Theorem mulassnq 11071

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 11009	. . . . . . 7 ⊢ (((1st ‘𝐴) ·N (1st ‘𝐵)) ·N (1st ‘𝐶)) = ((1st ‘𝐴) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))
2		mulasspi 11009	. . . . . . 7 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
3	1, 2	opeq12i 4848	. . . . . 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 11037	. . . . . . . . . 10 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
6		elpqn 11037	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
7	6	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
8		mulpipq2 11051	. . . . . . . . 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 596	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
10		relxp 5673	. . . . . . . . 9 ⊢ Rel (N × N)
11		elpqn 11037	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
12	11	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
13		1st2nd 8074	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
14	10, 12, 13	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
15	9, 14	oveq12d 7461	. . . . . . 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 8057	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
17	5, 16	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐴) ∈ N)
18		xp1st 8057	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
19	7, 18	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐵) ∈ N)
20		mulclpi 11005	. . . . . . . . 9 ⊢ (((1st ‘𝐴) ∈ N ∧ (1st ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
21	17, 19, 20	syl2anc 596	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
22		xp2nd 8058	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
23	5, 22	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐴) ∈ N)
24		xp2nd 8058	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
25	7, 24	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐵) ∈ N)
26		mulclpi 11005	. . . . . . . . 9 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
27	23, 25, 26	syl2anc 596	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
28		xp1st 8057	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
29	12, 28	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
30		xp2nd 8058	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
31	12, 30	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
32		mulpipq 11052	. . . . . . . 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 852	. . . . . . 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 2798	. . . . . 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 8074	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
36	10, 5, 35	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
37		mulpipq2 11051	. . . . . . . . 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 596	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
39	36, 38	oveq12d 7461	. . . . . . 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 11005	. . . . . . . . 9 ⊢ (((1st ‘𝐵) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
41	19, 29, 40	syl2anc 596	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
42		mulclpi 11005	. . . . . . . . 9 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
43	25, 31, 42	syl2anc 596	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
44		mulpipq 11052	. . . . . . . 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 852	. . . . . . 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 2798	. . . . . 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 2824	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (𝐴 ·pQ (𝐵 ·pQ 𝐶)))
48	47	fveq2d 6912	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶)) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶))))
49		mulerpq 11069	. . . 4 ⊢ (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶))
50		mulerpq 11069	. . . 4 ⊢ (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶)))
51	48, 49, 50	3eqtr4g 2823	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
52		mulpqnq 11053	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
53	52	3adant3 1150	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
54		nqerid 11045	. . . . . 6 ⊢ (𝐶 ∈ Q → ([Q]‘𝐶) = 𝐶)
55	54	eqcomd 2769	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 = ([Q]‘𝐶))
56	55	3ad2ant3 1153	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ([Q]‘𝐶))
57	53, 56	oveq12d 7461	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)))
58		nqerid 11045	. . . . . 6 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
59	58	eqcomd 2769	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 = ([Q]‘𝐴))
60	59	3ad2ant1 1151	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ([Q]‘𝐴))
61		mulpqnq 11053	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
62	61	3adant1 1148	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
63	60, 62	oveq12d 7461	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·Q (𝐵 ·Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
64	51, 57, 63	3eqtr4d 2808	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
65		mulnqf 11061	. . . 4 ⊢ ·Q :(Q × Q)⟶Q
66	65	fdmi 6743	. . 3 ⊢ dom ·Q = (Q × Q)
67		0nnq 11036	. . 3 ⊢ ¬ ∅ ∈ Q
68	66, 67	ndmovass 7633	. 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))

This proof depends on syntax axioms:   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ⟨cop 4838   × cxp 5653  Rel wrel 5660  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  Ncnpi 10956   ·_N cmi 10958   ·_pQ cmpq 10961  Qcnq 10964  [Q]cerq 10966   ·_Q cmq 10968

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-oadd 8508  df-omul 8509  df-er 8745  df-ni 10984  df-mi 10986  df-lti 10987  df-mpq 11021  df-enq 11023  df-nq 11024  df-erq 11025  df-mq 11027  df-1nq 11028

This theorem is used by:  recmulnq  11076  halfnq  11088  ltrnq  11091  addclprlem2  11129  mulclprlem  11131  mulasspr  11136  1idpr  11141  prlem934  11145  prlem936  11159  reclem3pr  11161