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Theorem distrnq 10376
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 10311 . . . . . . . . . . . . 13 ((1st𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (1st𝐴))
21oveq1i 7149 . . . . . . . . . . . 12 (((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((1st𝐵) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐶)))
3 fvex 6662 . . . . . . . . . . . . 13 (1st𝐵) ∈ V
4 fvex 6662 . . . . . . . . . . . . 13 (1st𝐴) ∈ V
5 fvex 6662 . . . . . . . . . . . . 13 (2nd𝐴) ∈ V
6 mulcompi 10311 . . . . . . . . . . . . 13 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
7 mulasspi 10312 . . . . . . . . . . . . 13 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
8 fvex 6662 . . . . . . . . . . . . 13 (2nd𝐶) ∈ V
93, 4, 5, 6, 7, 8caov411 7364 . . . . . . . . . . . 12 (((1st𝐵) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
102, 9eqtri 2824 . . . . . . . . . . 11 (((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
11 mulcompi 10311 . . . . . . . . . . . . 13 ((1st𝐴) ·N (1st𝐶)) = ((1st𝐶) ·N (1st𝐴))
1211oveq1i 7149 . . . . . . . . . . . 12 (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐵)))
13 fvex 6662 . . . . . . . . . . . . 13 (1st𝐶) ∈ V
14 fvex 6662 . . . . . . . . . . . . 13 (2nd𝐵) ∈ V
1513, 4, 5, 6, 7, 14caov411 7364 . . . . . . . . . . . 12 (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
1612, 15eqtri 2824 . . . . . . . . . . 11 (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
1710, 16oveq12i 7151 . . . . . . . . . 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 10313 . . . . . . . . . 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 10312 . . . . . . . . . 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 2830 . . . . . . . . 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 10312 . . . . . . . . . 10 (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = ((2nd𝐴) ·N ((2nd𝐵) ·N ((2nd𝐴) ·N (2nd𝐶))))
2214, 5, 8, 6, 7caov12 7360 . . . . . . . . . . 11 ((2nd𝐵) ·N ((2nd𝐴) ·N (2nd𝐶))) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
2322oveq2i 7150 . . . . . . . . . 10 ((2nd𝐴) ·N ((2nd𝐵) ·N ((2nd𝐴) ·N (2nd𝐶)))) = ((2nd𝐴) ·N ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))))
2421, 23eqtri 2824 . . . . . . . . 9 (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = ((2nd𝐴) ·N ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))))
2520, 24opeq12i 4773 . . . . . . . 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 10340 . . . . . . . . . . 11 (𝐴Q𝐴 ∈ (N × N))
27263ad2ant1 1130 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
28 xp2nd 7708 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
2927, 28syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
30 xp1st 7707 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
3127, 30syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
32 elpqn 10340 . . . . . . . . . . . . . 14 (𝐵Q𝐵 ∈ (N × N))
33323ad2ant2 1131 . . . . . . . . . . . . 13 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
34 xp1st 7707 . . . . . . . . . . . . 13 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
3533, 34syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
36 elpqn 10340 . . . . . . . . . . . . . 14 (𝐶Q𝐶 ∈ (N × N))
37363ad2ant3 1132 . . . . . . . . . . . . 13 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
38 xp2nd 7708 . . . . . . . . . . . . 13 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
3937, 38syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
40 mulclpi 10308 . . . . . . . . . . . 12 (((1st𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
4135, 39, 40syl2anc 587 . . . . . . . . . . 11 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
42 xp1st 7707 . . . . . . . . . . . . 13 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
4337, 42syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
44 xp2nd 7708 . . . . . . . . . . . . 13 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
4533, 44syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
46 mulclpi 10308 . . . . . . . . . . . 12 (((1st𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
4743, 45, 46syl2anc 587 . . . . . . . . . . 11 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
48 addclpi 10307 . . . . . . . . . . 11 ((((1st𝐵) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
4941, 47, 48syl2anc 587 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
50 mulclpi 10308 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N) → ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) ∈ N)
5131, 49, 50syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) ∈ N)
52 mulclpi 10308 . . . . . . . . . . 11 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
5345, 39, 52syl2anc 587 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
54 mulclpi 10308 . . . . . . . . . 10 (((2nd𝐴) ∈ N ∧ ((2nd𝐵) ·N (2nd𝐶)) ∈ N) → ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) ∈ N)
5529, 53, 54syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) ∈ N)
56 mulcanenq 10375 . . . . . . . . 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 1368 . . . . . . . 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, 57eqbrtrid 5068 . . . . . . 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 10354 . . . . . . . . . 10 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
6027, 33, 59syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
61 mulpipq2 10354 . . . . . . . . . 10 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
6227, 37, 61syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
6360, 62oveq12d 7157 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) = (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩))
64 mulclpi 10308 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
6531, 35, 64syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
66 mulclpi 10308 . . . . . . . . . 10 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
6729, 45, 66syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
68 mulclpi 10308 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
6931, 43, 68syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
70 mulclpi 10308 . . . . . . . . . 10 (((2nd𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
7129, 39, 70syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
72 addpipq 10352 . . . . . . . . 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 837 . . . . . . . 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 2836 . . . . . . 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 5541 . . . . . . . . . 10 Rel (N × N)
76 1st2nd 7724 . . . . . . . . . 10 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
7775, 27, 76sylancr 590 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
78 addpipq2 10351 . . . . . . . . . 10 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
7933, 37, 78syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
8077, 79oveq12d 7157 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩))
81 mulpipq 10355 . . . . . . . . 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 837 . . . . . . . 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 2836 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) = ⟨((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
8458, 74, 833brtr4d 5065 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ~Q (𝐴 ·pQ (𝐵 +pQ 𝐶)))
85 mulpqf 10361 . . . . . . . . . 10 ·pQ :((N × N) × (N × N))⟶(N × N)
8685fovcl 7262 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N))
8727, 33, 86syl2anc 587 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) ∈ (N × N))
8885fovcl 7262 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) ∈ (N × N))
8927, 37, 88syl2anc 587 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐶) ∈ (N × N))
90 addpqf 10359 . . . . . . . . 9 +pQ :((N × N) × (N × N))⟶(N × N)
9190fovcl 7262 . . . . . . . 8 (((𝐴 ·pQ 𝐵) ∈ (N × N) ∧ (𝐴 ·pQ 𝐶) ∈ (N × N)) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ∈ (N × N))
9287, 89, 91syl2anc 587 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ∈ (N × N))
9390fovcl 7262 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) ∈ (N × N))
9433, 37, 93syl2anc 587 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +pQ 𝐶) ∈ (N × N))
9585fovcl 7262 . . . . . . . 8 ((𝐴 ∈ (N × N) ∧ (𝐵 +pQ 𝐶) ∈ (N × N)) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) ∈ (N × N))
9627, 94, 95syl2anc 587 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) ∈ (N × N))
97 nqereq 10350 . . . . . . 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 587 . . . . . 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 235 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶))))
10099eqcomd 2807 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶))) = ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶))))
101 mulerpq 10372 . . . 4 (([Q]‘𝐴) ·Q ([Q]‘(𝐵 +pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶)))
102 adderpq 10371 . . . 4 (([Q]‘(𝐴 ·pQ 𝐵)) +Q ([Q]‘(𝐴 ·pQ 𝐶))) = ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)))
103100, 101, 1023eqtr4g 2861 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘𝐴) ·Q ([Q]‘(𝐵 +pQ 𝐶))) = (([Q]‘(𝐴 ·pQ 𝐵)) +Q ([Q]‘(𝐴 ·pQ 𝐶))))
104 nqerid 10348 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
105104eqcomd 2807 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
1061053ad2ant1 1130 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
107 addpqnq 10353 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
1081073adant1 1127 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
109106, 108oveq12d 7157 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 +pQ 𝐶))))
110 mulpqnq 10356 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
1111103adant3 1129 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
112 mulpqnq 10356 . . . . 5 ((𝐴Q𝐶Q) → (𝐴 ·Q 𝐶) = ([Q]‘(𝐴 ·pQ 𝐶)))
1131123adant2 1128 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐶) = ([Q]‘(𝐴 ·pQ 𝐶)))
114111, 113oveq12d 7157 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)) = (([Q]‘(𝐴 ·pQ 𝐵)) +Q ([Q]‘(𝐴 ·pQ 𝐶))))
115103, 109, 1143eqtr4d 2846 . 2 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)))
116 addnqf 10363 . . . 4 +Q :(Q × Q)⟶Q
117116fdmi 6502 . . 3 dom +Q = (Q × Q)
118 0nnq 10339 . . 3 ¬ ∅ ∈ Q
119 mulnqf 10364 . . . 4 ·Q :(Q × Q)⟶Q
120119fdmi 6502 . . 3 dom ·Q = (Q × Q)
121117, 118, 120ndmovdistr 7321 . 2 (¬ (𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)))
122115, 121pm2.61i 185 1 (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 209  w3a 1084   = wceq 1538  wcel 2112  cop 4534   class class class wbr 5033   × cxp 5521  Rel wrel 5528  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  Ncnpi 10259   +N cpli 10260   ·N cmi 10261   +pQ cplpq 10263   ·pQ cmpq 10264   ~Q ceq 10266  Qcnq 10267  [Q]cerq 10269   +Q cplq 10270   ·Q cmq 10271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-er 8276  df-ni 10287  df-pli 10288  df-mi 10289  df-lti 10290  df-plpq 10323  df-mpq 10324  df-enq 10326  df-nq 10327  df-erq 10328  df-plq 10329  df-mq 10330  df-1nq 10331
This theorem is referenced by:  ltaddnq  10389  halfnq  10391  addclprlem2  10432  distrlem1pr  10440  distrlem4pr  10441  prlem934  10448  prlem936  10462
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