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Theorem dfplpq2 6857
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-mpt2 5618 . 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 6847 . 2 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
3 1st2nd2 5902 . . . . . . . . . 10 (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
43eqeq1d 2093 . . . . . . . . 9 (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5 1st2nd2 5902 . . . . . . . . . 10 (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
65eqeq1d 2093 . . . . . . . . 9 (𝑦 ∈ (N × N) → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩))
74, 6bi2anan9 571 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩)))
87anbi1d 453 . . . . . . 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 5893 . . . . . . . . . . . . . 14 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
109ad2antlr 473 . . . . . . . . . . . . 13 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (1st𝑦) ∈ N)
117biimpa 290 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩))
1211simprd 112 . . . . . . . . . . . . . 14 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩)
13 vex 2618 . . . . . . . . . . . . . . . . 17 𝑢 ∈ V
14 vex 2618 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
1513, 14opth2 4041 . . . . . . . . . . . . . . . 16 (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩ ↔ ((1st𝑦) = 𝑢 ∧ (2nd𝑦) = 𝑓))
1615simplbi 268 . . . . . . . . . . . . . . 15 (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑢, 𝑓⟩ → (1st𝑦) = 𝑢)
1716eleq1d 2153 . . . . . . . . . . . . . 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 145 . . . . . . . . . . . 12 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → 𝑢N)
20 xp2nd 5894 . . . . . . . . . . . . . 14 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
2120ad2antrr 472 . . . . . . . . . . . . 13 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (2nd𝑥) ∈ N)
2211simpld 110 . . . . . . . . . . . . . 14 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩)
23 vex 2618 . . . . . . . . . . . . . . . . 17 𝑤 ∈ V
24 vex 2618 . . . . . . . . . . . . . . . . 17 𝑣 ∈ V
2523, 24opth2 4041 . . . . . . . . . . . . . . . 16 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ ↔ ((1st𝑥) = 𝑤 ∧ (2nd𝑥) = 𝑣))
2625simprbi 269 . . . . . . . . . . . . . . 15 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑤, 𝑣⟩ → (2nd𝑥) = 𝑣)
2726eleq1d 2153 . . . . . . . . . . . . . 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 145 . . . . . . . . . . . 12 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → 𝑣N)
30 mulcompig 6834 . . . . . . . . . . . 12 ((𝑢N𝑣N) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
3119, 29, 30syl2anc 403 . . . . . . . . . . 11 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
3231oveq2d 5629 . . . . . . . . . 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 3611 . . . . . . . . 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 2096 . . . . . . . 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 440 . . . . . . 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 188 . . . . . 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 1795 . . . . 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 5893 . . . . . . 7 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
3938, 20jca 300 . . . . . 6 (𝑥 ∈ (N × N) → ((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N))
40 xp2nd 5894 . . . . . . 7 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
419, 40jca 300 . . . . . 6 (𝑦 ∈ (N × N) → ((1st𝑦) ∈ N ∧ (2nd𝑦) ∈ N))
42 simpll 496 . . . . . . . . . . 11 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑤 = (1st𝑥))
43 simprr 499 . . . . . . . . . . 11 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑓 = (2nd𝑦))
4442, 43oveq12d 5631 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑤 ·N 𝑓) = ((1st𝑥) ·N (2nd𝑦)))
45 simprl 498 . . . . . . . . . . 11 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑢 = (1st𝑦))
46 simplr 497 . . . . . . . . . . 11 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → 𝑣 = (2nd𝑥))
4745, 46oveq12d 5631 . . . . . . . . . 10 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑢 ·N 𝑣) = ((1st𝑦) ·N (2nd𝑥)))
4844, 47oveq12d 5631 . . . . . . . . 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 5631 . . . . . . . . 9 (((𝑤 = (1st𝑥) ∧ 𝑣 = (2nd𝑥)) ∧ (𝑢 = (1st𝑦) ∧ 𝑓 = (2nd𝑦))) → (𝑣 ·N 𝑓) = ((2nd𝑥) ·N (2nd𝑦)))
5048, 49opeq12d 3613 . . . . . . . 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 2096 . . . . . . 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 4048 . . . . . 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 283 . . . . 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 188 . . . 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 442 . . 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 5661 . 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 2115 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 102  wb 103   = wceq 1287  wex 1424  wcel 1436  cop 3434   × cxp 4409  cfv 4981  (class class class)co 5613  {coprab 5614  cmpt2 5615  1st c1st 5866  2nd c2nd 5867  Ncnpi 6775   +N cpli 6776   ·N cmi 6777   +pQ cplpq 6779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-oadd 6139  df-omul 6140  df-ni 6807  df-mi 6809  df-plpq 6847
This theorem is referenced by:  addpipqqs  6873
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