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Theorem nq0nn 6297
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 6069 . . 3 (A ((𝜔 × N) / ~Q0 ) → 𝑎 (𝜔 × N)A = [𝑎] ~Q0 )
2 elxpi 4288 . . . . . . 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 1765 . . . . . 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 470 . . . . . . 7 (((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) → (w 𝜔 v N))
7 ax-ia2 100 . . . . . . . 8 (((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) → A = [𝑎] ~Q0 )
8 eceq1 6052 . . . . . . . . 9 (𝑎 = ⟨w, v⟩ → [𝑎] ~Q0 = [⟨w, v⟩] ~Q0 )
98ad2antrr 460 . . . . . . . 8 (((𝑎 = ⟨w, v (w 𝜔 v N)) A = [𝑎] ~Q0 ) → [𝑎] ~Q0 = [⟨w, v⟩] ~Q0 )
107, 9eqtrd 2054 . . . . . . 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 1474 . . . . 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 2406 . . 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 6280 . 2 Q0 = ((𝜔 × N) / ~Q0 )
1715, 16eleq2s 2114 1 (A Q0wv((w 𝜔 v N) A = [⟨w, v⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wex 1362   wcel 1374  wrex 2285  cop 3353  𝜔com 4240   × cxp 4270  [cec 6015   / cqs 6016  Ncnpi 6130   ~Q0 ceq0 6144  Q0cnq0 6145
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-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  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-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-ec 6019  df-qs 6023  df-nq0 6280
This theorem is referenced by:  nqpnq0nq  6308  nq0m0r  6311  nq0a0  6312  nq02m  6319
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