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Theorem ltanq 11011
Description: Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltanq (𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))

Proof of Theorem ltanq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addnqf 10988 . . 3 +Q :(Q × Q)⟶Q
21fdmi 6747 . 2 dom +Q = (Q × Q)
3 ltrelnq 10966 . 2 <Q ⊆ (Q × Q)
4 0nnq 10964 . 2 ¬ ∅ ∈ Q
5 ordpinq 10983 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
653adant3 1133 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
7 elpqn 10965 . . . . . . 7 (𝐶Q𝐶 ∈ (N × N))
873ad2ant3 1136 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
9 elpqn 10965 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
1093ad2ant1 1134 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
11 addpipq2 10976 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐶 +pQ 𝐴) = ⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩)
128, 10, 11syl2anc 584 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 +pQ 𝐴) = ⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩)
13 elpqn 10965 . . . . . . 7 (𝐵Q𝐵 ∈ (N × N))
14133ad2ant2 1135 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
15 addpipq2 10976 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐶 +pQ 𝐵) = ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩)
168, 14, 15syl2anc 584 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 +pQ 𝐵) = ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩)
1712, 16breq12d 5156 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵) ↔ ⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩))
18 addpqnq 10978 . . . . . . . 8 ((𝐶Q𝐴Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
1918ancoms 458 . . . . . . 7 ((𝐴Q𝐶Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
20193adant2 1132 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
21 addpqnq 10978 . . . . . . . 8 ((𝐶Q𝐵Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
2221ancoms 458 . . . . . . 7 ((𝐵Q𝐶Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
23223adant1 1131 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
2420, 23breq12d 5156 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵) ↔ ([Q]‘(𝐶 +pQ 𝐴)) <Q ([Q]‘(𝐶 +pQ 𝐵))))
25 lterpq 11010 . . . . 5 ((𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵) ↔ ([Q]‘(𝐶 +pQ 𝐴)) <Q ([Q]‘(𝐶 +pQ 𝐵)))
2624, 25bitr4di 289 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵) ↔ (𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵)))
27 xp2nd 8047 . . . . . . . . . 10 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
288, 27syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
29 mulclpi 10933 . . . . . . . . 9 (((2nd𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐶) ·N (2nd𝐶)) ∈ N)
3028, 28, 29syl2anc 584 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐶) ·N (2nd𝐶)) ∈ N)
31 ltmpi 10944 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ∈ N → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
3230, 31syl 17 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
33 xp2nd 8047 . . . . . . . . . . 11 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
3414, 33syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
35 mulclpi 10933 . . . . . . . . . 10 (((2nd𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐶) ·N (2nd𝐵)) ∈ N)
3628, 34, 35syl2anc 584 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐶) ·N (2nd𝐵)) ∈ N)
37 xp1st 8046 . . . . . . . . . . 11 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
388, 37syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
39 xp2nd 8047 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
4010, 39syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
41 mulclpi 10933 . . . . . . . . . 10 (((1st𝐶) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐶) ·N (2nd𝐴)) ∈ N)
4238, 40, 41syl2anc 584 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐴)) ∈ N)
43 mulclpi 10933 . . . . . . . . 9 ((((2nd𝐶) ·N (2nd𝐵)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐴)) ∈ N) → (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) ∈ N)
4436, 42, 43syl2anc 584 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) ∈ N)
45 ltapi 10943 . . . . . . . 8 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) ∈ N → ((((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) <N ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))))))
4644, 45syl 17 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) <N ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))))))
4732, 46bitrd 279 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) <N ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))))))
48 mulcompi 10936 . . . . . . . . . 10 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶)))
49 fvex 6919 . . . . . . . . . . 11 (1st𝐴) ∈ V
50 fvex 6919 . . . . . . . . . . 11 (2nd𝐵) ∈ V
51 fvex 6919 . . . . . . . . . . 11 (2nd𝐶) ∈ V
52 mulcompi 10936 . . . . . . . . . . 11 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
53 mulasspi 10937 . . . . . . . . . . 11 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
5449, 50, 51, 52, 53, 51caov411 7665 . . . . . . . . . 10 (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶))) = (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐴) ·N (2nd𝐶)))
5548, 54eqtri 2765 . . . . . . . . 9 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐴) ·N (2nd𝐶)))
5655oveq2i 7442 . . . . . . . 8 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) = ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐴) ·N (2nd𝐶))))
57 distrpi 10938 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐵)) ·N (((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶)))) = ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐴) ·N (2nd𝐶))))
58 mulcompi 10936 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐵)) ·N (((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶)))) = ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵)))
5956, 57, 583eqtr2i 2771 . . . . . . 7 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) = ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵)))
60 mulcompi 10936 . . . . . . . . . 10 (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵)))
61 fvex 6919 . . . . . . . . . . 11 (1st𝐶) ∈ V
62 fvex 6919 . . . . . . . . . . 11 (2nd𝐴) ∈ V
6361, 62, 51, 52, 53, 50caov411 7665 . . . . . . . . . 10 (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵))) = (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
6460, 63eqtri 2765 . . . . . . . . 9 (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) = (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
65 mulcompi 10936 . . . . . . . . . 10 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) ·N ((2nd𝐶) ·N (2nd𝐶)))
66 fvex 6919 . . . . . . . . . . 11 (1st𝐵) ∈ V
6766, 62, 51, 52, 53, 51caov411 7665 . . . . . . . . . 10 (((1st𝐵) ·N (2nd𝐴)) ·N ((2nd𝐶) ·N (2nd𝐶))) = (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
6865, 67eqtri 2765 . . . . . . . . 9 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
6964, 68oveq12i 7443 . . . . . . . 8 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) = ((((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd𝐶))))
70 distrpi 10938 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐴)) ·N (((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶)))) = ((((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd𝐶))))
71 mulcompi 10936 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐴)) ·N (((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶)))) = ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴)))
7269, 70, 713eqtr2i 2771 . . . . . . 7 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) = ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴)))
7359, 72breq12i 5152 . . . . . 6 (((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) <N ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) ↔ ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵))) <N ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴))))
7447, 73bitrdi 287 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵))) <N ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴)))))
75 ordpipq 10982 . . . . 5 (⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩ ↔ ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵))) <N ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴))))
7674, 75bitr4di 289 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩))
7717, 26, 763bitr4rd 312 . . 3 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
786, 77bitrd 279 . 2 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
792, 3, 4, 78ndmovord 7623 1 (𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  cop 4632   class class class wbr 5143   × cxp 5683  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Ncnpi 10884   +N cpli 10885   ·N cmi 10886   <N clti 10887   +pQ cplpq 10888   <pQ cltpq 10890  Qcnq 10892  [Q]cerq 10894   +Q cplq 10895   <Q cltq 10898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ni 10912  df-pli 10913  df-mi 10914  df-lti 10915  df-plpq 10948  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-plq 10954  df-1nq 10956  df-ltnq 10958
This theorem is referenced by:  ltaddnq  11014  ltbtwnnq  11018  addclpr  11058  distrlem4pr  11066  ltexprlem3  11078  ltexprlem4  11079  ltexprlem6  11081  prlem936  11087
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