anbi1d 465	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
11		eqeq1 2214	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩))
12	11	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
13	12	anbi1d 465	. . . . . 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 1894	. . . . 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 473	. . . 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 2270	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (ω × N) ↔ ⟨𝐶, 𝐷⟩ ∈ (ω × N)))
17	16	anbi2d 464	. . . . 5 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (ω × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (ω × N))))
18		eqeq1 2214	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩))
19	18	anbi2d 464	. . . . . . 7 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩)))
20	19	anbi1d 465	. . . . . 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 1894	. . . . 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 473	. . . 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 7572	. . . 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 4333	. . 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 289	. 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 4725	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (ω × N))
27		opelxpi 4725	. . . 4 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (ω × N))
28	26, 27	anim12i 338	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ ∈ (ω × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (ω × N)))
29	28	biantrurd 305	. 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 220	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 104   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2178  Vcvv 2776  ⟨cop 3646   class class class wbr 4059  ωcom 4656   × cxp 4691  (class class class)co 5967   ·_o comu 6523  Ncnpi 7420   ~_Q0 ceq0 7434

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-iota 5251  df-fv 5298  df-ov 5970  df-enq0 7572

This theorem is referenced by:  enq0eceq  7585  nqnq0pi  7586  addcmpblnq0  7591  mulcmpblnq0  7592  mulcanenq0ec  7593  nnnq0lem1  7594