Theorem mulassnq 10912

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 10850	. . . . . . 7 ⊢ (((1st ‘𝐴) ·N (1st ‘𝐵)) ·N (1st ‘𝐶)) = ((1st ‘𝐴) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))
2		mulasspi 10850	. . . . . . 7 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
3	1, 2	opeq12i 4842	. . . . . 6 ⊢ ⟨(((1st ‘𝐴) ·N (1st ‘𝐵)) ·N (1st ‘𝐶)), (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶))⟩ = ⟨((1st ‘𝐴) ·N ((1st ‘𝐵) ·N (1st ‘𝐶))), ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))⟩
4		elpqn 10878	. . . . . . . . . 10 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
6		elpqn 10878	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
7	6	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
8		mulpipq2 10892	. . . . . . . . 9 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
9	5, 7, 8	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
10		relxp 5656	. . . . . . . . 9 ⊢ Rel (N × N)
11		elpqn 10878	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
12	11	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
13		1st2nd 8018	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
14	10, 12, 13	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
15	9, 14	oveq12d 7405	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ·pQ ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩))
16		xp1st 8000	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
17	5, 16	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐴) ∈ N)
18		xp1st 8000	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
19	7, 18	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐵) ∈ N)
20		mulclpi 10846	. . . . . . . . 9 ⊢ (((1st ‘𝐴) ∈ N ∧ (1st ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
21	17, 19, 20	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
22		xp2nd 8001	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
23	5, 22	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐴) ∈ N)
24		xp2nd 8001	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
25	7, 24	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐵) ∈ N)
26		mulclpi 10846	. . . . . . . . 9 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
27	23, 25, 26	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
28		xp1st 8000	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
29	12, 28	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
30		xp2nd 8001	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
31	12, 30	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
32		mulpipq 10893	. . . . . . . 8 ⊢ (((((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N) ∧ ((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N)) → (⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ·pQ ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩) = ⟨(((1st ‘𝐴) ·N (1st ‘𝐵)) ·N (1st ‘𝐶)), (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶))⟩)
33	21, 27, 29, 31, 32	syl22anc 838	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ·pQ ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩) = ⟨(((1st ‘𝐴) ·N (1st ‘𝐵)) ·N (1st ‘𝐶)), (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶))⟩)
34	15, 33	eqtrd 2764	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = ⟨(((1st ‘𝐴) ·N (1st ‘𝐵)) ·N (1st ‘𝐶)), (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶))⟩)
35		1st2nd 8018	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
36	10, 5, 35	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
37		mulpipq2 10892	. . . . . . . . 9 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
38	7, 12, 37	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
39	36, 38	oveq12d 7405	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩))
40		mulclpi 10846	. . . . . . . . 9 ⊢ (((1st ‘𝐵) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
41	19, 29, 40	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
42		mulclpi 10846	. . . . . . . . 9 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
43	25, 31, 42	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
44		mulpipq 10893	. . . . . . . 8 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ (((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N ∧ ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩) = ⟨((1st ‘𝐴) ·N ((1st ‘𝐵) ·N (1st ‘𝐶))), ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))⟩)
45	17, 23, 41, 43, 44	syl22anc 838	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ·pQ ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩) = ⟨((1st ‘𝐴) ·N ((1st ‘𝐵) ·N (1st ‘𝐶))), ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))⟩)
46	39, 45	eqtrd 2764	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = ⟨((1st ‘𝐴) ·N ((1st ‘𝐵) ·N (1st ‘𝐶))), ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))⟩)
47	3, 34, 46	3eqtr4a 2790	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (𝐴 ·pQ (𝐵 ·pQ 𝐶)))
48	47	fveq2d 6862	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶)) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶))))
49		mulerpq 10910	. . . 4 ⊢ (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶))
50		mulerpq 10910	. . . 4 ⊢ (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶)))
51	48, 49, 50	3eqtr4g 2789	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
52		mulpqnq 10894	. . . . 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 10886	. . . . . 6 ⊢ (𝐶 ∈ Q → ([Q]‘𝐶) = 𝐶)
55	54	eqcomd 2735	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 = ([Q]‘𝐶))
56	55	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ([Q]‘𝐶))
57	53, 56	oveq12d 7405	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)))
58		nqerid 10886	. . . . . 6 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
59	58	eqcomd 2735	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 = ([Q]‘𝐴))
60	59	3ad2ant1 1133	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ([Q]‘𝐴))
61		mulpqnq 10894	. . . . 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 7405	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·Q (𝐵 ·Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
64	51, 57, 63	3eqtr4d 2774	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
65		mulnqf 10902	. . . 4 ⊢ ·Q :(Q × Q)⟶Q
66	65	fdmi 6699	. . 3 ⊢ dom ·Q = (Q × Q)
67		0nnq 10877	. . 3 ⊢ ¬ ∅ ∈ Q
68	66, 67	ndmovass 7577	. 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))

Syntax hints:   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   × cxp 5636  Rel wrel 5643  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Ncnpi 10797   ·_N cmi 10799   ·_pQ cmpq 10802  Qcnq 10805  [Q]cerq 10807   ·_Q cmq 10809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-omul 8439  df-er 8671  df-ni 10825  df-mi 10827  df-lti 10828  df-mpq 10862  df-enq 10864  df-nq 10865  df-erq 10866  df-mq 10868  df-1nq 10869

This theorem is referenced by:  recmulnq  10917  halfnq  10929  ltrnq  10932  addclprlem2  10970  mulclprlem  10972  mulasspr  10977  1idpr  10982  prlem934  10986  prlem936  11000  reclem3pr  11002