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Theorem nq0nn 6424
Description: Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
Assertion
Ref Expression
nq0nn (A Q0wv((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.
StepHypRef Expression
1 elqsi 6094 . . 3 (A ((𝜔 × N) / ~Q0 ) → 𝑎 (𝜔 × N)A = [𝑎] ~Q0 )
2 elxpi 4304 . . . . . . 7 (𝑎 (𝜔 × N) → wv(𝑎 = ⟨w, v (w 𝜔 v N)))
32anim1i 323 . . . . . 6 ((𝑎 (𝜔 × N) A = [𝑎] ~Q0 ) → (wv(𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ))
4 19.41vv 1780 . . . . . 6 (wv((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) ↔ (wv(𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ))
53, 4sylibr 137 . . . . 5 ((𝑎 (𝜔 × N) A = [𝑎] ~Q0 ) → wv((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ))
6 simplr 482 . . . . . . 7 (((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) → (w 𝜔 v N))
7 simpr 103 . . . . . . . 8 (((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) → A = [𝑎] ~Q0 )
8 eceq1 6077 . . . . . . . . 9 (𝑎 = ⟨w, v⟩ → [𝑎] ~Q0 = [⟨w, v⟩] ~Q0 )
98ad2antrr 457 . . . . . . . 8 (((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) → [𝑎] ~Q0 = [⟨w, v⟩] ~Q0 )
107, 9eqtrd 2069 . . . . . . 7 (((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) → A = [⟨w, v⟩] ~Q0 )
116, 10jca 290 . . . . . 6 (((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) → ((w 𝜔 v N) A = [⟨w, v⟩] ~Q0 ))
12112eximi 1489 . . . . 5 (wv((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) → wv((w 𝜔 v N) A = [⟨w, v⟩] ~Q0 ))
135, 12syl 14 . . . 4 ((𝑎 (𝜔 × N) A = [𝑎] ~Q0 ) → wv((w 𝜔 v N) A = [⟨w, v⟩] ~Q0 ))
1413rexlimiva 2422 . . 3 (𝑎 (𝜔 × N)A = [𝑎] ~Q0wv((w 𝜔 v N) A = [⟨w, v⟩] ~Q0 ))
151, 14syl 14 . 2 (A ((𝜔 × N) / ~Q0 ) → wv((w 𝜔 v N) A = [⟨w, v⟩] ~Q0 ))
16 df-nq0 6407 . 2 Q0 = ((𝜔 × N) / ~Q0 )
1715, 16eleq2s 2129 1 (A Q0wv((w 𝜔 v N) A = [⟨w, v⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  wrex 2301  cop 3370  𝜔com 4256   × cxp 4286  [cec 6040   / cqs 6041  Ncnpi 6256   ~Q0 ceq0 6270  Q0cnq0 6271
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  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-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-ec 6044  df-qs 6048  df-nq0 6407
This theorem is referenced by:  nqpnq0nq  6435  nq0m0r  6438  nq0a0  6439  nq02m  6447
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