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 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))

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addasspi 10355 . . . . . . . 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 7183 . . . . . . . . . . 11 ((1st𝐴) ·N (2nd𝐵)) ∈ V
3 ovex 7183 . . . . . . . . . . 11 ((1st𝐵) ·N (2nd𝐴)) ∈ V
4 fvex 6671 . . . . . . . . . . 11 (2nd𝐶) ∈ V
5 mulcompi 10356 . . . . . . . . . . 11 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
6 distrpi 10358 . . . . . . . . . . 11 (𝑥 ·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N (𝑥 ·N 𝑧))
72, 3, 4, 5, 6caovdir 7378 . . . . . . . . . 10 ((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = ((((1st𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)))
8 mulasspi 10357 . . . . . . . . . . 11 (((1st𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
98oveq1i 7160 . . . . . . . . . 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 2781 . . . . . . . . 9 ((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = (((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)))
1110oveq1i 7160 . . . . . . . 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 7183 . . . . . . . . . . 11 ((1st𝐵) ·N (2nd𝐶)) ∈ V
13 ovex 7183 . . . . . . . . . . 11 ((1st𝐶) ·N (2nd𝐵)) ∈ V
14 fvex 6671 . . . . . . . . . . 11 (2nd𝐴) ∈ V
1512, 13, 14, 5, 6caovdir 7378 . . . . . . . . . 10 ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴)) = ((((1st𝐵) ·N (2nd𝐶)) ·N (2nd𝐴)) +N (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴)))
16 fvex 6671 . . . . . . . . . . . 12 (1st𝐵) ∈ V
17 mulasspi 10357 . . . . . . . . . . . 12 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
1816, 4, 14, 5, 17caov32 7371 . . . . . . . . . . 11 (((1st𝐵) ·N (2nd𝐶)) ·N (2nd𝐴)) = (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶))
19 mulasspi 10357 . . . . . . . . . . . 12 (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴)) = ((1st𝐶) ·N ((2nd𝐵) ·N (2nd𝐴)))
20 mulcompi 10356 . . . . . . . . . . . . 13 ((2nd𝐵) ·N (2nd𝐴)) = ((2nd𝐴) ·N (2nd𝐵))
2120oveq2i 7161 . . . . . . . . . . . 12 ((1st𝐶) ·N ((2nd𝐵) ·N (2nd𝐴))) = ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))
2219, 21eqtri 2781 . . . . . . . . . . 11 (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴)) = ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))
2318, 22oveq12i 7162 . . . . . . . . . 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 2781 . . . . . . . . 9 ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴)) = ((((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵))))
2524oveq2i 7161 . . . . . . . 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 2791 . . . . . . 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 10357 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
2826, 27opeq12i 4768 . . . . . 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 10385 . . . . . . . . . 10 (𝐴Q𝐴 ∈ (N × N))
30293ad2ant1 1130 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
31 elpqn 10385 . . . . . . . . . 10 (𝐵Q𝐵 ∈ (N × N))
32313ad2ant2 1131 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
33 addpipq2 10396 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
3430, 32, 33syl2anc 587 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
35 relxp 5542 . . . . . . . . 9 Rel (N × N)
36 elpqn 10385 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
37363ad2ant3 1132 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
38 1st2nd 7742 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
3935, 37, 38sylancr 590 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
4034, 39oveq12d 7168 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = (⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨(1st𝐶), (2nd𝐶)⟩))
41 xp1st 7725 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
4230, 41syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
43 xp2nd 7726 . . . . . . . . . . 11 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
4432, 43syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
45 mulclpi 10353 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
4642, 44, 45syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
47 xp1st 7725 . . . . . . . . . . 11 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
4832, 47syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
49 xp2nd 7726 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
5030, 49syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
51 mulclpi 10353 . . . . . . . . . 10 (((1st𝐵) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
5248, 50, 51syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
53 addclpi 10352 . . . . . . . . 9 ((((1st𝐴) ·N (2nd𝐵)) ∈ N ∧ ((1st𝐵) ·N (2nd𝐴)) ∈ N) → (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ∈ N)
5446, 52, 53syl2anc 587 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ∈ N)
55 mulclpi 10353 . . . . . . . . 9 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
5650, 44, 55syl2anc 587 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
57 xp1st 7725 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
5837, 57syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
59 xp2nd 7726 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
6037, 59syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
61 addpipq 10397 . . . . . . . 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 837 . . . . . . 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 2793 . . . . . 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 7742 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
6535, 30, 64sylancr 590 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
66 addpipq2 10396 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
6732, 37, 66syl2anc 587 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
6865, 67oveq12d 7168 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +pQ (𝐵 +pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ +pQ ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩))
69 mulclpi 10353 . . . . . . . . . 10 (((1st𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
7048, 60, 69syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
71 mulclpi 10353 . . . . . . . . . 10 (((1st𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
7258, 44, 71syl2anc 587 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
73 addclpi 10352 . . . . . . . . 9 ((((1st𝐵) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
7470, 72, 73syl2anc 587 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
75 mulclpi 10353 . . . . . . . . 9 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
7644, 60, 75syl2anc 587 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
77 addpipq 10397 . . . . . . . 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 837 . . . . . . 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 2793 . . . . . 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 2819 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = (𝐴 +pQ (𝐵 +pQ 𝐶)))
8180fveq2d 6662 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 +pQ 𝐵) +pQ 𝐶)) = ([Q]‘(𝐴 +pQ (𝐵 +pQ 𝐶))))
82 adderpq 10416 . . . 4 (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)) = ([Q]‘((𝐴 +pQ 𝐵) +pQ 𝐶))
83 adderpq 10416 . . . 4 (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))) = ([Q]‘(𝐴 +pQ (𝐵 +pQ 𝐶)))
8481, 82, 833eqtr4g 2818 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)) = (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))))
85 addpqnq 10398 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
86853adant3 1129 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
87 nqerid 10393 . . . . . 6 (𝐶Q → ([Q]‘𝐶) = 𝐶)
8887eqcomd 2764 . . . . 5 (𝐶Q𝐶 = ([Q]‘𝐶))
89883ad2ant3 1132 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ([Q]‘𝐶))
9086, 89oveq12d 7168 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)))
91 nqerid 10393 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
9291eqcomd 2764 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
93923ad2ant1 1130 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
94 addpqnq 10398 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
95943adant1 1127 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
9693, 95oveq12d 7168 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +Q (𝐵 +Q 𝐶)) = (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))))
9784, 90, 963eqtr4d 2803 . 2 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶)))
98 addnqf 10408 . . . 4 +Q :(Q × Q)⟶Q
9998fdmi 6509 . . 3 dom +Q = (Q × Q)
100 0nnq 10384 . . 3 ¬ ∅ ∈ Q
10199, 100ndmovass 7332 . 2 (¬ (𝐴Q𝐵Q𝐶Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶)))
10297, 101pm2.61i 185 1 ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ⟨cop 4528   × cxp 5522  Rel wrel 5529  ‘cfv 6335  (class class class)co 7150  1st c1st 7691  2nd c2nd 7692  Ncnpi 10304   +N cpli 10305   ·N cmi 10306   +pQ cplpq 10308  Qcnq 10312  [Q]cerq 10314   +Q cplq 10315 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-oadd 8116  df-omul 8117  df-er 8299  df-ni 10332  df-pli 10333  df-mi 10334  df-lti 10335  df-plpq 10368  df-enq 10371  df-nq 10372  df-erq 10373  df-plq 10374  df-1nq 10376 This theorem is referenced by:  ltaddnq  10434  addasspr  10482  prlem934  10493  ltexprlem7  10502
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