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Theorem addassnq 9992
Description: Addition of positive fractions is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addassnq ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶))

Proof of Theorem addassnq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addasspi 9929 . . . . . . . 8 ((((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶))) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))) = (((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N ((((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))))
2 ovex 6842 . . . . . . . . . . 11 ((1st𝐴) ·N (2nd𝐵)) ∈ V
3 ovex 6842 . . . . . . . . . . 11 ((1st𝐵) ·N (2nd𝐴)) ∈ V
4 fvex 6363 . . . . . . . . . . 11 (2nd𝐶) ∈ V
5 mulcompi 9930 . . . . . . . . . . 11 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
6 distrpi 9932 . . . . . . . . . . 11 (𝑥 ·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N (𝑥 ·N 𝑧))
72, 3, 4, 5, 6caovdir 7034 . . . . . . . . . 10 ((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = ((((1st𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)))
8 mulasspi 9931 . . . . . . . . . . 11 (((1st𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
98oveq1i 6824 . . . . . . . . . 10 ((((1st𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶))) = (((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)))
107, 9eqtri 2782 . . . . . . . . 9 ((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = (((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)))
1110oveq1i 6824 . . . . . . . 8 (((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))) = ((((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶))) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵))))
12 ovex 6842 . . . . . . . . . . 11 ((1st𝐵) ·N (2nd𝐶)) ∈ V
13 ovex 6842 . . . . . . . . . . 11 ((1st𝐶) ·N (2nd𝐵)) ∈ V
14 fvex 6363 . . . . . . . . . . 11 (2nd𝐴) ∈ V
1512, 13, 14, 5, 6caovdir 7034 . . . . . . . . . 10 ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴)) = ((((1st𝐵) ·N (2nd𝐶)) ·N (2nd𝐴)) +N (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴)))
16 fvex 6363 . . . . . . . . . . . 12 (1st𝐵) ∈ V
17 mulasspi 9931 . . . . . . . . . . . 12 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
1816, 4, 14, 5, 17caov32 7027 . . . . . . . . . . 11 (((1st𝐵) ·N (2nd𝐶)) ·N (2nd𝐴)) = (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶))
19 mulasspi 9931 . . . . . . . . . . . 12 (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴)) = ((1st𝐶) ·N ((2nd𝐵) ·N (2nd𝐴)))
20 mulcompi 9930 . . . . . . . . . . . . 13 ((2nd𝐵) ·N (2nd𝐴)) = ((2nd𝐴) ·N (2nd𝐵))
2120oveq2i 6825 . . . . . . . . . . . 12 ((1st𝐶) ·N ((2nd𝐵) ·N (2nd𝐴))) = ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))
2219, 21eqtri 2782 . . . . . . . . . . 11 (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴)) = ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))
2318, 22oveq12i 6826 . . . . . . . . . 10 ((((1st𝐵) ·N (2nd𝐶)) ·N (2nd𝐴)) +N (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴))) = ((((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵))))
2415, 23eqtri 2782 . . . . . . . . 9 ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴)) = ((((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵))))
2524oveq2i 6825 . . . . . . . 8 (((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴))) = (((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N ((((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))))
261, 11, 253eqtr4i 2792 . . . . . . 7 (((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))) = (((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴)))
27 mulasspi 9931 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
2826, 27opeq12i 4558 . . . . . 6 ⟨(((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩ = ⟨(((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩
29 elpqn 9959 . . . . . . . . . 10 (𝐴Q𝐴 ∈ (N × N))
30293ad2ant1 1128 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
31 elpqn 9959 . . . . . . . . . 10 (𝐵Q𝐵 ∈ (N × N))
32313ad2ant2 1129 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
33 addpipq2 9970 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
3430, 32, 33syl2anc 696 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
35 relxp 5283 . . . . . . . . 9 Rel (N × N)
36 elpqn 9959 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
37363ad2ant3 1130 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
38 1st2nd 7382 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
3935, 37, 38sylancr 698 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
4034, 39oveq12d 6832 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = (⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨(1st𝐶), (2nd𝐶)⟩))
41 xp1st 7366 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
4230, 41syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
43 xp2nd 7367 . . . . . . . . . . 11 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
4432, 43syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
45 mulclpi 9927 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
4642, 44, 45syl2anc 696 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
47 xp1st 7366 . . . . . . . . . . 11 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
4832, 47syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
49 xp2nd 7367 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
5030, 49syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
51 mulclpi 9927 . . . . . . . . . 10 (((1st𝐵) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
5248, 50, 51syl2anc 696 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
53 addclpi 9926 . . . . . . . . 9 ((((1st𝐴) ·N (2nd𝐵)) ∈ N ∧ ((1st𝐵) ·N (2nd𝐴)) ∈ N) → (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ∈ N)
5446, 52, 53syl2anc 696 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ∈ N)
55 mulclpi 9927 . . . . . . . . 9 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
5650, 44, 55syl2anc 696 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
57 xp1st 7366 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
5837, 57syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
59 xp2nd 7367 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
6037, 59syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
61 addpipq 9971 . . . . . . . 8 ((((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐵)) ∈ N) ∧ ((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N)) → (⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨(1st𝐶), (2nd𝐶)⟩) = ⟨(((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
6254, 56, 58, 60, 61syl22anc 1478 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨(1st𝐶), (2nd𝐶)⟩) = ⟨(((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
6340, 62eqtrd 2794 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = ⟨(((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
64 1st2nd 7382 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
6535, 30, 64sylancr 698 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
66 addpipq2 9970 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
6732, 37, 66syl2anc 696 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
6865, 67oveq12d 6832 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +pQ (𝐵 +pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ +pQ ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩))
69 mulclpi 9927 . . . . . . . . . 10 (((1st𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
7048, 60, 69syl2anc 696 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
71 mulclpi 9927 . . . . . . . . . 10 (((1st𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
7258, 44, 71syl2anc 696 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
73 addclpi 9926 . . . . . . . . 9 ((((1st𝐵) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
7470, 72, 73syl2anc 696 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
75 mulclpi 9927 . . . . . . . . 9 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
7644, 60, 75syl2anc 696 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
77 addpipq 9971 . . . . . . . 8 ((((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 ((2nd𝐵) ·N (2nd𝐶))) +N ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
7842, 50, 74, 76, 77syl22anc 1478 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨(1st𝐴), (2nd𝐴)⟩ +pQ ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩) = ⟨(((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
7968, 78eqtrd 2794 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +pQ (𝐵 +pQ 𝐶)) = ⟨(((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
8028, 63, 793eqtr4a 2820 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = (𝐴 +pQ (𝐵 +pQ 𝐶)))
8180fveq2d 6357 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 +pQ 𝐵) +pQ 𝐶)) = ([Q]‘(𝐴 +pQ (𝐵 +pQ 𝐶))))
82 adderpq 9990 . . . 4 (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)) = ([Q]‘((𝐴 +pQ 𝐵) +pQ 𝐶))
83 adderpq 9990 . . . 4 (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))) = ([Q]‘(𝐴 +pQ (𝐵 +pQ 𝐶)))
8481, 82, 833eqtr4g 2819 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)) = (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))))
85 addpqnq 9972 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
86853adant3 1127 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
87 nqerid 9967 . . . . . 6 (𝐶Q → ([Q]‘𝐶) = 𝐶)
8887eqcomd 2766 . . . . 5 (𝐶Q𝐶 = ([Q]‘𝐶))
89883ad2ant3 1130 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ([Q]‘𝐶))
9086, 89oveq12d 6832 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)))
91 nqerid 9967 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
9291eqcomd 2766 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
93923ad2ant1 1128 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
94 addpqnq 9972 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
95943adant1 1125 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
9693, 95oveq12d 6832 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +Q (𝐵 +Q 𝐶)) = (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))))
9784, 90, 963eqtr4d 2804 . 2 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶)))
98 addnqf 9982 . . . 4 +Q :(Q × Q)⟶Q
9998fdmi 6213 . . 3 dom +Q = (Q × Q)
100 0nnq 9958 . . 3 ¬ ∅ ∈ Q
10199, 100ndmovass 6988 . 2 (¬ (𝐴Q𝐵Q𝐶Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶)))
10297, 101pm2.61i 176 1 ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶))
Colors of variables: wff setvar class
Syntax hints:  w3a 1072   = wceq 1632  wcel 2139  cop 4327   × cxp 5264  Rel wrel 5271  cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  Ncnpi 9878   +N cpli 9879   ·N cmi 9880   +pQ cplpq 9882  Qcnq 9886  [Q]cerq 9888   +Q cplq 9889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-omul 7735  df-er 7913  df-ni 9906  df-pli 9907  df-mi 9908  df-lti 9909  df-plpq 9942  df-enq 9945  df-nq 9946  df-erq 9947  df-plq 9948  df-1nq 9950
This theorem is referenced by:  ltaddnq  10008  addasspr  10056  prlem934  10067  ltexprlem7  10076
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