Theorem mulassnq 10984

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 10922	. . . . . . 7 ⊢ (((1st ‘𝐴) ·N (1st ‘𝐵)) ·N (1st ‘𝐶)) = ((1st ‘𝐴) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))
2		mulasspi 10922	. . . . . . 7 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
3	1, 2	opeq12i 4880	. . . . . 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 10950	. . . . . . . . . 10 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
6		elpqn 10950	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
7	6	3ad2ant2 1131	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
8		mulpipq2 10964	. . . . . . . . 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 582	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
10		relxp 5696	. . . . . . . . 9 ⊢ Rel (N × N)
11		elpqn 10950	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
12	11	3ad2ant3 1132	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
13		1st2nd 8044	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
14	10, 12, 13	sylancr 585	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
15	9, 14	oveq12d 7437	. . . . . . 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 8026	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
17	5, 16	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐴) ∈ N)
18		xp1st 8026	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
19	7, 18	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐵) ∈ N)
20		mulclpi 10918	. . . . . . . . 9 ⊢ (((1st ‘𝐴) ∈ N ∧ (1st ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
21	17, 19, 20	syl2anc 582	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
22		xp2nd 8027	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
23	5, 22	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐴) ∈ N)
24		xp2nd 8027	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
25	7, 24	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐵) ∈ N)
26		mulclpi 10918	. . . . . . . . 9 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
27	23, 25, 26	syl2anc 582	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
28		xp1st 8026	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
29	12, 28	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
30		xp2nd 8027	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
31	12, 30	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
32		mulpipq 10965	. . . . . . . 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 837	. . . . . . 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 2765	. . . . . 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 8044	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
36	10, 5, 35	sylancr 585	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
37		mulpipq2 10964	. . . . . . . . 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 582	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
39	36, 38	oveq12d 7437	. . . . . . 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 10918	. . . . . . . . 9 ⊢ (((1st ‘𝐵) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
41	19, 29, 40	syl2anc 582	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
42		mulclpi 10918	. . . . . . . . 9 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
43	25, 31, 42	syl2anc 582	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
44		mulpipq 10965	. . . . . . . 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 837	. . . . . . 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 2765	. . . . . 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 2791	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (𝐴 ·pQ (𝐵 ·pQ 𝐶)))
48	47	fveq2d 6900	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶)) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶))))
49		mulerpq 10982	. . . 4 ⊢ (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶))
50		mulerpq 10982	. . . 4 ⊢ (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶)))
51	48, 49, 50	3eqtr4g 2790	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
52		mulpqnq 10966	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
53	52	3adant3 1129	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
54		nqerid 10958	. . . . . 6 ⊢ (𝐶 ∈ Q → ([Q]‘𝐶) = 𝐶)
55	54	eqcomd 2731	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 = ([Q]‘𝐶))
56	55	3ad2ant3 1132	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ([Q]‘𝐶))
57	53, 56	oveq12d 7437	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)))
58		nqerid 10958	. . . . . 6 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
59	58	eqcomd 2731	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 = ([Q]‘𝐴))
60	59	3ad2ant1 1130	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ([Q]‘𝐴))
61		mulpqnq 10966	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
62	61	3adant1 1127	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
63	60, 62	oveq12d 7437	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·Q (𝐵 ·Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
64	51, 57, 63	3eqtr4d 2775	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
65		mulnqf 10974	. . . 4 ⊢ ·Q :(Q × Q)⟶Q
66	65	fdmi 6734	. . 3 ⊢ dom ·Q = (Q × Q)
67		0nnq 10949	. . 3 ⊢ ¬ ∅ ∈ Q
68	66, 67	ndmovass 7609	. 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 1084   = wceq 1533   ∈ wcel 2098  ⟨cop 4636   × cxp 5676  Rel wrel 5683  ‘cfv 6549  (class class class)co 7419  1^st c1st 7992  2^nd c2nd 7993  Ncnpi 10869   ·_N cmi 10871   ·_pQ cmpq 10874  Qcnq 10877  [Q]cerq 10879   ·_Q cmq 10881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-er 8725  df-ni 10897  df-mi 10899  df-lti 10900  df-mpq 10934  df-enq 10936  df-nq 10937  df-erq 10938  df-mq 10940  df-1nq 10941

This theorem is referenced by:  recmulnq  10989  halfnq  11001  ltrnq  11004  addclprlem2  11042  mulclprlem  11044  mulasspr  11049  1idpr  11054  prlem934  11058  prlem936  11072  reclem3pr  11074