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 Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6356. (Contributed by Jim Kingdon, 15-Sep-2019.)
Assertion
Ref Expression
dmaddpqlem (x Qwv x = [⟨w, v⟩] ~Q )
Distinct variable group:   w,v,x

Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 elqsi 6087 . . 3 (x ((N × N) / ~Q ) → 𝑎 (N × N)x = [𝑎] ~Q )
2 elxpi 4303 . . . . . . . 8 (𝑎 (N × N) → wv(𝑎 = ⟨w, v (w N v N)))
3 simpl 102 . . . . . . . . 9 ((𝑎 = ⟨w, v (w N v N)) → 𝑎 = ⟨w, v⟩)
432eximi 1489 . . . . . . . 8 (wv(𝑎 = ⟨w, v (w N v N)) → wv 𝑎 = ⟨w, v⟩)
52, 4syl 14 . . . . . . 7 (𝑎 (N × N) → wv 𝑎 = ⟨w, v⟩)
65anim1i 323 . . . . . 6 ((𝑎 (N × N) x = [𝑎] ~Q ) → (wv 𝑎 = ⟨w, v x = [𝑎] ~Q ))
7 19.41vv 1780 . . . . . 6 (wv(𝑎 = ⟨w, v x = [𝑎] ~Q ) ↔ (wv 𝑎 = ⟨w, v x = [𝑎] ~Q ))
86, 7sylibr 137 . . . . 5 ((𝑎 (N × N) x = [𝑎] ~Q ) → wv(𝑎 = ⟨w, v x = [𝑎] ~Q ))
9 simpr 103 . . . . . . 7 ((𝑎 = ⟨w, v x = [𝑎] ~Q ) → x = [𝑎] ~Q )
10 eceq1 6070 . . . . . . . 8 (𝑎 = ⟨w, v⟩ → [𝑎] ~Q = [⟨w, v⟩] ~Q )
1110adantr 261 . . . . . . 7 ((𝑎 = ⟨w, v x = [𝑎] ~Q ) → [𝑎] ~Q = [⟨w, v⟩] ~Q )
129, 11eqtrd 2069 . . . . . 6 ((𝑎 = ⟨w, v x = [𝑎] ~Q ) → x = [⟨w, v⟩] ~Q )
13122eximi 1489 . . . . 5 (wv(𝑎 = ⟨w, v x = [𝑎] ~Q ) → wv x = [⟨w, v⟩] ~Q )
148, 13syl 14 . . . 4 ((𝑎 (N × N) x = [𝑎] ~Q ) → wv x = [⟨w, v⟩] ~Q )
1514rexlimiva 2422 . . 3 (𝑎 (N × N)x = [𝑎] ~Qwv x = [⟨w, v⟩] ~Q )
161, 15syl 14 . 2 (x ((N × N) / ~Q ) → wv x = [⟨w, v⟩] ~Q )
17 df-nqqs 6325 . 2 Q = ((N × N) / ~Q )
1816, 17eleq2s 2129 1 (x Qwv x = [⟨w, v⟩] ~Q )
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  ⟨cop 3369   × cxp 4285  [cec 6033   / cqs 6034  Ncnpi 6249   ~Q ceq 6256  Qcnq 6257 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 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-xp 4293  df-cnv 4295  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-ec 6037  df-qs 6041  df-nqqs 6325 This theorem is referenced by:  dmaddpq  6356  dmmulpq  6357
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