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Theorem dfmpq2 7011
 Description: Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)
Assertion
Ref Expression
dfmpq2 ·pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Proof of Theorem dfmpq2
StepHypRef Expression
1 df-mpt2 5695 . 2 (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)}
2 df-mpq 7001 . 2 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
3 1st2nd2 5983 . . . . . . . . . 10 (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
43eqeq1d 2103 . . . . . . . . 9 (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5 1st2nd2 5983 . . . . . . . . . 10 (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
65eqeq1d 2103 . . . . . . . . 9 (𝑦 ∈ (N × N) → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩))
74, 6bi2anan9 574 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩)))
87anbi1d 454 . . . . . . 7 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩) ↔ ((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩)))
98bicomd 140 . . . . . 6 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩) ↔ ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩)))
1094exbidv 1805 . . . . 5 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤𝑣𝑢𝑓((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩) ↔ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩)))
11 xp1st 5974 . . . . . . 7 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
12 xp2nd 5975 . . . . . . 7 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
1311, 12jca 301 . . . . . 6 (𝑥 ∈ (N × N) → ((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N))
14 xp1st 5974 . . . . . . 7 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
15 xp2nd 5975 . . . . . . 7 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
1614, 15jca 301 . . . . . 6 (𝑦 ∈ (N × N) → ((1st𝑦) ∈ N ∧ (2nd𝑦) ∈ N))
17 simpll 497 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑤 = (1st𝑥))
18 simprl 499 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑢 = (1st𝑦))
1917, 18oveq12d 5708 . . . . . . . . 9 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑤 ·N 𝑢) = ((1st𝑥) ·N (1st𝑦)))
20 simplr 498 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑣 = (2nd𝑥))
21 simprr 500 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑓 = (2nd𝑦))
2220, 21oveq12d 5708 . . . . . . . . 9 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑣 ·N 𝑓) = ((2nd𝑥) ·N (2nd𝑦)))
2319, 22opeq12d 3652 . . . . . . . 8 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
2423eqeq2d 2106 . . . . . . 7 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ ↔ 𝑧 = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩))
2524copsex4g 4098 . . . . . 6 ((((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) ∧ ((1st𝑦) ∈ N ∧ (2nd𝑦) ∈ N)) → (∃𝑤𝑣𝑢𝑓((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩))
2613, 16, 25syl2an 284 . . . . 5 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤𝑣𝑢𝑓((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩))
2710, 26bitr3d 189 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩))
2827pm5.32i 443 . . 3 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩)) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩))
2928oprabbii 5742 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)}
301, 2, 293eqtr4i 2125 1 ·pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))}
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ⟨cop 3469   × cxp 4465  ‘cfv 5049  (class class class)co 5690  {coprab 5691   ↦ cmpt2 5692  1st c1st 5947  2nd c2nd 5948  Ncnpi 6928   ·N cmi 6930   ·pQ cmpq 6933 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-mpq 7001 This theorem is referenced by:  mulpipqqs  7029
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