Theorem nq0nn 7404

Description: Decomposition of a nonnegative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)

Assertion

Ref	Expression
nq0nn	⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))

Distinct variable group:   𝑣,𝐴,𝑤

Proof of Theorem nq0nn

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsi 6565	. . 3 ⊢ (𝐴 ∈ ((ω × N) / ~Q0 ) → ∃𝑎 ∈ (ω × N)𝐴 = [𝑎] ~Q0 )
2		elxpi 4627	. . . . . . 7 ⊢ (𝑎 ∈ (ω × N) → ∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)))
3	2	anim1i 338	. . . . . 6 ⊢ ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ))
4		19.41vv 1896	. . . . . 6 ⊢ (∃𝑤∃𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ) ↔ (∃𝑤∃𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ))
5	3, 4	sylibr 133	. . . . 5 ⊢ ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤∃𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ))
6		simplr 525	. . . . . . 7 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ) → (𝑤 ∈ ω ∧ 𝑣 ∈ N))
7		simpr 109	. . . . . . . 8 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ) → 𝐴 = [𝑎] ~Q0 )
8		eceq1 6548	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
9	8	ad2antrr 485	. . . . . . . 8 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ) → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
10	7, 9	eqtrd 2203	. . . . . . 7 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 )
11	6, 10	jca 304	. . . . . 6 ⊢ (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ) → ((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
12	11	2eximi 1594	. . . . 5 ⊢ (∃𝑤∃𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣 ∈ N)) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
13	5, 12	syl 14	. . . 4 ⊢ ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
14	13	rexlimiva 2582	. . 3 ⊢ (∃𝑎 ∈ (ω × N)𝐴 = [𝑎] ~Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
15	1, 14	syl 14	. 2 ⊢ (𝐴 ∈ ((ω × N) / ~Q0 ) → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
16		df-nq0 7387	. 2 ⊢ Q0 = ((ω × N) / ~Q0 )
17	15, 16	eleq2s 2265	1 ⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∃wrex 2449  ⟨cop 3586  ωcom 4574   × cxp 4609  [cec 6511   / cqs 6512  Ncnpi 7234   ~_Q0 ceq0 7248  Q₀cnq0 7249

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-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-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515  df-qs 6519  df-nq0 7387

This theorem is referenced by:  nqpnq0nq  7415  nq0m0r  7418  nq0a0  7419  nq02m  7427