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Theorem distrnq 9821
Description: Multiplication of positive fractions is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
distrnq (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶))

Proof of Theorem distrnq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulcompi 9756 . . . . . . . . . . . . 13 ((1st𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (1st𝐴))
21oveq1i 6700 . . . . . . . . . . . 12 (((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((1st𝐵) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐶)))
3 fvex 6239 . . . . . . . . . . . . 13 (1st𝐵) ∈ V
4 fvex 6239 . . . . . . . . . . . . 13 (1st𝐴) ∈ V
5 fvex 6239 . . . . . . . . . . . . 13 (2nd𝐴) ∈ V
6 mulcompi 9756 . . . . . . . . . . . . 13 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
7 mulasspi 9757 . . . . . . . . . . . . 13 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
8 fvex 6239 . . . . . . . . . . . . 13 (2nd𝐶) ∈ V
93, 4, 5, 6, 7, 8caov411 6908 . . . . . . . . . . . 12 (((1st𝐵) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
102, 9eqtri 2673 . . . . . . . . . . 11 (((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
11 mulcompi 9756 . . . . . . . . . . . . 13 ((1st𝐴) ·N (1st𝐶)) = ((1st𝐶) ·N (1st𝐴))
1211oveq1i 6700 . . . . . . . . . . . 12 (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐵)))
13 fvex 6239 . . . . . . . . . . . . 13 (1st𝐶) ∈ V
14 fvex 6239 . . . . . . . . . . . . 13 (2nd𝐵) ∈ V
1513, 4, 5, 6, 7, 14caov411 6908 . . . . . . . . . . . 12 (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
1612, 15eqtri 2673 . . . . . . . . . . 11 (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
1710, 16oveq12i 6702 . . . . . . . . . 10 ((((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) +N (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵)))) = ((((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐵) ·N (2nd𝐶))) +N (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐶) ·N (2nd𝐵))))
18 distrpi 9758 . . . . . . . . . 10 (((2nd𝐴) ·N (1st𝐴)) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) = ((((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐵) ·N (2nd𝐶))) +N (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐶) ·N (2nd𝐵))))
19 mulasspi 9757 . . . . . . . . . 10 (((2nd𝐴) ·N (1st𝐴)) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) = ((2nd𝐴) ·N ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))))
2017, 18, 193eqtr2i 2679 . . . . . . . . 9 ((((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) +N (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵)))) = ((2nd𝐴) ·N ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))))
21 mulasspi 9757 . . . . . . . . . 10 (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = ((2nd𝐴) ·N ((2nd𝐵) ·N ((2nd𝐴) ·N (2nd𝐶))))
2214, 5, 8, 6, 7caov12 6904 . . . . . . . . . . 11 ((2nd𝐵) ·N ((2nd𝐴) ·N (2nd𝐶))) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
2322oveq2i 6701 . . . . . . . . . 10 ((2nd𝐴) ·N ((2nd𝐵) ·N ((2nd𝐴) ·N (2nd𝐶)))) = ((2nd𝐴) ·N ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))))
2421, 23eqtri 2673 . . . . . . . . 9 (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = ((2nd𝐴) ·N ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))))
2520, 24opeq12i 4438 . . . . . . . 8 ⟨((((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) +N (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵)))), (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶)))⟩ = ⟨((2nd𝐴) ·N ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))))), ((2nd𝐴) ·N ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))))⟩
26 elpqn 9785 . . . . . . . . . . 11 (𝐴Q𝐴 ∈ (N × N))
27263ad2ant1 1102 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
28 xp2nd 7243 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
2927, 28syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
30 xp1st 7242 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
3127, 30syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
32 elpqn 9785 . . . . . . . . . . . . . 14 (𝐵Q𝐵 ∈ (N × N))
33323ad2ant2 1103 . . . . . . . . . . . . 13 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
34 xp1st 7242 . . . . . . . . . . . . 13 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
3533, 34syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
36 elpqn 9785 . . . . . . . . . . . . . 14 (𝐶Q𝐶 ∈ (N × N))
37363ad2ant3 1104 . . . . . . . . . . . . 13 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
38 xp2nd 7243 . . . . . . . . . . . . 13 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
3937, 38syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
40 mulclpi 9753 . . . . . . . . . . . 12 (((1st𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
4135, 39, 40syl2anc 694 . . . . . . . . . . 11 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
42 xp1st 7242 . . . . . . . . . . . . 13 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
4337, 42syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
44 xp2nd 7243 . . . . . . . . . . . . 13 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
4533, 44syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
46 mulclpi 9753 . . . . . . . . . . . 12 (((1st𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
4743, 45, 46syl2anc 694 . . . . . . . . . . 11 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
48 addclpi 9752 . . . . . . . . . . 11 ((((1st𝐵) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
4941, 47, 48syl2anc 694 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
50 mulclpi 9753 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N) → ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) ∈ N)
5131, 49, 50syl2anc 694 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) ∈ N)
52 mulclpi 9753 . . . . . . . . . . 11 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
5345, 39, 52syl2anc 694 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
54 mulclpi 9753 . . . . . . . . . 10 (((2nd𝐴) ∈ N ∧ ((2nd𝐵) ·N (2nd𝐶)) ∈ N) → ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) ∈ N)
5529, 53, 54syl2anc 694 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) ∈ N)
56 mulcanenq 9820 . . . . . . . . 9 (((2nd𝐴) ∈ N ∧ ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) ∈ N ∧ ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) ∈ N) → ⟨((2nd𝐴) ·N ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))))), ((2nd𝐴) ·N ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))))⟩ ~Q ⟨((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
5729, 51, 55, 56syl3anc 1366 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ⟨((2nd𝐴) ·N ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))))), ((2nd𝐴) ·N ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))))⟩ ~Q ⟨((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
5825, 57syl5eqbr 4720 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ⟨((((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) +N (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵)))), (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶)))⟩ ~Q ⟨((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
59 mulpipq2 9799 . . . . . . . . . 10 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
6027, 33, 59syl2anc 694 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
61 mulpipq2 9799 . . . . . . . . . 10 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
6227, 37, 61syl2anc 694 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
6360, 62oveq12d 6708 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) = (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩))
64 mulclpi 9753 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
6531, 35, 64syl2anc 694 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
66 mulclpi 9753 . . . . . . . . . 10 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
6729, 45, 66syl2anc 694 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
68 mulclpi 9753 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
6931, 43, 68syl2anc 694 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
70 mulclpi 9753 . . . . . . . . . 10 (((2nd𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
7129, 39, 70syl2anc 694 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
72 addpipq 9797 . . . . . . . . 9 (((((1st𝐴) ·N (1st𝐵)) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐵)) ∈ N) ∧ (((1st𝐴) ·N (1st𝐶)) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐶)) ∈ N)) → (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩) = ⟨((((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) +N (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵)))), (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶)))⟩)
7365, 67, 69, 71, 72syl22anc 1367 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩) = ⟨((((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) +N (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵)))), (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶)))⟩)
7463, 73eqtrd 2685 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) = ⟨((((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) +N (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵)))), (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶)))⟩)
75 relxp 5160 . . . . . . . . . 10 Rel (N × N)
76 1st2nd 7258 . . . . . . . . . 10 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
7775, 27, 76sylancr 696 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
78 addpipq2 9796 . . . . . . . . . 10 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
7933, 37, 78syl2anc 694 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
8077, 79oveq12d 6708 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩))
81 mulpipq 9800 . . . . . . . . 9 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N ∧ ((2nd𝐵) ·N (2nd𝐶)) ∈ N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩) = ⟨((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
8231, 29, 49, 53, 81syl22anc 1367 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩) = ⟨((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
8380, 82eqtrd 2685 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) = ⟨((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
8458, 74, 833brtr4d 4717 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ~Q (𝐴 ·pQ (𝐵 +pQ 𝐶)))
85 mulpqf 9806 . . . . . . . . . 10 ·pQ :((N × N) × (N × N))⟶(N × N)
8685fovcl 6807 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N))
8727, 33, 86syl2anc 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) ∈ (N × N))
8885fovcl 6807 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) ∈ (N × N))
8927, 37, 88syl2anc 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐶) ∈ (N × N))
90 addpqf 9804 . . . . . . . . 9 +pQ :((N × N) × (N × N))⟶(N × N)
9190fovcl 6807 . . . . . . . 8 (((𝐴 ·pQ 𝐵) ∈ (N × N) ∧ (𝐴 ·pQ 𝐶) ∈ (N × N)) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ∈ (N × N))
9287, 89, 91syl2anc 694 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ∈ (N × N))
9390fovcl 6807 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) ∈ (N × N))
9433, 37, 93syl2anc 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +pQ 𝐶) ∈ (N × N))
9585fovcl 6807 . . . . . . . 8 ((𝐴 ∈ (N × N) ∧ (𝐵 +pQ 𝐶) ∈ (N × N)) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) ∈ (N × N))
9627, 94, 95syl2anc 694 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) ∈ (N × N))
97 nqereq 9795 . . . . . . 7 ((((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ∈ (N × N) ∧ (𝐴 ·pQ (𝐵 +pQ 𝐶)) ∈ (N × N)) → (((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ~Q (𝐴 ·pQ (𝐵 +pQ 𝐶)) ↔ ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶)))))
9892, 96, 97syl2anc 694 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ~Q (𝐴 ·pQ (𝐵 +pQ 𝐶)) ↔ ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶)))))
9984, 98mpbid 222 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶))))
10099eqcomd 2657 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶))) = ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶))))
101 mulerpq 9817 . . . 4 (([Q]‘𝐴) ·Q ([Q]‘(𝐵 +pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶)))
102 adderpq 9816 . . . 4 (([Q]‘(𝐴 ·pQ 𝐵)) +Q ([Q]‘(𝐴 ·pQ 𝐶))) = ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)))
103100, 101, 1023eqtr4g 2710 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘𝐴) ·Q ([Q]‘(𝐵 +pQ 𝐶))) = (([Q]‘(𝐴 ·pQ 𝐵)) +Q ([Q]‘(𝐴 ·pQ 𝐶))))
104 nqerid 9793 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
105104eqcomd 2657 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
1061053ad2ant1 1102 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
107 addpqnq 9798 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
1081073adant1 1099 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
109106, 108oveq12d 6708 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 +pQ 𝐶))))
110 mulpqnq 9801 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
1111103adant3 1101 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
112 mulpqnq 9801 . . . . 5 ((𝐴Q𝐶Q) → (𝐴 ·Q 𝐶) = ([Q]‘(𝐴 ·pQ 𝐶)))
1131123adant2 1100 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐶) = ([Q]‘(𝐴 ·pQ 𝐶)))
114111, 113oveq12d 6708 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)) = (([Q]‘(𝐴 ·pQ 𝐵)) +Q ([Q]‘(𝐴 ·pQ 𝐶))))
115103, 109, 1143eqtr4d 2695 . 2 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)))
116 addnqf 9808 . . . 4 +Q :(Q × Q)⟶Q
117116fdmi 6090 . . 3 dom +Q = (Q × Q)
118 0nnq 9784 . . 3 ¬ ∅ ∈ Q
119 mulnqf 9809 . . . 4 ·Q :(Q × Q)⟶Q
120119fdmi 6090 . . 3 dom ·Q = (Q × Q)
121117, 118, 120ndmovdistr 6865 . 2 (¬ (𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)))
122115, 121pm2.61i 176 1 (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1054   = wceq 1523  wcel 2030  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   +pQ cplpq 9708   ·pQ cmpq 9709   ~Q ceq 9711  Qcnq 9712  [Q]cerq 9714   +Q cplq 9715   ·Q cmq 9716
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-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-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776
This theorem is referenced by:  ltaddnq  9834  halfnq  9836  addclprlem2  9877  distrlem1pr  9885  distrlem4pr  9886  prlem934  9893  prlem936  9907
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