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Theorem dfplpq2 7378
Description: Alternate definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)
Assertion
Ref Expression
dfplpq2 +pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Proof of Theorem dfplpq2
StepHypRef Expression
1 df-mpo 5897 . 2 (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)}
2 df-plpq 7368 . 2 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
3 1st2nd2 6195 . . . . . . . . . 10 (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
43eqeq1d 2198 . . . . . . . . 9 (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5 1st2nd2 6195 . . . . . . . . . 10 (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
65eqeq1d 2198 . . . . . . . . 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 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ ((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩)))
9 xp1st 6185 . . . . . . . . . . . . . 14 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
109ad2antlr 489 . . . . . . . . . . . . 13 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (1st𝑦) ∈ N)
117biimpa 296 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩))
1211simprd 114 . . . . . . . . . . . . . 14 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩)
13 vex 2755 . . . . . . . . . . . . . . . . 17 𝑢 ∈ V
14 vex 2755 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
1513, 14opth2 4255 . . . . . . . . . . . . . . . 16 (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩ ↔ ((1st𝑦) = 𝑢 ∧ (2nd𝑦) = 𝑓))
1615simplbi 274 . . . . . . . . . . . . . . 15 (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩ → (1st𝑦) = 𝑢)
1716eleq1d 2258 . . . . . . . . . . . . . 14 (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩ → ((1st𝑦) ∈ N𝑢N))
1812, 17syl 14 . . . . . . . . . . . . 13 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ((1st𝑦) ∈ N𝑢N))
1910, 18mpbid 147 . . . . . . . . . . . 12 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → 𝑢N)
20 xp2nd 6186 . . . . . . . . . . . . . 14 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
2120ad2antrr 488 . . . . . . . . . . . . 13 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (2nd𝑥) ∈ N)
2211simpld 112 . . . . . . . . . . . . . 14 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩)
23 vex 2755 . . . . . . . . . . . . . . . . 17 𝑤 ∈ V
24 vex 2755 . . . . . . . . . . . . . . . . 17 𝑣 ∈ V
2523, 24opth2 4255 . . . . . . . . . . . . . . . 16 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ↔ ((1st𝑥) = 𝑤 ∧ (2nd𝑥) = 𝑣))
2625simprbi 275 . . . . . . . . . . . . . . 15 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ → (2nd𝑥) = 𝑣)
2726eleq1d 2258 . . . . . . . . . . . . . 14 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ → ((2nd𝑥) ∈ N𝑣N))
2822, 27syl 14 . . . . . . . . . . . . 13 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ((2nd𝑥) ∈ N𝑣N))
2921, 28mpbid 147 . . . . . . . . . . . 12 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → 𝑣N)
30 mulcompig 7355 . . . . . . . . . . . 12 ((𝑢N𝑣N) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
3119, 29, 30syl2anc 411 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
3231oveq2d 5908 . . . . . . . . . 10 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)) = ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)))
3332opeq1d 3799 . . . . . . . . 9 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩ = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)
3433eqeq2d 2201 . . . . . . . 8 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩ ↔ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))
3534pm5.32da 452 . . . . . . 7 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)))
368, 35bitr3d 190 . . . . . 6 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)))
37364exbidv 1881 . . . . 5 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤𝑣𝑢𝑓((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)))
38 xp1st 6185 . . . . . . 7 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
3938, 20jca 306 . . . . . 6 (𝑥 ∈ (N × N) → ((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N))
40 xp2nd 6186 . . . . . . 7 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
419, 40jca 306 . . . . . 6 (𝑦 ∈ (N × N) → ((1st𝑦) ∈ N ∧ (2nd𝑦) ∈ N))
42 simpll 527 . . . . . . . . . . 11 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑤 = (1st𝑥))
43 simprr 531 . . . . . . . . . . 11 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑓 = (2nd𝑦))
4442, 43oveq12d 5910 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑤 ·N 𝑓) = ((1st𝑥) ·N (2nd𝑦)))
45 simprl 529 . . . . . . . . . . 11 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑢 = (1st𝑦))
46 simplr 528 . . . . . . . . . . 11 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑣 = (2nd𝑥))
4745, 46oveq12d 5910 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑢 ·N 𝑣) = ((1st𝑦) ·N (2nd𝑥)))
4844, 47oveq12d 5910 . . . . . . . . 9 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → ((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)) = (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))))
4946, 43oveq12d 5910 . . . . . . . . 9 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑣 ·N 𝑓) = ((2nd𝑥) ·N (2nd𝑦)))
5048, 49opeq12d 3801 . . . . . . . 8 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩ = ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
5150eqeq2d 2201 . . . . . . 7 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩ ↔ 𝑧 = ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩))
5251copsex4g 4262 . . . . . 6 ((((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) ∧ ((1st𝑦) ∈ N ∧ (2nd𝑦) ∈ N)) → (∃𝑤𝑣𝑢𝑓((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩))
5339, 41, 52syl2an 289 . . . . 5 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤𝑣𝑢𝑓((⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩))
5437, 53bitr3d 190 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩))
5554pm5.32i 454 . . 3 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩))
5655oprabbii 5947 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)}
571, 2, 563eqtr4i 2220 1 +pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))}
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  cop 3610   × cxp 4639  cfv 5232  (class class class)co 5892  {coprab 5893  cmpo 5894  1st c1st 6158  2nd c2nd 6159  Ncnpi 7296   +N cpli 7297   ·N cmi 7298   +pQ cplpq 7300
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-oadd 6440  df-omul 6441  df-ni 7328  df-mi 7330  df-plpq 7368
This theorem is referenced by:  addpipqqs  7394
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