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Theorem mulassnq 11000

Description: Multiplication of positive fractions is associative. (Contributed by NM, 1-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
mulassnq	⊢ ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (𝐴 ·_Q (𝐵 ·_Q 𝐶))

**Proof of Theorem mulassnq**
Step	Hyp	Ref	Expression
1		mulasspi 10938	. . . . . . 7 ⊢ (((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ·_N (1^st ‘𝐶)) = ((1^st ‘𝐴) ·_N ((1^st ‘𝐵) ·_N (1^st ‘𝐶)))
2		mulasspi 10938	. . . . . . 7 ⊢ (((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ·_N (2^nd ‘𝐶)) = ((2^nd ‘𝐴) ·_N ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)))
3	1, 2	opeq12i 4877	. . . . . 6 ⊢ ⟨(((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ·_N (1^st ‘𝐶)), (((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ·_N (2^nd ‘𝐶))⟩ = ⟨((1^st ‘𝐴) ·_N ((1^st ‘𝐵) ·_N (1^st ‘𝐶))), ((2^nd ‘𝐴) ·_N ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)))⟩
4		elpqn 10966	. . . . . . . . . 10 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
6		elpqn 10966	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
7	6	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
8		mulpipq2 10980	. . . . . . . . 9 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·_pQ 𝐵) = ⟨((1^st ‘𝐴) ·_N (1^st ‘𝐵)), ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵))⟩)
9	5, 7, 8	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_pQ 𝐵) = ⟨((1^st ‘𝐴) ·_N (1^st ‘𝐵)), ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵))⟩)
10		relxp 5702	. . . . . . . . 9 ⊢ Rel (N × N)
11		elpqn 10966	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
12	11	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
13		1st2nd 8065	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
14	10, 12, 13	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
15	9, 14	oveq12d 7450	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_pQ 𝐵) ·_pQ 𝐶) = (⟨((1^st ‘𝐴) ·_N (1^st ‘𝐵)), ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵))⟩ ·_pQ ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩))
16		xp1st 8047	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (1^st ‘𝐴) ∈ N)
17	5, 16	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1^st ‘𝐴) ∈ N)
18		xp1st 8047	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (1^st ‘𝐵) ∈ N)
19	7, 18	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1^st ‘𝐵) ∈ N)
20		mulclpi 10934	. . . . . . . . 9 ⊢ (((1^st ‘𝐴) ∈ N ∧ (1^st ‘𝐵) ∈ N) → ((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ∈ N)
21	17, 19, 20	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ∈ N)
22		xp2nd 8048	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (2^nd ‘𝐴) ∈ N)
23	5, 22	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2^nd ‘𝐴) ∈ N)
24		xp2nd 8048	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (2^nd ‘𝐵) ∈ N)
25	7, 24	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2^nd ‘𝐵) ∈ N)
26		mulclpi 10934	. . . . . . . . 9 ⊢ (((2^nd ‘𝐴) ∈ N ∧ (2^nd ‘𝐵) ∈ N) → ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ∈ N)
27	23, 25, 26	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ∈ N)
28		xp1st 8047	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1^st ‘𝐶) ∈ N)
29	12, 28	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1^st ‘𝐶) ∈ N)
30		xp2nd 8048	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2^nd ‘𝐶) ∈ N)
31	12, 30	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2^nd ‘𝐶) ∈ N)
32		mulpipq 10981	. . . . . . . 8 ⊢ (((((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ∈ N ∧ ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ∈ N) ∧ ((1^st ‘𝐶) ∈ N ∧ (2^nd ‘𝐶) ∈ N)) → (⟨((1^st ‘𝐴) ·_N (1^st ‘𝐵)), ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵))⟩ ·_pQ ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩) = ⟨(((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ·_N (1^st ‘𝐶)), (((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ·_N (2^nd ‘𝐶))⟩)
33	21, 27, 29, 31, 32	syl22anc 838	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (⟨((1^st ‘𝐴) ·_N (1^st ‘𝐵)), ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵))⟩ ·_pQ ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩) = ⟨(((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ·_N (1^st ‘𝐶)), (((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ·_N (2^nd ‘𝐶))⟩)
34	15, 33	eqtrd 2776	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_pQ 𝐵) ·_pQ 𝐶) = ⟨(((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ·_N (1^st ‘𝐶)), (((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ·_N (2^nd ‘𝐶))⟩)
35		1st2nd 8065	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
36	10, 5, 35	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
37		mulpipq2 10980	. . . . . . . . 9 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·_pQ 𝐶) = ⟨((1^st ‘𝐵) ·_N (1^st ‘𝐶)), ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶))⟩)
38	7, 12, 37	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·_pQ 𝐶) = ⟨((1^st ‘𝐵) ·_N (1^st ‘𝐶)), ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶))⟩)
39	36, 38	oveq12d 7450	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_pQ (𝐵 ·_pQ 𝐶)) = (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ·_pQ ⟨((1^st ‘𝐵) ·_N (1^st ‘𝐶)), ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶))⟩))
40		mulclpi 10934	. . . . . . . . 9 ⊢ (((1^st ‘𝐵) ∈ N ∧ (1^st ‘𝐶) ∈ N) → ((1^st ‘𝐵) ·_N (1^st ‘𝐶)) ∈ N)
41	19, 29, 40	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1^st ‘𝐵) ·_N (1^st ‘𝐶)) ∈ N)
42		mulclpi 10934	. . . . . . . . 9 ⊢ (((2^nd ‘𝐵) ∈ N ∧ (2^nd ‘𝐶) ∈ N) → ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)) ∈ N)
43	25, 31, 42	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)) ∈ N)
44		mulpipq 10981	. . . . . . . 8 ⊢ ((((1^st ‘𝐴) ∈ N ∧ (2^nd ‘𝐴) ∈ N) ∧ (((1^st ‘𝐵) ·_N (1^st ‘𝐶)) ∈ N ∧ ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)) ∈ N)) → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ·_pQ ⟨((1^st ‘𝐵) ·_N (1^st ‘𝐶)), ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶))⟩) = ⟨((1^st ‘𝐴) ·_N ((1^st ‘𝐵) ·_N (1^st ‘𝐶))), ((2^nd ‘𝐴) ·_N ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)))⟩)
45	17, 23, 41, 43, 44	syl22anc 838	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ·_pQ ⟨((1^st ‘𝐵) ·_N (1^st ‘𝐶)), ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶))⟩) = ⟨((1^st ‘𝐴) ·_N ((1^st ‘𝐵) ·_N (1^st ‘𝐶))), ((2^nd ‘𝐴) ·_N ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)))⟩)
46	39, 45	eqtrd 2776	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_pQ (𝐵 ·_pQ 𝐶)) = ⟨((1^st ‘𝐴) ·_N ((1^st ‘𝐵) ·_N (1^st ‘𝐶))), ((2^nd ‘𝐴) ·_N ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)))⟩)
47	3, 34, 46	3eqtr4a 2802	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_pQ 𝐵) ·_pQ 𝐶) = (𝐴 ·_pQ (𝐵 ·_pQ 𝐶)))
48	47	fveq2d 6909	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ([Q]‘((𝐴 ·_pQ 𝐵) ·_pQ 𝐶)) = ([Q]‘(𝐴 ·_pQ (𝐵 ·_pQ 𝐶))))
49		mulerpq 10998	. . . 4 ⊢ (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·_pQ 𝐵) ·_pQ 𝐶))
50		mulerpq 10998	. . . 4 ⊢ (([Q]‘𝐴) ·_Q ([Q]‘(𝐵 ·_pQ 𝐶))) = ([Q]‘(𝐴 ·_pQ (𝐵 ·_pQ 𝐶)))
51	48, 49, 50	3eqtr4g 2801	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·_Q ([Q]‘(𝐵 ·_pQ 𝐶))))
52		mulpqnq 10982	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·_Q 𝐵) = ([Q]‘(𝐴 ·_pQ 𝐵)))
53	52	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_Q 𝐵) = ([Q]‘(𝐴 ·_pQ 𝐵)))
54		nqerid 10974	. . . . . 6 ⊢ (𝐶 ∈ Q → ([Q]‘𝐶) = 𝐶)
55	54	eqcomd 2742	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 = ([Q]‘𝐶))
56	55	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ([Q]‘𝐶))
57	53, 56	oveq12d 7450	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)))
58		nqerid 10974	. . . . . 6 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
59	58	eqcomd 2742	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 = ([Q]‘𝐴))
60	59	3ad2ant1 1133	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ([Q]‘𝐴))
61		mulpqnq 10982	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·_Q 𝐶) = ([Q]‘(𝐵 ·_pQ 𝐶)))
62	61	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·_Q 𝐶) = ([Q]‘(𝐵 ·_pQ 𝐶)))
63	60, 62	oveq12d 7450	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_Q (𝐵 ·_Q 𝐶)) = (([Q]‘𝐴) ·_Q ([Q]‘(𝐵 ·_pQ 𝐶))))
64	51, 57, 63	3eqtr4d 2786	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (𝐴 ·_Q (𝐵 ·_Q 𝐶)))
65		mulnqf 10990	. . . 4 ⊢ ·_Q :(Q × Q)⟶Q
66	65	fdmi 6746	. . 3 ⊢ dom ·_Q = (Q × Q)
67		0nnq 10965	. . 3 ⊢ ¬ ∅ ∈ Q
68	66, 67	ndmovass 7622	. 2 ⊢ (¬ (𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (𝐴 ·_Q (𝐵 ·_Q 𝐶)))
69	64, 68	pm2.61i 182	1 ⊢ ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (𝐴 ·_Q (𝐵 ·_Q 𝐶))

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ⟨cop 4631 × cxp 5682 Rel wrel 5689 ‘cfv 6560 (class class class)co 7432 1^st c1st 8013 2^nd c2nd 8014 Ncnpi 10885 ·_N cmi 10887 ·_pQ cmpq 10890 Qcnq 10893 [Q]cerq 10895 ·_Q cmq 10897

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-omul 8512 df-er 8746 df-ni 10913 df-mi 10915 df-lti 10916 df-mpq 10950 df-enq 10952 df-nq 10953 df-erq 10954 df-mq 10956 df-1nq 10957

This theorem is referenced by: recmulnq 11005 halfnq 11017 ltrnq 11020 addclprlem2 11058 mulclprlem 11060 mulasspr 11065 1idpr 11070 prlem934 11074 prlem936 11088 reclem3pr 11090

W3C validator