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Theorem dfmpq2 6510
Description: Alternative 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 5544 . 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 6500 . 2 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
3 1st2nd2 5828 . . . . . . . . . 10 (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
43eqeq1d 2064 . . . . . . . . 9 (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5 1st2nd2 5828 . . . . . . . . . 10 (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
65eqeq1d 2064 . . . . . . . . 9 (𝑦 ∈ (N × N) → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩))
74, 6bi2anan9 548 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩)))
87anbi1d 446 . . . . . . 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 133 . . . . . 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 1766 . . . . 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 5819 . . . . . . 7 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
12 xp2nd 5820 . . . . . . 7 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
1311, 12jca 294 . . . . . 6 (𝑥 ∈ (N × N) → ((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N))
14 xp1st 5819 . . . . . . 7 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
15 xp2nd 5820 . . . . . . 7 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
1614, 15jca 294 . . . . . 6 (𝑦 ∈ (N × N) → ((1st𝑦) ∈ N ∧ (2nd𝑦) ∈ N))
17 simpll 489 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑤 = (1st𝑥))
18 simprl 491 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑢 = (1st𝑦))
1917, 18oveq12d 5557 . . . . . . . . 9 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑤 ·N 𝑢) = ((1st𝑥) ·N (1st𝑦)))
20 simplr 490 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑣 = (2nd𝑥))
21 simprr 492 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑓 = (2nd𝑦))
2220, 21oveq12d 5557 . . . . . . . . 9 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑣 ·N 𝑓) = ((2nd𝑥) ·N (2nd𝑦)))
2319, 22opeq12d 3584 . . . . . . . 8 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
2423eqeq2d 2067 . . . . . . 7 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ ↔ 𝑧 = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩))
2524copsex4g 4011 . . . . . 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 277 . . . . 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 183 . . . 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 435 . . 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 5587 . 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 2086 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 101  wb 102   = wceq 1259  wex 1397  wcel 1409  cop 3405   × cxp 4370  cfv 4929  (class class class)co 5539  {coprab 5540  cmpt2 5541  1st c1st 5792  2nd c2nd 5793  Ncnpi 6427   ·N cmi 6429   ·pQ cmpq 6432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-mpq 6500
This theorem is referenced by:  mulpipqqs  6528
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