MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addassnq Structured version   Visualization version   GIF version

Theorem addassnq 9740
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 9677 . . . . . . . 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 6643 . . . . . . . . . . 11 ((1st𝐴) ·N (2nd𝐵)) ∈ V
3 ovex 6643 . . . . . . . . . . 11 ((1st𝐵) ·N (2nd𝐴)) ∈ V
4 fvex 6168 . . . . . . . . . . 11 (2nd𝐶) ∈ V
5 mulcompi 9678 . . . . . . . . . . 11 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
6 distrpi 9680 . . . . . . . . . . 11 (𝑥 ·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N (𝑥 ·N 𝑧))
72, 3, 4, 5, 6caovdir 6833 . . . . . . . . . 10 ((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = ((((1st𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)))
8 mulasspi 9679 . . . . . . . . . . 11 (((1st𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
98oveq1i 6625 . . . . . . . . . 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 2643 . . . . . . . . 9 ((((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ·N (2nd𝐶)) = (((1st𝐴) ·N ((2nd𝐵) ·N (2nd𝐶))) +N (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)))
1110oveq1i 6625 . . . . . . . 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 6643 . . . . . . . . . . 11 ((1st𝐵) ·N (2nd𝐶)) ∈ V
13 ovex 6643 . . . . . . . . . . 11 ((1st𝐶) ·N (2nd𝐵)) ∈ V
14 fvex 6168 . . . . . . . . . . 11 (2nd𝐴) ∈ V
1512, 13, 14, 5, 6caovdir 6833 . . . . . . . . . 10 ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴)) = ((((1st𝐵) ·N (2nd𝐶)) ·N (2nd𝐴)) +N (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴)))
16 fvex 6168 . . . . . . . . . . . 12 (1st𝐵) ∈ V
17 mulasspi 9679 . . . . . . . . . . . 12 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
1816, 4, 14, 5, 17caov32 6826 . . . . . . . . . . 11 (((1st𝐵) ·N (2nd𝐶)) ·N (2nd𝐴)) = (((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶))
19 mulasspi 9679 . . . . . . . . . . . 12 (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴)) = ((1st𝐶) ·N ((2nd𝐵) ·N (2nd𝐴)))
20 mulcompi 9678 . . . . . . . . . . . . 13 ((2nd𝐵) ·N (2nd𝐴)) = ((2nd𝐴) ·N (2nd𝐵))
2120oveq2i 6626 . . . . . . . . . . . 12 ((1st𝐶) ·N ((2nd𝐵) ·N (2nd𝐴))) = ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))
2219, 21eqtri 2643 . . . . . . . . . . 11 (((1st𝐶) ·N (2nd𝐵)) ·N (2nd𝐴)) = ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵)))
2318, 22oveq12i 6627 . . . . . . . . . 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 2643 . . . . . . . . 9 ((((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ·N (2nd𝐴)) = ((((1st𝐵) ·N (2nd𝐴)) ·N (2nd𝐶)) +N ((1st𝐶) ·N ((2nd𝐴) ·N (2nd𝐵))))
2524oveq2i 6626 . . . . . . . 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 2653 . . . . . . 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 9679 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
2826, 27opeq12i 4382 . . . . . 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 9707 . . . . . . . . . 10 (𝐴Q𝐴 ∈ (N × N))
30293ad2ant1 1080 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
31 elpqn 9707 . . . . . . . . . 10 (𝐵Q𝐵 ∈ (N × N))
32313ad2ant2 1081 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
33 addpipq2 9718 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
3430, 32, 33syl2anc 692 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
35 relxp 5198 . . . . . . . . 9 Rel (N × N)
36 elpqn 9707 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
37363ad2ant3 1082 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
38 1st2nd 7174 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
3935, 37, 38sylancr 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
4034, 39oveq12d 6633 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = (⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ +pQ ⟨(1st𝐶), (2nd𝐶)⟩))
41 xp1st 7158 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
4230, 41syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
43 xp2nd 7159 . . . . . . . . . . 11 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
4432, 43syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
45 mulclpi 9675 . . . . . . . . . 10 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
4642, 44, 45syl2anc 692 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
47 xp1st 7158 . . . . . . . . . . 11 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
4832, 47syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
49 xp2nd 7159 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
5030, 49syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
51 mulclpi 9675 . . . . . . . . . 10 (((1st𝐵) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
5248, 50, 51syl2anc 692 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (2nd𝐴)) ∈ N)
53 addclpi 9674 . . . . . . . . 9 ((((1st𝐴) ·N (2nd𝐵)) ∈ N ∧ ((1st𝐵) ·N (2nd𝐴)) ∈ N) → (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ∈ N)
5446, 52, 53syl2anc 692 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) ∈ N)
55 mulclpi 9675 . . . . . . . . 9 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
5650, 44, 55syl2anc 692 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
57 xp1st 7158 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
5837, 57syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
59 xp2nd 7159 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
6037, 59syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
61 addpipq 9719 . . . . . . . 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 1324 . . . . . . 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 2655 . . . . . 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 7174 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
6535, 30, 64sylancr 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
66 addpipq2 9718 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
6732, 37, 66syl2anc 692 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +pQ 𝐶) = ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩)
6865, 67oveq12d 6633 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +pQ (𝐵 +pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ +pQ ⟨(((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐶))⟩))
69 mulclpi 9675 . . . . . . . . . 10 (((1st𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
7048, 60, 69syl2anc 692 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (2nd𝐶)) ∈ N)
71 mulclpi 9675 . . . . . . . . . 10 (((1st𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
7258, 44, 71syl2anc 692 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐵)) ∈ N)
73 addclpi 9674 . . . . . . . . 9 ((((1st𝐵) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐵)) ∈ N) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
7470, 72, 73syl2anc 692 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐵) ·N (2nd𝐶)) +N ((1st𝐶) ·N (2nd𝐵))) ∈ N)
75 mulclpi 9675 . . . . . . . . 9 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
7644, 60, 75syl2anc 692 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
77 addpipq 9719 . . . . . . . 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 1324 . . . . . . 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 2655 . . . . . 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 2681 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = (𝐴 +pQ (𝐵 +pQ 𝐶)))
8180fveq2d 6162 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 +pQ 𝐵) +pQ 𝐶)) = ([Q]‘(𝐴 +pQ (𝐵 +pQ 𝐶))))
82 adderpq 9738 . . . 4 (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)) = ([Q]‘((𝐴 +pQ 𝐵) +pQ 𝐶))
83 adderpq 9738 . . . 4 (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))) = ([Q]‘(𝐴 +pQ (𝐵 +pQ 𝐶)))
8481, 82, 833eqtr4g 2680 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)) = (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))))
85 addpqnq 9720 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
86853adant3 1079 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
87 nqerid 9715 . . . . . 6 (𝐶Q → ([Q]‘𝐶) = 𝐶)
8887eqcomd 2627 . . . . 5 (𝐶Q𝐶 = ([Q]‘𝐶))
89883ad2ant3 1082 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ([Q]‘𝐶))
9086, 89oveq12d 6633 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)))
91 nqerid 9715 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
9291eqcomd 2627 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
93923ad2ant1 1080 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
94 addpqnq 9720 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
95943adant1 1077 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
9693, 95oveq12d 6633 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 +Q (𝐵 +Q 𝐶)) = (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))))
9784, 90, 963eqtr4d 2665 . 2 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶)))
98 addnqf 9730 . . . 4 +Q :(Q × Q)⟶Q
9998fdmi 6019 . . 3 dom +Q = (Q × Q)
100 0nnq 9706 . . 3 ¬ ∅ ∈ Q
10199, 100ndmovass 6787 . 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 1036   = wceq 1480  wcel 1987  cop 4161   × cxp 5082  Rel wrel 5089  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  Ncnpi 9626   +N cpli 9627   ·N cmi 9628   +pQ cplpq 9630  Qcnq 9634  [Q]cerq 9636   +Q cplq 9637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-ni 9654  df-pli 9655  df-mi 9656  df-lti 9657  df-plpq 9690  df-enq 9693  df-nq 9694  df-erq 9695  df-plq 9696  df-1nq 9698
This theorem is referenced by:  ltaddnq  9756  addasspr  9804  prlem934  9815  ltexprlem7  9824
  Copyright terms: Public domain W3C validator