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Theorem dmaddpqlem 6365
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6367. (Contributed by Jim Kingdon, 15-Sep-2019.)
Assertion
Ref Expression
dmaddpqlem (x Qwv x = [⟨w, v⟩] ~Q )
Distinct variable group:   w,v,x

Proof of Theorem dmaddpqlem
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 elqsi 6098 . . 3 (x ((N × N) / ~Q ) → 𝑎 (N × N)x = [𝑎] ~Q )
2 elxpi 4307 . . . . . . . 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 1492 . . . . . . . 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 1783 . . . . . 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 6081 . . . . . . . 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 2072 . . . . . 6 ((𝑎 = ⟨w, v x = [𝑎] ~Q ) → x = [⟨w, v⟩] ~Q )
13122eximi 1492 . . . . 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 2425 . . 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 6336 . 2 Q = ((N × N) / ~Q )
1816, 17eleq2s 2132 1 (x Qwv x = [⟨w, v⟩] ~Q )
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1243  wex 1381   wcel 1393  wrex 2304  cop 3373   × cxp 4289  [cec 6044   / cqs 6045  Ncnpi 6260   ~Q ceq 6267  Qcnq 6268
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-sn 3376  df-pr 3377  df-op 3379  df-br 3759  df-opab 3813  df-xp 4297  df-cnv 4299  df-dm 4301  df-rn 4302  df-res 4303  df-ima 4304  df-ec 6048  df-qs 6052  df-nqqs 6336
This theorem is referenced by:  dmaddpq  6367  dmmulpq  6368
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