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Theorem mulassnq 10895

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 10833	. . . . . . 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 10833	. . . . . . 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 4835	. . . . . 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 10861	. . . . . . . . . 10 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
6		elpqn 10861	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
7	6	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
8		mulpipq2 10875	. . . . . . . . 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 5651	. . . . . . . . 9 ⊢ Rel (N × N)
11		elpqn 10861	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
12	11	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
13		1st2nd 7971	. . . . . . . . 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 7375	. . . . . . 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 7953	. . . . . . . . . 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 7953	. . . . . . . . . 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 10829	. . . . . . . . 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 7954	. . . . . . . . . 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 7954	. . . . . . . . . 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 10829	. . . . . . . . 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 7953	. . . . . . . . 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 7954	. . . . . . . . 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 10876	. . . . . . . 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 837	. . . . . . 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 7971	. . . . . . . . 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 10875	. . . . . . . . 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 7375	. . . . . . 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 10829	. . . . . . . . 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 10829	. . . . . . . . 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 10876	. . . . . . . 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 837	. . . . . . 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 6846	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ([Q]‘((𝐴 ·_pQ 𝐵) ·_pQ 𝐶)) = ([Q]‘(𝐴 ·_pQ (𝐵 ·_pQ 𝐶))))
49		mulerpq 10893	. . . 4 ⊢ (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·_pQ 𝐵) ·_pQ 𝐶))
50		mulerpq 10893	. . . 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 10877	. . . . 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 10869	. . . . . 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 7375	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)))
58		nqerid 10869	. . . . . 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 10877	. . . . 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 7375	. . 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 10885	. . . 4 ⊢ ·_Q :(Q × Q)⟶Q
66	65	fdmi 6680	. . 3 ⊢ dom ·_Q = (Q × Q)
67		0nnq 10860	. . 3 ⊢ ¬ ∅ ∈ Q
68	66, 67	ndmovass 7542	. 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 1087 = wceq 1541 ∈ wcel 2106 ⟨cop 4592 × cxp 5631 Rel wrel 5638 ‘cfv 6496 (class class class)co 7357 1^st c1st 7919 2^nd c2nd 7920 Ncnpi 10780 ·_N cmi 10782 ·_pQ cmpq 10785 Qcnq 10788 [Q]cerq 10790 ·_Q cmq 10792

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7672

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-omul 8417 df-er 8648 df-ni 10808 df-mi 10810 df-lti 10811 df-mpq 10845 df-enq 10847 df-nq 10848 df-erq 10849 df-mq 10851 df-1nq 10852

This theorem is referenced by: recmulnq 10900 halfnq 10912 ltrnq 10915 addclprlem2 10953 mulclprlem 10955 mulasspr 10960 1idpr 10965 prlem934 10969 prlem936 10983 reclem3pr 10985

W3C validator