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Theorem mulassnq 9469
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
StepHypRef Expression
1 mulasspi 9407 . . . . . . 7 (((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)) = ((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶)))
2 mulasspi 9407 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
31, 2opeq12i 4201 . . . . . 6 ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩ = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩
4 elpqn 9435 . . . . . . . . . 10 (𝐴Q𝐴 ∈ (N × N))
543ad2ant1 1065 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
6 elpqn 9435 . . . . . . . . . 10 (𝐵Q𝐵 ∈ (N × N))
763ad2ant2 1066 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
8 mulpipq2 9449 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
95, 7, 8syl2anc 682 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
10 relxp 4989 . . . . . . . . 9 Rel (N × N)
11 elpqn 9435 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
12113ad2ant3 1067 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
13 1st2nd 6916 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
1410, 12, 13sylancr 685 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
159, 14oveq12d 6381 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ·pQ ⟨(1st𝐶), (2nd𝐶)⟩))
16 xp1st 6900 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
175, 16syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
18 xp1st 6900 . . . . . . . . . 10 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
197, 18syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
20 mulclpi 9403 . . . . . . . . 9 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
2117, 19, 20syl2anc 682 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
22 xp2nd 6901 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
235, 22syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
24 xp2nd 6901 . . . . . . . . . 10 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
257, 24syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
26 mulclpi 9403 . . . . . . . . 9 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
2723, 25, 26syl2anc 682 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
28 xp1st 6900 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
2912, 28syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
30 xp2nd 6901 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
3112, 30syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
32 mulpipq 9450 . . . . . . . 8 (((((1st𝐴) ·N (1st𝐵)) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐵)) ∈ N) ∧ ((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N)) → (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ·pQ ⟨(1st𝐶), (2nd𝐶)⟩) = ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
3321, 27, 29, 31, 32syl22anc 1309 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ·pQ ⟨(1st𝐶), (2nd𝐶)⟩) = ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
3415, 33eqtrd 2539 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
35 1st2nd 6916 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
3610, 5, 35sylancr 685 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
37 mulpipq2 9449 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
387, 12, 37syl2anc 682 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
3936, 38oveq12d 6381 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩))
40 mulclpi 9403 . . . . . . . . 9 (((1st𝐵) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
4119, 29, 40syl2anc 682 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
42 mulclpi 9403 . . . . . . . . 9 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
4325, 31, 42syl2anc 682 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
44 mulpipq 9450 . . . . . . . 8 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ (((1st𝐵) ·N (1st𝐶)) ∈ N ∧ ((2nd𝐵) ·N (2nd𝐶)) ∈ N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩) = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
4517, 23, 41, 43, 44syl22anc 1309 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩) = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
4639, 45eqtrd 2539 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
473, 34, 463eqtr4a 2565 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (𝐴 ·pQ (𝐵 ·pQ 𝐶)))
4847fveq2d 5931 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶)) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶))))
49 mulerpq 9467 . . . 4 (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶))
50 mulerpq 9467 . . . 4 (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶)))
5148, 49, 503eqtr4g 2564 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
52 mulpqnq 9451 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
53523adant3 1064 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
54 nqerid 9443 . . . . . 6 (𝐶Q → ([Q]‘𝐶) = 𝐶)
5554eqcomd 2511 . . . . 5 (𝐶Q𝐶 = ([Q]‘𝐶))
56553ad2ant3 1067 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ([Q]‘𝐶))
5753, 56oveq12d 6381 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)))
58 nqerid 9443 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
5958eqcomd 2511 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
60593ad2ant1 1065 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
61 mulpqnq 9451 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
62613adant1 1062 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
6360, 62oveq12d 6381 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 ·Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
6451, 57, 633eqtr4d 2549 . 2 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
65 mulnqf 9459 . . . 4 ·Q :(Q × Q)⟶Q
6665fdmi 5791 . . 3 dom ·Q = (Q × Q)
67 0nnq 9434 . . 3 ¬ ∅ ∈ Q
6866, 67ndmovass 6532 . 2 (¬ (𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
6964, 68pm2.61i 171 1 ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶))
Colors of variables: wff setvar class
Syntax hints:  w3a 1021   = wceq 1468  wcel 1937  cop 4001   × cxp 4878  Rel wrel 4885  cfv 5633  (class class class)co 6363  1st c1st 6868  2nd c2nd 6869  Ncnpi 9354   ·N cmi 9356   ·pQ cmpq 9359  Qcnq 9362  [Q]cerq 9364   ·Q cmq 9366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-ov 6366  df-oprab 6367  df-mpt2 6368  df-om 6770  df-1st 6870  df-2nd 6871  df-wrecs 7105  df-recs 7167  df-rdg 7205  df-1o 7259  df-oadd 7263  df-omul 7264  df-er 7440  df-ni 9382  df-mi 9384  df-lti 9385  df-mpq 9419  df-enq 9421  df-nq 9422  df-erq 9423  df-mq 9425  df-1nq 9426
This theorem is referenced by:  recmulnq  9474  halfnq  9486  ltrnq  9489  addclprlem2  9527  mulclprlem  9529  mulasspr  9534  1idpr  9539  prlem934  9543  prlem936  9557  reclem3pr  9559
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