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Theorem ltexnq 9835
Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltexnq (𝐵Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexnq
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 9786 . . . 4 <Q ⊆ (Q × Q)
21brel 5202 . . 3 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
3 ordpinq 9803 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
4 elpqn 9785 . . . . . . . . 9 (𝐴Q𝐴 ∈ (N × N))
54adantr 480 . . . . . . . 8 ((𝐴Q𝐵Q) → 𝐴 ∈ (N × N))
6 xp1st 7242 . . . . . . . 8 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
75, 6syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (1st𝐴) ∈ N)
8 elpqn 9785 . . . . . . . . 9 (𝐵Q𝐵 ∈ (N × N))
98adantl 481 . . . . . . . 8 ((𝐴Q𝐵Q) → 𝐵 ∈ (N × N))
10 xp2nd 7243 . . . . . . . 8 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
119, 10syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (2nd𝐵) ∈ N)
12 mulclpi 9753 . . . . . . 7 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
137, 11, 12syl2anc 694 . . . . . 6 ((𝐴Q𝐵Q) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
14 xp1st 7242 . . . . . . . 8 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
159, 14syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (1st𝐵) ∈ N)
16 xp2nd 7243 . . . . . . . 8 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
175, 16syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (2nd𝐴) ∈ N)
18 mulclpi 9753 . . . . . . 7 (((1st𝐵) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
1915, 17, 18syl2anc 694 . . . . . 6 ((𝐴Q𝐵Q) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
20 ltexpi 9762 . . . . . 6 ((((1st𝐴) ·N (2nd𝐵)) ∈ N ∧ ((1st𝐵) ·N (2nd𝐴)) ∈ N) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ∃𝑦N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴))))
2113, 19, 20syl2anc 694 . . . . 5 ((𝐴Q𝐵Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ∃𝑦N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴))))
22 relxp 5160 . . . . . . . . . . . 12 Rel (N × N)
234ad2antrr 762 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐴 ∈ (N × N))
24 1st2nd 7258 . . . . . . . . . . . 12 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2522, 23, 24sylancr 696 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2625oveq1d 6705 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = (⟨(1st𝐴), (2nd𝐴)⟩ +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩))
277adantr 480 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → (1st𝐴) ∈ N)
2817adantr 480 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → (2nd𝐴) ∈ N)
29 simpr 476 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝑦N)
30 mulclpi 9753 . . . . . . . . . . . . 13 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
3117, 11, 30syl2anc 694 . . . . . . . . . . . 12 ((𝐴Q𝐵Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
3231adantr 480 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
33 addpipq 9797 . . . . . . . . . . 11 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (𝑦N ∧ ((2nd𝐴) ·N (2nd𝐵)) ∈ N)) → (⟨(1st𝐴), (2nd𝐴)⟩ +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩)
3427, 28, 29, 32, 33syl22anc 1367 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → (⟨(1st𝐴), (2nd𝐴)⟩ +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩)
3526, 34eqtrd 2685 . . . . . . . . 9 (((𝐴Q𝐵Q) ∧ 𝑦N) → (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩)
36 oveq2 6698 . . . . . . . . . . . 12 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ((2nd𝐴) ·N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦)) = ((2nd𝐴) ·N ((1st𝐵) ·N (2nd𝐴))))
37 distrpi 9758 . . . . . . . . . . . . 13 ((2nd𝐴) ·N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦)) = (((2nd𝐴) ·N ((1st𝐴) ·N (2nd𝐵))) +N ((2nd𝐴) ·N 𝑦))
38 fvex 6239 . . . . . . . . . . . . . . 15 (2nd𝐴) ∈ V
39 fvex 6239 . . . . . . . . . . . . . . 15 (1st𝐴) ∈ V
40 fvex 6239 . . . . . . . . . . . . . . 15 (2nd𝐵) ∈ V
41 mulcompi 9756 . . . . . . . . . . . . . . 15 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
42 mulasspi 9757 . . . . . . . . . . . . . . 15 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
4338, 39, 40, 41, 42caov12 6904 . . . . . . . . . . . . . 14 ((2nd𝐴) ·N ((1st𝐴) ·N (2nd𝐵))) = ((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))
44 mulcompi 9756 . . . . . . . . . . . . . 14 ((2nd𝐴) ·N 𝑦) = (𝑦 ·N (2nd𝐴))
4543, 44oveq12i 6702 . . . . . . . . . . . . 13 (((2nd𝐴) ·N ((1st𝐴) ·N (2nd𝐵))) +N ((2nd𝐴) ·N 𝑦)) = (((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴)))
4637, 45eqtr2i 2674 . . . . . . . . . . . 12 (((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))) = ((2nd𝐴) ·N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦))
47 mulasspi 9757 . . . . . . . . . . . . 13 (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((2nd𝐴) ·N (1st𝐵)))
48 mulcompi 9756 . . . . . . . . . . . . . 14 ((2nd𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (2nd𝐴))
4948oveq2i 6701 . . . . . . . . . . . . 13 ((2nd𝐴) ·N ((2nd𝐴) ·N (1st𝐵))) = ((2nd𝐴) ·N ((1st𝐵) ·N (2nd𝐴)))
5047, 49eqtri 2673 . . . . . . . . . . . 12 (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((1st𝐵) ·N (2nd𝐴)))
5136, 46, 503eqtr4g 2710 . . . . . . . . . . 11 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → (((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)))
52 mulasspi 9757 . . . . . . . . . . . . 13 (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) = ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))
5352eqcomi 2660 . . . . . . . . . . . 12 ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))
5453a1i 11 . . . . . . . . . . 11 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)))
5551, 54opeq12d 4441 . . . . . . . . . 10 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩ = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩)
5655eqeq2d 2661 . . . . . . . . 9 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ((𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩ ↔ (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩))
5735, 56syl5ibcom 235 . . . . . . . 8 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩))
58 fveq2 6229 . . . . . . . . 9 ((𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ → ([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩))
59 adderpq 9816 . . . . . . . . . . 11 (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = ([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩))
60 nqerid 9793 . . . . . . . . . . . . 13 (𝐴Q → ([Q]‘𝐴) = 𝐴)
6160ad2antrr 762 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘𝐴) = 𝐴)
6261oveq1d 6705 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)))
6359, 62syl5eqr 2699 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)))
64 mulclpi 9753 . . . . . . . . . . . . . . . 16 (((2nd𝐴) ∈ N ∧ (2nd𝐴) ∈ N) → ((2nd𝐴) ·N (2nd𝐴)) ∈ N)
6517, 17, 64syl2anc 694 . . . . . . . . . . . . . . 15 ((𝐴Q𝐵Q) → ((2nd𝐴) ·N (2nd𝐴)) ∈ N)
6665adantr 480 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((2nd𝐴) ·N (2nd𝐴)) ∈ N)
6715adantr 480 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (1st𝐵) ∈ N)
6811adantr 480 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (2nd𝐵) ∈ N)
69 mulcanenq 9820 . . . . . . . . . . . . . 14 ((((2nd𝐴) ·N (2nd𝐴)) ∈ N ∧ (1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩)
7066, 67, 68, 69syl3anc 1366 . . . . . . . . . . . . 13 (((𝐴Q𝐵Q) ∧ 𝑦N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩)
718ad2antlr 763 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐵 ∈ (N × N))
72 1st2nd 7258 . . . . . . . . . . . . . 14 ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
7322, 71, 72sylancr 696 . . . . . . . . . . . . 13 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
7470, 73breqtrrd 4713 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q 𝐵)
75 mulclpi 9753 . . . . . . . . . . . . . . 15 ((((2nd𝐴) ·N (2nd𝐴)) ∈ N ∧ (1st𝐵) ∈ N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) ∈ N)
7666, 67, 75syl2anc 694 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) ∈ N)
77 mulclpi 9753 . . . . . . . . . . . . . . 15 ((((2nd𝐴) ·N (2nd𝐴)) ∈ N ∧ (2nd𝐵) ∈ N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) ∈ N)
7866, 68, 77syl2anc 694 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) ∈ N)
79 opelxpi 5182 . . . . . . . . . . . . . 14 (((((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) ∈ N ∧ (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) ∈ N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ∈ (N × N))
8076, 78, 79syl2anc 694 . . . . . . . . . . . . 13 (((𝐴Q𝐵Q) ∧ 𝑦N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ∈ (N × N))
81 nqereq 9795 . . . . . . . . . . . . 13 ((⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) = ([Q]‘𝐵)))
8280, 71, 81syl2anc 694 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → (⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) = ([Q]‘𝐵)))
8374, 82mpbid 222 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) = ([Q]‘𝐵))
84 nqerid 9793 . . . . . . . . . . . 12 (𝐵Q → ([Q]‘𝐵) = 𝐵)
8584ad2antlr 763 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘𝐵) = 𝐵)
8683, 85eqtrd 2685 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) = 𝐵)
8763, 86eqeq12d 2666 . . . . . . . . 9 (((𝐴Q𝐵Q) ∧ 𝑦N) → (([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵))
8858, 87syl5ib 234 . . . . . . . 8 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵))
8957, 88syld 47 . . . . . . 7 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵))
90 fvex 6239 . . . . . . . 8 ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) ∈ V
91 oveq2 6698 . . . . . . . . 9 (𝑥 = ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) → (𝐴 +Q 𝑥) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)))
9291eqeq1d 2653 . . . . . . . 8 (𝑥 = ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) → ((𝐴 +Q 𝑥) = 𝐵 ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵))
9390, 92spcev 3331 . . . . . . 7 ((𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
9489, 93syl6 35 . . . . . 6 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
9594rexlimdva 3060 . . . . 5 ((𝐴Q𝐵Q) → (∃𝑦N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
9621, 95sylbid 230 . . . 4 ((𝐴Q𝐵Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
973, 96sylbid 230 . . 3 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
982, 97mpcom 38 . 2 (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
99 eleq1 2718 . . . . . . 7 ((𝐴 +Q 𝑥) = 𝐵 → ((𝐴 +Q 𝑥) ∈ Q𝐵Q))
10099biimparc 503 . . . . . 6 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) ∈ Q)
101 addnqf 9808 . . . . . . . 8 +Q :(Q × Q)⟶Q
102101fdmi 6090 . . . . . . 7 dom +Q = (Q × Q)
103 0nnq 9784 . . . . . . 7 ¬ ∅ ∈ Q
104102, 103ndmovrcl 6862 . . . . . 6 ((𝐴 +Q 𝑥) ∈ Q → (𝐴Q𝑥Q))
105 ltaddnq 9834 . . . . . 6 ((𝐴Q𝑥Q) → 𝐴 <Q (𝐴 +Q 𝑥))
106100, 104, 1053syl 18 . . . . 5 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q (𝐴 +Q 𝑥))
107 simpr 476 . . . . 5 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) = 𝐵)
108106, 107breqtrd 4711 . . . 4 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q 𝐵)
109108ex 449 . . 3 (𝐵Q → ((𝐴 +Q 𝑥) = 𝐵𝐴 <Q 𝐵))
110109exlimdv 1901 . 2 (𝐵Q → (∃𝑥(𝐴 +Q 𝑥) = 𝐵𝐴 <Q 𝐵))
11198, 110impbid2 216 1 (𝐵Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  cop 4216   class class class wbr 4685   × cxp 5141  Rel wrel 5148  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Ncnpi 9704   +N cpli 9705   ·N cmi 9706   <N clti 9707   +pQ cplpq 9708   ~Q ceq 9711  Qcnq 9712  [Q]cerq 9714   +Q cplq 9715   <Q cltq 9718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-ltnq 9778
This theorem is referenced by:  ltbtwnnq  9838  prnmadd  9857  ltexprlem4  9899  ltexprlem7  9902  prlem936  9907
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