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Theorem ltexnq 9757
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 9708 . . . 4 <Q ⊆ (Q × Q)
21brel 5138 . . 3 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
3 ordpinq 9725 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
4 elpqn 9707 . . . . . . . . 9 (𝐴Q𝐴 ∈ (N × N))
54adantr 481 . . . . . . . 8 ((𝐴Q𝐵Q) → 𝐴 ∈ (N × N))
6 xp1st 7158 . . . . . . . 8 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
75, 6syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (1st𝐴) ∈ N)
8 elpqn 9707 . . . . . . . . 9 (𝐵Q𝐵 ∈ (N × N))
98adantl 482 . . . . . . . 8 ((𝐴Q𝐵Q) → 𝐵 ∈ (N × N))
10 xp2nd 7159 . . . . . . . 8 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
119, 10syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (2nd𝐵) ∈ N)
12 mulclpi 9675 . . . . . . 7 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
137, 11, 12syl2anc 692 . . . . . 6 ((𝐴Q𝐵Q) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
14 xp1st 7158 . . . . . . . 8 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
159, 14syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (1st𝐵) ∈ N)
16 xp2nd 7159 . . . . . . . 8 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
175, 16syl 17 . . . . . . 7 ((𝐴Q𝐵Q) → (2nd𝐴) ∈ N)
18 mulclpi 9675 . . . . . . 7 (((1st𝐵) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
1915, 17, 18syl2anc 692 . . . . . 6 ((𝐴Q𝐵Q) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
20 ltexpi 9684 . . . . . 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 692 . . . . 5 ((𝐴Q𝐵Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ∃𝑦N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴))))
22 relxp 5198 . . . . . . . . . . . 12 Rel (N × N)
234ad2antrr 761 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐴 ∈ (N × N))
24 1st2nd 7174 . . . . . . . . . . . 12 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2522, 23, 24sylancr 694 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2625oveq1d 6630 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = (⟨(1st𝐴), (2nd𝐴)⟩ +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩))
277adantr 481 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → (1st𝐴) ∈ N)
2817adantr 481 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → (2nd𝐴) ∈ N)
29 simpr 477 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝑦N)
30 mulclpi 9675 . . . . . . . . . . . . 13 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
3117, 11, 30syl2anc 692 . . . . . . . . . . . 12 ((𝐴Q𝐵Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
3231adantr 481 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
33 addpipq 9719 . . . . . . . . . . 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 1324 . . . . . . . . . 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 2655 . . . . . . . . 9 (((𝐴Q𝐵Q) ∧ 𝑦N) → (𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))⟩)
36 oveq2 6623 . . . . . . . . . . . 12 ((((1st𝐴) ·N (2nd𝐵)) +N 𝑦) = ((1st𝐵) ·N (2nd𝐴)) → ((2nd𝐴) ·N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦)) = ((2nd𝐴) ·N ((1st𝐵) ·N (2nd𝐴))))
37 distrpi 9680 . . . . . . . . . . . . 13 ((2nd𝐴) ·N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦)) = (((2nd𝐴) ·N ((1st𝐴) ·N (2nd𝐵))) +N ((2nd𝐴) ·N 𝑦))
38 fvex 6168 . . . . . . . . . . . . . . 15 (2nd𝐴) ∈ V
39 fvex 6168 . . . . . . . . . . . . . . 15 (1st𝐴) ∈ V
40 fvex 6168 . . . . . . . . . . . . . . 15 (2nd𝐵) ∈ V
41 mulcompi 9678 . . . . . . . . . . . . . . 15 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
42 mulasspi 9679 . . . . . . . . . . . . . . 15 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
4338, 39, 40, 41, 42caov12 6827 . . . . . . . . . . . . . 14 ((2nd𝐴) ·N ((1st𝐴) ·N (2nd𝐵))) = ((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))
44 mulcompi 9678 . . . . . . . . . . . . . 14 ((2nd𝐴) ·N 𝑦) = (𝑦 ·N (2nd𝐴))
4543, 44oveq12i 6627 . . . . . . . . . . . . 13 (((2nd𝐴) ·N ((1st𝐴) ·N (2nd𝐵))) +N ((2nd𝐴) ·N 𝑦)) = (((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴)))
4637, 45eqtr2i 2644 . . . . . . . . . . . 12 (((1st𝐴) ·N ((2nd𝐴) ·N (2nd𝐵))) +N (𝑦 ·N (2nd𝐴))) = ((2nd𝐴) ·N (((1st𝐴) ·N (2nd𝐵)) +N 𝑦))
47 mulasspi 9679 . . . . . . . . . . . . 13 (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((2nd𝐴) ·N (1st𝐵)))
48 mulcompi 9678 . . . . . . . . . . . . . 14 ((2nd𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (2nd𝐴))
4948oveq2i 6626 . . . . . . . . . . . . 13 ((2nd𝐴) ·N ((2nd𝐴) ·N (1st𝐵))) = ((2nd𝐴) ·N ((1st𝐵) ·N (2nd𝐴)))
5047, 49eqtri 2643 . . . . . . . . . . . 12 (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) = ((2nd𝐴) ·N ((1st𝐵) ·N (2nd𝐴)))
5136, 46, 503eqtr4g 2680 . . . . . . . . . . 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 9679 . . . . . . . . . . . . 13 (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) = ((2nd𝐴) ·N ((2nd𝐴) ·N (2nd𝐵)))
5352eqcomi 2630 . . . . . . . . . . . 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 4385 . . . . . . . . . 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 2631 . . . . . . . . 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 6158 . . . . . . . . 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 9738 . . . . . . . . . . 11 (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = ([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩))
60 nqerid 9715 . . . . . . . . . . . . 13 (𝐴Q → ([Q]‘𝐴) = 𝐴)
6160ad2antrr 761 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘𝐴) = 𝐴)
6261oveq1d 6630 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)))
6359, 62syl5eqr 2669 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘(𝐴 +pQ𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)))
64 mulclpi 9675 . . . . . . . . . . . . . . . 16 (((2nd𝐴) ∈ N ∧ (2nd𝐴) ∈ N) → ((2nd𝐴) ·N (2nd𝐴)) ∈ N)
6517, 17, 64syl2anc 692 . . . . . . . . . . . . . . 15 ((𝐴Q𝐵Q) → ((2nd𝐴) ·N (2nd𝐴)) ∈ N)
6665adantr 481 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → ((2nd𝐴) ·N (2nd𝐴)) ∈ N)
6715adantr 481 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (1st𝐵) ∈ N)
6811adantr 481 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (2nd𝐵) ∈ N)
69 mulcanenq 9742 . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . 13 (((𝐴Q𝐵Q) ∧ 𝑦N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩)
718ad2antlr 762 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐵 ∈ (N × N))
72 1st2nd 7174 . . . . . . . . . . . . . 14 ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
7322, 71, 72sylancr 694 . . . . . . . . . . . . 13 (((𝐴Q𝐵Q) ∧ 𝑦N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
7470, 73breqtrrd 4651 . . . . . . . . . . . 12 (((𝐴Q𝐵Q) ∧ 𝑦N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ~Q 𝐵)
75 mulclpi 9675 . . . . . . . . . . . . . . 15 ((((2nd𝐴) ·N (2nd𝐴)) ∈ N ∧ (1st𝐵) ∈ N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) ∈ N)
7666, 67, 75syl2anc 692 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)) ∈ N)
77 mulclpi 9675 . . . . . . . . . . . . . . 15 ((((2nd𝐴) ·N (2nd𝐴)) ∈ N ∧ (2nd𝐵) ∈ N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) ∈ N)
7866, 68, 77syl2anc 692 . . . . . . . . . . . . . 14 (((𝐴Q𝐵Q) ∧ 𝑦N) → (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵)) ∈ N)
79 opelxpi 5118 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . 13 (((𝐴Q𝐵Q) ∧ 𝑦N) → ⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩ ∈ (N × N))
81 nqereq 9717 . . . . . . . . . . . . 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 692 . . . . . . . . . . . 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 9715 . . . . . . . . . . . 12 (𝐵Q → ([Q]‘𝐵) = 𝐵)
8584ad2antlr 762 . . . . . . . . . . 11 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘𝐵) = 𝐵)
8683, 85eqtrd 2655 . . . . . . . . . 10 (((𝐴Q𝐵Q) ∧ 𝑦N) → ([Q]‘⟨(((2nd𝐴) ·N (2nd𝐴)) ·N (1st𝐵)), (((2nd𝐴) ·N (2nd𝐴)) ·N (2nd𝐵))⟩) = 𝐵)
8763, 86eqeq12d 2636 . . . . . . . . 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 6168 . . . . . . . 8 ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) ∈ V
91 oveq2 6623 . . . . . . . . 9 (𝑥 = ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) → (𝐴 +Q 𝑥) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)))
9291eqeq1d 2623 . . . . . . . 8 (𝑥 = ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩) → ((𝐴 +Q 𝑥) = 𝐵 ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd𝐴) ·N (2nd𝐵))⟩)) = 𝐵))
9390, 92spcev 3290 . . . . . . 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 3026 . . . . 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 2686 . . . . . . 7 ((𝐴 +Q 𝑥) = 𝐵 → ((𝐴 +Q 𝑥) ∈ Q𝐵Q))
10099biimparc 504 . . . . . 6 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) ∈ Q)
101 addnqf 9730 . . . . . . . 8 +Q :(Q × Q)⟶Q
102101fdmi 6019 . . . . . . 7 dom +Q = (Q × Q)
103 0nnq 9706 . . . . . . 7 ¬ ∅ ∈ Q
104102, 103ndmovrcl 6785 . . . . . 6 ((𝐴 +Q 𝑥) ∈ Q → (𝐴Q𝑥Q))
105 ltaddnq 9756 . . . . . 6 ((𝐴Q𝑥Q) → 𝐴 <Q (𝐴 +Q 𝑥))
106100, 104, 1053syl 18 . . . . 5 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q (𝐴 +Q 𝑥))
107 simpr 477 . . . . 5 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) = 𝐵)
108106, 107breqtrd 4649 . . . 4 ((𝐵Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q 𝐵)
109108ex 450 . . 3 (𝐵Q → ((𝐴 +Q 𝑥) = 𝐵𝐴 <Q 𝐵))
110109exlimdv 1858 . 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 384   = wceq 1480  wex 1701  wcel 1987  wrex 2909  cop 4161   class class class wbr 4623   × cxp 5082  Rel wrel 5089  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  Ncnpi 9626   +N cpli 9627   ·N cmi 9628   <N clti 9629   +pQ cplpq 9630   ~Q ceq 9633  Qcnq 9634  [Q]cerq 9636   +Q cplq 9637   <Q cltq 9640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-ni 9654  df-pli 9655  df-mi 9656  df-lti 9657  df-plpq 9690  df-mpq 9691  df-ltpq 9692  df-enq 9693  df-nq 9694  df-erq 9695  df-plq 9696  df-mq 9697  df-1nq 9698  df-ltnq 9700
This theorem is referenced by:  ltbtwnnq  9760  prnmadd  9779  ltexprlem4  9821  ltexprlem7  9824  prlem936  9829
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