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Theorem distrnq 10064
 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 9999 . . . . . . . . . . . . 13 ((1st𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (1st𝐴))
21oveq1i 6880 . . . . . . . . . . . 12 (((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((1st𝐵) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐶)))
3 fvex 6417 . . . . . . . . . . . . 13 (1st𝐵) ∈ V
4 fvex 6417 . . . . . . . . . . . . 13 (1st𝐴) ∈ V
5 fvex 6417 . . . . . . . . . . . . 13 (2nd𝐴) ∈ V
6 mulcompi 9999 . . . . . . . . . . . . 13 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
7 mulasspi 10000 . . . . . . . . . . . . 13 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
8 fvex 6417 . . . . . . . . . . . . 13 (2nd𝐶) ∈ V
93, 4, 5, 6, 7, 8caov411 7092 . . . . . . . . . . . 12 (((1st𝐵) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
102, 9eqtri 2828 . . . . . . . . . . 11 (((1st𝐴) ·N (1st𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
11 mulcompi 9999 . . . . . . . . . . . . 13 ((1st𝐴) ·N (1st𝐶)) = ((1st𝐶) ·N (1st𝐴))
1211oveq1i 6880 . . . . . . . . . . . 12 (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐵)))
13 fvex 6417 . . . . . . . . . . . . 13 (1st𝐶) ∈ V
14 fvex 6417 . . . . . . . . . . . . 13 (2nd𝐵) ∈ V
1513, 4, 5, 6, 7, 14caov411 7092 . . . . . . . . . . . 12 (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
1612, 15eqtri 2828 . . . . . . . . . . 11 (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐵))) = (((2nd𝐴) ·N (1st𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
1710, 16oveq12i 6882 . . . . . . . . . 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 10001 . . . . . . . . . 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 10000 . . . . . . . . . 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 2834 . . . . . . . . 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 10000 . . . . . . . . . 10 (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = ((2nd𝐴) ·N ((2nd𝐵) ·N ((2nd𝐴) ·N (2nd𝐶))))
2214, 5, 8, 6, 7caov12 7088 . . . . . . . . . . 11 ((2nd𝐵) ·N ((2nd𝐴) ·N (2nd𝐶))) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
2322oveq2i 6881 . . . . . . . . . 10 ((2nd𝐴) ·N ((2nd𝐵) ·N ((2nd𝐴) ·N (2nd𝐶)))) = ((2nd𝐴) ·N ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))))
2421, 23eqtri 2828 . . . . . . . . 9 (((2nd𝐴) ·N (2nd𝐵)) ·N ((2nd𝐴) ·N (2nd𝐶))) = ((2nd𝐴) ·N ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))))
2520, 24opeq12i 4600 . . . . . . . 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 10028 . . . . . . . . . . 11 (𝐴Q𝐴 ∈ (N × N))
27263ad2ant1 1156 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
28 xp2nd 7427 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
2927, 28syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
30 xp1st 7426 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
3127, 30syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
32 elpqn 10028 . . . . . . . . . . . . . 14 (𝐵Q𝐵 ∈ (N × N))
33323ad2ant2 1157 . . . . . . . . . . . . 13 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
34 xp1st 7426 . . . . . . . . . . . . 13 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
3533, 34syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
36 elpqn 10028 . . . . . . . . . . . . . 14 (𝐶Q𝐶 ∈ (N × N))
37363ad2ant3 1158 . . . . . . . . . . . . 13 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
38 xp2nd 7427 . . . . . . . . . . . . 13 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
3937, 38syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
40 mulclpi 9996 . . . . . . . . . . . 12 (((1st𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
4135, 39, 40syl2anc 575 . . . . . . . . . . 11 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
42 xp1st 7426 . . . . . . . . . . . . 13 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
4337, 42syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
44 xp2nd 7427 . . . . . . . . . . . . 13 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
4533, 44syl 17 . . . . . . . . . . . 12 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
46 mulclpi 9996 . . . . . . . . . . . 12 (((1st𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
4743, 45, 46syl2anc 575 . . . . . . . . . . 11 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
48 addclpi 9995 . . . . . . . . . . 11 ((((1st𝐵) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
4941, 47, 48syl2anc 575 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
50 mulclpi 9996 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N) → ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) ∈ N)
5131, 49, 50syl2anc 575 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))) ∈ N)
52 mulclpi 9996 . . . . . . . . . . 11 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
5345, 39, 52syl2anc 575 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
54 mulclpi 9996 . . . . . . . . . 10 (((2nd𝐴) ∈ N ∧ ((2nd𝐵) ·N (2nd𝐶)) ∈ N) → ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) ∈ N)
5529, 53, 54syl2anc 575 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) ∈ N)
56 mulcanenq 10063 . . . . . . . . 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 1483 . . . . . . . 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 4879 . . . . . . 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 10042 . . . . . . . . . 10 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
6027, 33, 59syl2anc 575 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
61 mulpipq2 10042 . . . . . . . . . 10 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
6227, 37, 61syl2anc 575 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
6360, 62oveq12d 6888 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) = (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩))
64 mulclpi 9996 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
6531, 35, 64syl2anc 575 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
66 mulclpi 9996 . . . . . . . . . 10 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
6729, 45, 66syl2anc 575 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
68 mulclpi 9996 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
6931, 43, 68syl2anc 575 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
70 mulclpi 9996 . . . . . . . . . 10 (((2nd𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
7129, 39, 70syl2anc 575 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
72 addpipq 10040 . . . . . . . . 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 858 . . . . . . . 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 2840 . . . . . . 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 5328 . . . . . . . . . 10 Rel (N × N)
76 1st2nd 7442 . . . . . . . . . 10 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
7775, 27, 76sylancr 577 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
78 addpipq2 10039 . . . . . . . . . 10 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
7933, 37, 78syl2anc 575 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
8077, 79oveq12d 6888 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩))
81 mulpipq 10043 . . . . . . . . 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 858 . . . . . . . 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 2840 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) = ⟨((1st𝐴) ·N (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵)))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
8458, 74, 833brtr4d 4876 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ~Q (𝐴 ·pQ (𝐵 +pQ 𝐶)))
85 mulpqf 10049 . . . . . . . . . 10 ·pQ :((N × N) × (N × N))⟶(N × N)
8685fovcl 6991 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) ∈ (N × N))
8727, 33, 86syl2anc 575 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) ∈ (N × N))
8885fovcl 6991 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) ∈ (N × N))
8927, 37, 88syl2anc 575 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐶) ∈ (N × N))
90 addpqf 10047 . . . . . . . . 9 +pQ :((N × N) × (N × N))⟶(N × N)
9190fovcl 6991 . . . . . . . 8 (((𝐴 ·pQ 𝐵) ∈ (N × N) ∧ (𝐴 ·pQ 𝐶) ∈ (N × N)) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ∈ (N × N))
9287, 89, 91syl2anc 575 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)) ∈ (N × N))
9390fovcl 6991 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) ∈ (N × N))
9433, 37, 93syl2anc 575 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +pQ 𝐶) ∈ (N × N))
9585fovcl 6991 . . . . . . . 8 ((𝐴 ∈ (N × N) ∧ (𝐵 +pQ 𝐶) ∈ (N × N)) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) ∈ (N × N))
9627, 94, 95syl2anc 575 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 +pQ 𝐶)) ∈ (N × N))
97 nqereq 10038 . . . . . . 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 575 . . . . . 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 223 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶))))
10099eqcomd 2812 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶))) = ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶))))
101 mulerpq 10060 . . . 4 (([Q]‘𝐴) ·Q ([Q]‘(𝐵 +pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 +pQ 𝐶)))
102 adderpq 10059 . . . 4 (([Q]‘(𝐴 ·pQ 𝐵)) +Q ([Q]‘(𝐴 ·pQ 𝐶))) = ([Q]‘((𝐴 ·pQ 𝐵) +pQ (𝐴 ·pQ 𝐶)))
103100, 101, 1023eqtr4g 2865 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘𝐴) ·Q ([Q]‘(𝐵 +pQ 𝐶))) = (([Q]‘(𝐴 ·pQ 𝐵)) +Q ([Q]‘(𝐴 ·pQ 𝐶))))
104 nqerid 10036 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
105104eqcomd 2812 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
1061053ad2ant1 1156 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
107 addpqnq 10041 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
1081073adant1 1153 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
109106, 108oveq12d 6888 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 +pQ 𝐶))))
110 mulpqnq 10044 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
1111103adant3 1155 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
112 mulpqnq 10044 . . . . 5 ((𝐴Q𝐶Q) → (𝐴 ·Q 𝐶) = ([Q]‘(𝐴 ·pQ 𝐶)))
1131123adant2 1154 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐶) = ([Q]‘(𝐴 ·pQ 𝐶)))
114111, 113oveq12d 6888 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)) = (([Q]‘(𝐴 ·pQ 𝐵)) +Q ([Q]‘(𝐴 ·pQ 𝐶))))
115103, 109, 1143eqtr4d 2850 . 2 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)))
116 addnqf 10051 . . . 4 +Q :(Q × Q)⟶Q
117116fdmi 6262 . . 3 dom +Q = (Q × Q)
118 0nnq 10027 . . 3 ¬ ∅ ∈ Q
119 mulnqf 10052 . . . 4 ·Q :(Q × Q)⟶Q
120119fdmi 6262 . . 3 dom ·Q = (Q × Q)
121117, 118, 120ndmovdistr 7049 . 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 197   ∧ w3a 1100   = wceq 1637   ∈ wcel 2156  ⟨cop 4376   class class class wbr 4844   × cxp 5309  Rel wrel 5316  ‘cfv 6097  (class class class)co 6870  1st c1st 7392  2nd c2nd 7393  Ncnpi 9947   +N cpli 9948   ·N cmi 9949   +pQ cplpq 9951   ·pQ cmpq 9952   ~Q ceq 9954  Qcnq 9955  [Q]cerq 9957   +Q cplq 9958   ·Q cmq 9959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-oadd 7796  df-omul 7797  df-er 7975  df-ni 9975  df-pli 9976  df-mi 9977  df-lti 9978  df-plpq 10011  df-mpq 10012  df-enq 10014  df-nq 10015  df-erq 10016  df-plq 10017  df-mq 10018  df-1nq 10019 This theorem is referenced by:  ltaddnq  10077  halfnq  10079  addclprlem2  10120  distrlem1pr  10128  distrlem4pr  10129  prlem934  10136  prlem936  10150
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