mulassnq - Metamath Proof Explorer

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

Theorem mulassnq 10383

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 10321	. . . . . . 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 10321	. . . . . . 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 4810	. . . . . 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 10349	. . . . . . . . . 10 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	3ad2ant1 1129	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
6		elpqn 10349	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
7	6	3ad2ant2 1130	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
8		mulpipq2 10363	. . . . . . . . 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 586	. . . . . . . 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 5575	. . . . . . . . 9 ⊢ Rel (N × N)
11		elpqn 10349	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
12	11	3ad2ant3 1131	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
13		1st2nd 7740	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
14	10, 12, 13	sylancr 589	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩)
15	9, 14	oveq12d 7176	. . . . . . 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 7723	. . . . . . . . . 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 7723	. . . . . . . . . 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 10317	. . . . . . . . 9 ⊢ (((1^st ‘𝐴) ∈ N ∧ (1^st ‘𝐵) ∈ N) → ((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ∈ N)
21	17, 19, 20	syl2anc 586	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1^st ‘𝐴) ·_N (1^st ‘𝐵)) ∈ N)
22		xp2nd 7724	. . . . . . . . . 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 7724	. . . . . . . . . 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 10317	. . . . . . . . 9 ⊢ (((2^nd ‘𝐴) ∈ N ∧ (2^nd ‘𝐵) ∈ N) → ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ∈ N)
27	23, 25, 26	syl2anc 586	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2^nd ‘𝐴) ·_N (2^nd ‘𝐵)) ∈ N)
28		xp1st 7723	. . . . . . . . 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 7724	. . . . . . . . 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 10364	. . . . . . . 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 836	. . . . . . 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 2858	. . . . . 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 7740	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
36	10, 5, 35	sylancr 589	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
37		mulpipq2 10363	. . . . . . . . 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 586	. . . . . . . 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 7176	. . . . . . 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 10317	. . . . . . . . 9 ⊢ (((1^st ‘𝐵) ∈ N ∧ (1^st ‘𝐶) ∈ N) → ((1^st ‘𝐵) ·_N (1^st ‘𝐶)) ∈ N)
41	19, 29, 40	syl2anc 586	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1^st ‘𝐵) ·_N (1^st ‘𝐶)) ∈ N)
42		mulclpi 10317	. . . . . . . . 9 ⊢ (((2^nd ‘𝐵) ∈ N ∧ (2^nd ‘𝐶) ∈ N) → ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)) ∈ N)
43	25, 31, 42	syl2anc 586	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2^nd ‘𝐵) ·_N (2^nd ‘𝐶)) ∈ N)
44		mulpipq 10364	. . . . . . . 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 836	. . . . . . 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 2858	. . . . . 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 2884	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_pQ 𝐵) ·_pQ 𝐶) = (𝐴 ·_pQ (𝐵 ·_pQ 𝐶)))
48	47	fveq2d 6676	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ([Q]‘((𝐴 ·_pQ 𝐵) ·_pQ 𝐶)) = ([Q]‘(𝐴 ·_pQ (𝐵 ·_pQ 𝐶))))
49		mulerpq 10381	. . . 4 ⊢ (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·_pQ 𝐵) ·_pQ 𝐶))
50		mulerpq 10381	. . . 4 ⊢ (([Q]‘𝐴) ·_Q ([Q]‘(𝐵 ·_pQ 𝐶))) = ([Q]‘(𝐴 ·_pQ (𝐵 ·_pQ 𝐶)))
51	48, 49, 50	3eqtr4g 2883	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·_Q ([Q]‘(𝐵 ·_pQ 𝐶))))
52		mulpqnq 10365	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·_Q 𝐵) = ([Q]‘(𝐴 ·_pQ 𝐵)))
53	52	3adant3 1128	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_Q 𝐵) = ([Q]‘(𝐴 ·_pQ 𝐵)))
54		nqerid 10357	. . . . . 6 ⊢ (𝐶 ∈ Q → ([Q]‘𝐶) = 𝐶)
55	54	eqcomd 2829	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 = ([Q]‘𝐶))
56	55	3ad2ant3 1131	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ([Q]‘𝐶))
57	53, 56	oveq12d 7176	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (([Q]‘(𝐴 ·_pQ 𝐵)) ·_Q ([Q]‘𝐶)))
58		nqerid 10357	. . . . . 6 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
59	58	eqcomd 2829	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 = ([Q]‘𝐴))
60	59	3ad2ant1 1129	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ([Q]‘𝐴))
61		mulpqnq 10365	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·_Q 𝐶) = ([Q]‘(𝐵 ·_pQ 𝐶)))
62	61	3adant1 1126	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·_Q 𝐶) = ([Q]‘(𝐵 ·_pQ 𝐶)))
63	60, 62	oveq12d 7176	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·_Q (𝐵 ·_Q 𝐶)) = (([Q]‘𝐴) ·_Q ([Q]‘(𝐵 ·_pQ 𝐶))))
64	51, 57, 63	3eqtr4d 2868	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (𝐴 ·_Q (𝐵 ·_Q 𝐶)))
65		mulnqf 10373	. . . 4 ⊢ ·_Q :(Q × Q)⟶Q
66	65	fdmi 6526	. . 3 ⊢ dom ·_Q = (Q × Q)
67		0nnq 10348	. . 3 ⊢ ¬ ∅ ∈ Q
68	66, 67	ndmovass 7338	. 2 ⊢ (¬ (𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (𝐴 ·_Q (𝐵 ·_Q 𝐶)))
69	64, 68	pm2.61i 184	1 ⊢ ((𝐴 ·_Q 𝐵) ·_Q 𝐶) = (𝐴 ·_Q (𝐵 ·_Q 𝐶))

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⟨cop 4575 × cxp 5555 Rel wrel 5562 ‘cfv 6357 (class class class)co 7158 1^st c1st 7689 2^nd c2nd 7690 Ncnpi 10268 ·_N cmi 10270 ·_pQ cmpq 10273 Qcnq 10276 [Q]cerq 10278 ·_Q cmq 10280

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-ni 10296 df-mi 10298 df-lti 10299 df-mpq 10333 df-enq 10335 df-nq 10336 df-erq 10337 df-mq 10339 df-1nq 10340

This theorem is referenced by: recmulnq 10388 halfnq 10400 ltrnq 10403 addclprlem2 10441 mulclprlem 10443 mulasspr 10448 1idpr 10453 prlem934 10457 prlem936 10471 reclem3pr 10473

W3C validator