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Theorem dfmpq2 7351
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-mpo 5877 . 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 7341 . 2 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
3 1st2nd2 6173 . . . . . . . . . 10 (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
43eqeq1d 2186 . . . . . . . . 9 (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5 1st2nd2 6173 . . . . . . . . . 10 (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
65eqeq1d 2186 . . . . . . . . 9 (𝑦 ∈ (N × N) → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩))
74, 6bi2anan9 606 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩)))
87anbi1d 465 . . . . . . 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 141 . . . . . 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 1870 . . . . 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 6163 . . . . . . 7 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
12 xp2nd 6164 . . . . . . 7 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
1311, 12jca 306 . . . . . 6 (𝑥 ∈ (N × N) → ((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N))
14 xp1st 6163 . . . . . . 7 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
15 xp2nd 6164 . . . . . . 7 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
1614, 15jca 306 . . . . . 6 (𝑦 ∈ (N × N) → ((1st𝑦) ∈ N ∧ (2nd𝑦) ∈ N))
17 simpll 527 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑤 = (1st𝑥))
18 simprl 529 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑢 = (1st𝑦))
1917, 18oveq12d 5890 . . . . . . . . 9 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑤 ·N 𝑢) = ((1st𝑥) ·N (1st𝑦)))
20 simplr 528 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑣 = (2nd𝑥))
21 simprr 531 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑓 = (2nd𝑦))
2220, 21oveq12d 5890 . . . . . . . . 9 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑣 ·N 𝑓) = ((2nd𝑥) ·N (2nd𝑦)))
2319, 22opeq12d 3786 . . . . . . . 8 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
2423eqeq2d 2189 . . . . . . 7 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ ↔ 𝑧 = ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩))
2524copsex4g 4246 . . . . . 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 289 . . . . 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 190 . . . 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 454 . . 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 5927 . 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 2208 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 104  wb 105   = wceq 1353  wex 1492  wcel 2148  cop 3595   × cxp 4623  cfv 5215  (class class class)co 5872  {coprab 5873  cmpo 5874  1st c1st 6136  2nd c2nd 6137  Ncnpi 7268   ·N cmi 7270   ·pQ cmpq 7273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-mpq 7341
This theorem is referenced by:  mulpipqqs  7369
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