Theorem addassnq 10723

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.

Step	Hyp	Ref	Expression
1		addasspi 10660	. . . . . . . 8 ⊢ ((((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶))) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))) = (((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N ((((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))))
2		ovex 7317	. . . . . . . . . . 11 ⊢ ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ V
3		ovex 7317	. . . . . . . . . . 11 ⊢ ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ V
4		fvex 6796	. . . . . . . . . . 11 ⊢ (2nd ‘𝐶) ∈ V
5		mulcompi 10661	. . . . . . . . . . 11 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
6		distrpi 10663	. . . . . . . . . . 11 ⊢ (𝑥 ·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N (𝑥 ·N 𝑧))
7	2, 3, 4, 5, 6	caovdir 7515	. . . . . . . . . 10 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶)) = ((((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶)) +N (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶)))
8		mulasspi 10662	. . . . . . . . . . 11 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶)) = ((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
9	8	oveq1i 7294	. . . . . . . . . 10 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶)) +N (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶))) = (((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶)))
10	7, 9	eqtri 2767	. . . . . . . . 9 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶)) = (((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶)))
11	10	oveq1i 7294	. . . . . . . 8 ⊢ (((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))) = ((((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶))) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))))
12		ovex 7317	. . . . . . . . . . 11 ⊢ ((1st ‘𝐵) ·N (2nd ‘𝐶)) ∈ V
13		ovex 7317	. . . . . . . . . . 11 ⊢ ((1st ‘𝐶) ·N (2nd ‘𝐵)) ∈ V
14		fvex 6796	. . . . . . . . . . 11 ⊢ (2nd ‘𝐴) ∈ V
15	12, 13, 14, 5, 6	caovdir 7515	. . . . . . . . . 10 ⊢ ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ·N (2nd ‘𝐴)) = ((((1st ‘𝐵) ·N (2nd ‘𝐶)) ·N (2nd ‘𝐴)) +N (((1st ‘𝐶) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐴)))
16		fvex 6796	. . . . . . . . . . . 12 ⊢ (1st ‘𝐵) ∈ V
17		mulasspi 10662	. . . . . . . . . . . 12 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
18	16, 4, 14, 5, 17	caov32 7508	. . . . . . . . . . 11 ⊢ (((1st ‘𝐵) ·N (2nd ‘𝐶)) ·N (2nd ‘𝐴)) = (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶))
19		mulasspi 10662	. . . . . . . . . . . 12 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐴)) = ((1st ‘𝐶) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐴)))
20		mulcompi 10661	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐵) ·N (2nd ‘𝐴)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵))
21	20	oveq2i 7295	. . . . . . . . . . . 12 ⊢ ((1st ‘𝐶) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐴))) = ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
22	19, 21	eqtri 2767	. . . . . . . . . . 11 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐴)) = ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
23	18, 22	oveq12i 7296	. . . . . . . . . 10 ⊢ ((((1st ‘𝐵) ·N (2nd ‘𝐶)) ·N (2nd ‘𝐴)) +N (((1st ‘𝐶) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐴))) = ((((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))))
24	15, 23	eqtri 2767	. . . . . . . . 9 ⊢ ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ·N (2nd ‘𝐴)) = ((((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))))
25	24	oveq2i 7295	. . . . . . . 8 ⊢ (((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ·N (2nd ‘𝐴))) = (((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N ((((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))))
26	1, 11, 25	3eqtr4i 2777	. . . . . . 7 ⊢ (((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))) = (((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ·N (2nd ‘𝐴)))
27		mulasspi 10662	. . . . . . 7 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
28	26, 27	opeq12i 4810	. . . . . 6 ⊢ ⟨(((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))), (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶))⟩ = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))⟩
29		elpqn 10690	. . . . . . . . . 10 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
30	29	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
31		elpqn 10690	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
32	31	3ad2ant2 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
33		addpipq2 10701	. . . . . . . . 9 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
34	30, 32, 33	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 +pQ 𝐵) = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
35		relxp 5608	. . . . . . . . 9 ⊢ Rel (N × N)
36		elpqn 10690	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
37	36	3ad2ant3 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
38		1st2nd 7889	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
39	35, 37, 38	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
40	34, 39	oveq12d 7302	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = (⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ +pQ ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩))
41		xp1st 7872	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
42	30, 41	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐴) ∈ N)
43		xp2nd 7873	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
44	32, 43	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐵) ∈ N)
45		mulclpi 10658	. . . . . . . . . 10 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
46	42, 44, 45	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
47		xp1st 7872	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
48	32, 47	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐵) ∈ N)
49		xp2nd 7873	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
50	30, 49	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐴) ∈ N)
51		mulclpi 10658	. . . . . . . . . 10 ⊢ (((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
52	48, 50, 51	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
53		addclpi 10657	. . . . . . . . 9 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N ∧ ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ∈ N)
54	46, 52, 53	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ∈ N)
55		mulclpi 10658	. . . . . . . . 9 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
56	50, 44, 55	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
57		xp1st 7872	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
58	37, 57	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
59		xp2nd 7873	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
60	37, 59	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
61		addpipq 10702	. . . . . . . 8 ⊢ ((((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N) ∧ ((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N)) → (⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ +pQ ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩) = ⟨(((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))), (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶))⟩)
62	54, 56, 58, 60, 61	syl22anc 836	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ +pQ ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩) = ⟨(((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))), (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶))⟩)
63	40, 62	eqtrd 2779	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = ⟨(((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))), (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶))⟩)
64		1st2nd 7889	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
65	35, 30, 64	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
66		addpipq2 10701	. . . . . . . . 9 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 +pQ 𝐶) = ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
67	32, 37, 66	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 +pQ 𝐶) = ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
68	65, 67	oveq12d 7302	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 +pQ (𝐵 +pQ 𝐶)) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩))
69		mulclpi 10658	. . . . . . . . . 10 ⊢ (((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
70	48, 60, 69	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
71		mulclpi 10658	. . . . . . . . . 10 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐵)) ∈ N)
72	58, 44, 71	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐶) ·N (2nd ‘𝐵)) ∈ N)
73		addclpi 10657	. . . . . . . . 9 ⊢ ((((1st ‘𝐵) ·N (2nd ‘𝐶)) ∈ N ∧ ((1st ‘𝐶) ·N (2nd ‘𝐵)) ∈ N) → (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ∈ N)
74	70, 72, 73	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ∈ N)
75		mulclpi 10658	. . . . . . . . 9 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
76	44, 60, 75	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
77		addpipq 10702	. . . . . . . 8 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ∈ N ∧ ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))⟩)
78	42, 50, 74, 76, 77	syl22anc 836	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨(((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))⟩)
79	68, 78	eqtrd 2779	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 +pQ (𝐵 +pQ 𝐶)) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶))) +N ((((1st ‘𝐵) ·N (2nd ‘𝐶)) +N ((1st ‘𝐶) ·N (2nd ‘𝐵))) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))⟩)
80	28, 63, 79	3eqtr4a 2805	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 +pQ 𝐵) +pQ 𝐶) = (𝐴 +pQ (𝐵 +pQ 𝐶)))
81	80	fveq2d 6787	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ([Q]‘((𝐴 +pQ 𝐵) +pQ 𝐶)) = ([Q]‘(𝐴 +pQ (𝐵 +pQ 𝐶))))
82		adderpq 10721	. . . 4 ⊢ (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)) = ([Q]‘((𝐴 +pQ 𝐵) +pQ 𝐶))
83		adderpq 10721	. . . 4 ⊢ (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))) = ([Q]‘(𝐴 +pQ (𝐵 +pQ 𝐶)))
84	81, 82, 83	3eqtr4g 2804	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)) = (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))))
85		addpqnq 10703	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
86	85	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
87		nqerid 10698	. . . . . 6 ⊢ (𝐶 ∈ Q → ([Q]‘𝐶) = 𝐶)
88	87	eqcomd 2745	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 = ([Q]‘𝐶))
89	88	3ad2ant3 1134	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ([Q]‘𝐶))
90	86, 89	oveq12d 7302	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (([Q]‘(𝐴 +pQ 𝐵)) +Q ([Q]‘𝐶)))
91		nqerid 10698	. . . . . 6 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
92	91	eqcomd 2745	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 = ([Q]‘𝐴))
93	92	3ad2ant1 1132	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ([Q]‘𝐴))
94		addpqnq 10703	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
95	94	3adant1 1129	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 +Q 𝐶) = ([Q]‘(𝐵 +pQ 𝐶)))
96	93, 95	oveq12d 7302	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 +Q (𝐵 +Q 𝐶)) = (([Q]‘𝐴) +Q ([Q]‘(𝐵 +pQ 𝐶))))
97	84, 90, 96	3eqtr4d 2789	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶)))
98		addnqf 10713	. . . 4 ⊢ +Q :(Q × Q)⟶Q
99	98	fdmi 6621	. . 3 ⊢ dom +Q = (Q × Q)
100		0nnq 10689	. . 3 ⊢ ¬ ∅ ∈ Q
101	99, 100	ndmovass 7469	. 2 ⊢ (¬ (𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶)))
102	97, 101	pm2.61i 182	1 ⊢ ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶))

Syntax hints:   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ⟨cop 4568   × cxp 5588  Rel wrel 5595  ‘cfv 6437  (class class class)co 7284  1^st c1st 7838  2^nd c2nd 7839  Ncnpi 10609   +_N cpli 10610   ·_N cmi 10611   +_pQ cplpq 10613  Qcnq 10617  [Q]cerq 10619   +_Q cplq 10620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pr 5353  ax-un 7597

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-oadd 8310  df-omul 8311  df-er 8507  df-ni 10637  df-pli 10638  df-mi 10639  df-lti 10640  df-plpq 10673  df-enq 10676  df-nq 10677  df-erq 10678  df-plq 10679  df-1nq 10681

This theorem is referenced by:  ltaddnq  10739  addasspr  10787  prlem934  10798  ltexprlem7  10807