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Theorem mulassnq 10743

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 10681	. . . . . . 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 10681	. . . . . . 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 4811	. . . . . 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 10709	. . . . . . . . . 10 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
6		elpqn 10709	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
7	6	3ad2ant2 1132	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
8		mulpipq2 10723	. . . . . . . . 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 583	. . . . . . . 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 5609	. . . . . . . . 9 ⊢ Rel (N × N)
11		elpqn 10709	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
12	11	3ad2ant3 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
13		1st2nd 7900	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
14	10, 12, 13	sylancr 586	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
15	9, 14	oveq12d 7313	. . . . . . 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 7883	. . . . . . . . . 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 7883	. . . . . . . . . 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 10677	. . . . . . . . 9 ⊢ (((1^st ‘𝐴) ∈ N ∧ (1^st ‘𝐵) ∈ N) → ((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ∈ N)
21	17, 19, 20	syl2anc 583	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ∈ N)
22		xp2nd 7884	. . . . . . . . . 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 7884	. . . . . . . . . 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 10677	. . . . . . . . 9 ⊢ (((2^nd ‘𝐴) ∈ N ∧ (2^nd ‘𝐵) ∈ N) → ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ∈ N)
27	23, 25, 26	syl2anc 583	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ∈ N)
28		xp1st 7883	. . . . . . . . 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 7884	. . . . . . . . 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 10724	. . . . . . . 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 835	. . . . . . 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 2773	. . . . . 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 7900	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
36	10, 5, 35	sylancr 586	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
37		mulpipq2 10723	. . . . . . . . 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 583	. . . . . . . 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 7313	. . . . . . 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 10677	. . . . . . . . 9 ⊢ (((1^st ‘𝐵) ∈ N ∧ (1^st ‘𝐶) ∈ N) → ((1^st ‘𝐵) ·_N (1^st ‘𝐶)) ∈ N)
41	19, 29, 40	syl2anc 583	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1^st ‘𝐵) ·_N (1^st ‘𝐶)) ∈ N)
42		mulclpi 10677	. . . . . . . . 9 ⊢ (((2^nd ‘𝐵) ∈ N ∧ (2^nd ‘𝐶) ∈ N) → ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)) ∈ N)
43	25, 31, 42	syl2anc 583	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)) ∈ N)
44		mulpipq 10724	. . . . . . . 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 835	. . . . . . 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 2773	. . . . . 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 2799	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_pQ 𝐵) ·_pQ 𝐶) = (𝐴 ·_pQ (𝐵 ·_pQ 𝐶)))
48	47	fveq2d 6796	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ([Q]‘((𝐴 ·_pQ 𝐵) ·_pQ 𝐶)) = ([Q]‘(𝐴 ·_pQ (𝐵 ·_pQ 𝐶))))
49		mulerpq 10741	. . . 4 ⊢ (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·_pQ 𝐵) ·_pQ 𝐶))
50		mulerpq 10741	. . . 4 ⊢ (([Q]‘𝐴) ·_Q ([Q]‘(𝐵 ·_pQ 𝐶))) = ([Q]‘(𝐴 ·_pQ (𝐵 ·_pQ 𝐶)))
51	48, 49, 50	3eqtr4g 2798	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·_Q ([Q]‘(𝐵 ·_pQ 𝐶))))
52		mulpqnq 10725	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·_Q 𝐵) = ([Q]‘(𝐴 ·_pQ 𝐵)))
53	52	3adant3 1130	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_Q 𝐵) = ([Q]‘(𝐴 ·_pQ 𝐵)))
54		nqerid 10717	. . . . . 6 ⊢ (𝐶 ∈ Q → ([Q]‘𝐶) = 𝐶)
55	54	eqcomd 2739	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 = ([Q]‘𝐶))
56	55	3ad2ant3 1133	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ([Q]‘𝐶))
57	53, 56	oveq12d 7313	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)))
58		nqerid 10717	. . . . . 6 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
59	58	eqcomd 2739	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 = ([Q]‘𝐴))
60	59	3ad2ant1 1131	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ([Q]‘𝐴))
61		mulpqnq 10725	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·_Q 𝐶) = ([Q]‘(𝐵 ·_pQ 𝐶)))
62	61	3adant1 1128	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·_Q 𝐶) = ([Q]‘(𝐵 ·_pQ 𝐶)))
63	60, 62	oveq12d 7313	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_Q (𝐵 ·_Q 𝐶)) = (([Q]‘𝐴) ·_Q ([Q]‘(𝐵 ·_pQ 𝐶))))
64	51, 57, 63	3eqtr4d 2783	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (𝐴 ·_Q (𝐵 ·_Q 𝐶)))
65		mulnqf 10733	. . . 4 ⊢ ·_Q :(Q × Q)⟶Q
66	65	fdmi 6630	. . 3 ⊢ dom ·_Q = (Q × Q)
67		0nnq 10708	. . 3 ⊢ ¬ ∅ ∈ Q
68	66, 67	ndmovass 7480	. 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 1085 = wceq 1537 ∈ wcel 2101 ⟨cop 4570 × cxp 5589 Rel wrel 5596 ‘cfv 6447 (class class class)co 7295 1^st c1st 7849 2^nd c2nd 7850 Ncnpi 10628 ·_N cmi 10630 ·_pQ cmpq 10633 Qcnq 10636 [Q]cerq 10638 ·_Q cmq 10640

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7608

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-oadd 8321 df-omul 8322 df-er 8518 df-ni 10656 df-mi 10658 df-lti 10659 df-mpq 10693 df-enq 10695 df-nq 10696 df-erq 10697 df-mq 10699 df-1nq 10700

This theorem is referenced by: recmulnq 10748 halfnq 10760 ltrnq 10763 addclprlem2 10801 mulclprlem 10803 mulasspr 10808 1idpr 10813 prlem934 10817 prlem936 10831 reclem3pr 10833

W3C validator