Theorem mulassnq 10932

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 10870	. . . . . . 7 ⊢ (((1st ‘𝐴) ·N (1st ‘𝐵)) ·N (1st ‘𝐶)) = ((1st ‘𝐴) ·N ((1st ‘𝐵) ·N (1st ‘𝐶)))
2		mulasspi 10870	. . . . . . 7 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐵)) ·N (2nd ‘𝐶)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐵) ·N (2nd ‘𝐶)))
3	1, 2	opeq12i 4839	. . . . . 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 10898	. . . . . . . . . 10 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	3ad2ant1 1149	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
6		elpqn 10898	. . . . . . . . . 10 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
7	6	3ad2ant2 1150	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
8		mulpipq2 10912	. . . . . . . . 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 595	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
10		relxp 5670	. . . . . . . . 9 ⊢ Rel (N × N)
11		elpqn 10898	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
12	11	3ad2ant3 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
13		1st2nd 8024	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
14	10, 12, 13	sylancr 598	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
15	9, 14	oveq12d 7418	. . . . . . 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 8006	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
17	5, 16	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐴) ∈ N)
18		xp1st 8006	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
19	7, 18	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐵) ∈ N)
20		mulclpi 10866	. . . . . . . . 9 ⊢ (((1st ‘𝐴) ∈ N ∧ (1st ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
21	17, 19, 20	syl2anc 595	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
22		xp2nd 8007	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
23	5, 22	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐴) ∈ N)
24		xp2nd 8007	. . . . . . . . . 10 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
25	7, 24	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐵) ∈ N)
26		mulclpi 10866	. . . . . . . . 9 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
27	23, 25, 26	syl2anc 595	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
28		xp1st 8006	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
29	12, 28	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
30		xp2nd 8007	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
31	12, 30	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
32		mulpipq 10913	. . . . . . . 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 851	. . . . . . 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 2800	. . . . . 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 8024	. . . . . . . . 9 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
36	10, 5, 35	sylancr 598	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
37		mulpipq2 10912	. . . . . . . . 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 595	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·pQ 𝐶) = ⟨((1st ‘𝐵) ·N (1st ‘𝐶)), ((2nd ‘𝐵) ·N (2nd ‘𝐶))⟩)
39	36, 38	oveq12d 7418	. . . . . . 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 10866	. . . . . . . . 9 ⊢ (((1st ‘𝐵) ∈ N ∧ (1st ‘𝐶) ∈ N) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
41	19, 29, 40	syl2anc 595	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐵) ·N (1st ‘𝐶)) ∈ N)
42		mulclpi 10866	. . . . . . . . 9 ⊢ (((2nd ‘𝐵) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
43	25, 31, 42	syl2anc 595	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐵) ·N (2nd ‘𝐶)) ∈ N)
44		mulpipq 10913	. . . . . . . 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 851	. . . . . . 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 2800	. . . . . 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 2826	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (𝐴 ·pQ (𝐵 ·pQ 𝐶)))
48	47	fveq2d 6875	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶)) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶))))
49		mulerpq 10930	. . . 4 ⊢ (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶))
50		mulerpq 10930	. . . 4 ⊢ (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶)))
51	48, 49, 50	3eqtr4g 2825	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
52		mulpqnq 10914	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
53	52	3adant3 1148	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
54		nqerid 10906	. . . . . 6 ⊢ (𝐶 ∈ Q → ([Q]‘𝐶) = 𝐶)
55	54	eqcomd 2771	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 = ([Q]‘𝐶))
56	55	3ad2ant3 1151	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 = ([Q]‘𝐶))
57	53, 56	oveq12d 7418	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)))
58		nqerid 10906	. . . . . 6 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
59	58	eqcomd 2771	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 = ([Q]‘𝐴))
60	59	3ad2ant1 1149	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 = ([Q]‘𝐴))
61		mulpqnq 10914	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
62	61	3adant1 1146	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
63	60, 62	oveq12d 7418	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·Q (𝐵 ·Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
64	51, 57, 63	3eqtr4d 2810	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
65		mulnqf 10922	. . . 4 ⊢ ·Q :(Q × Q)⟶Q
66	65	fdmi 6707	. . 3 ⊢ dom ·Q = (Q × Q)
67		0nnq 10897	. . 3 ⊢ ¬ ∅ ∈ Q
68	66, 67	ndmovass 7588	. 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))

Syntax hints:   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ⟨cop 4591   × cxp 5650  Rel wrel 5657  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  Ncnpi 10817   ·_N cmi 10819   ·_pQ cmpq 10822  Qcnq 10825  [Q]cerq 10827   ·_Q cmq 10829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-omul 8446  df-er 8682  df-ni 10845  df-mi 10847  df-lti 10848  df-mpq 10882  df-enq 10884  df-nq 10885  df-erq 10886  df-mq 10888  df-1nq 10889

This theorem is referenced by:  recmulnq  10937  halfnq  10949  ltrnq  10952  addclprlem2  10990  mulclprlem  10992  mulasspr  10997  1idpr  11002  prlem934  11006  prlem936  11020  reclem3pr  11022