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Theorem mulassnq 9819
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 9757 . . . . . . 7 (((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)) = ((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶)))
2 mulasspi 9757 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
31, 2opeq12i 4438 . . . . . 6 ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩ = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩
4 elpqn 9785 . . . . . . . . . 10 (𝐴Q𝐴 ∈ (N × N))
543ad2ant1 1102 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
6 elpqn 9785 . . . . . . . . . 10 (𝐵Q𝐵 ∈ (N × N))
763ad2ant2 1103 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
8 mulpipq2 9799 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
95, 7, 8syl2anc 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
10 relxp 5160 . . . . . . . . 9 Rel (N × N)
11 elpqn 9785 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
12113ad2ant3 1104 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
13 1st2nd 7258 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
1410, 12, 13sylancr 696 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
159, 14oveq12d 6708 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ·pQ ⟨(1st𝐶), (2nd𝐶)⟩))
16 xp1st 7242 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
175, 16syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
18 xp1st 7242 . . . . . . . . . 10 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
197, 18syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
20 mulclpi 9753 . . . . . . . . 9 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
2117, 19, 20syl2anc 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
22 xp2nd 7243 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
235, 22syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
24 xp2nd 7243 . . . . . . . . . 10 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
257, 24syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
26 mulclpi 9753 . . . . . . . . 9 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
2723, 25, 26syl2anc 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
28 xp1st 7242 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
2912, 28syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
30 xp2nd 7243 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
3112, 30syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
32 mulpipq 9800 . . . . . . . 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 1367 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ·pQ ⟨(1st𝐶), (2nd𝐶)⟩) = ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
3415, 33eqtrd 2685 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
35 1st2nd 7258 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
3610, 5, 35sylancr 696 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
37 mulpipq2 9799 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
387, 12, 37syl2anc 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
3936, 38oveq12d 6708 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩))
40 mulclpi 9753 . . . . . . . . 9 (((1st𝐵) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
4119, 29, 40syl2anc 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
42 mulclpi 9753 . . . . . . . . 9 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
4325, 31, 42syl2anc 694 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
44 mulpipq 9800 . . . . . . . 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 1367 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩) = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
4639, 45eqtrd 2685 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
473, 34, 463eqtr4a 2711 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (𝐴 ·pQ (𝐵 ·pQ 𝐶)))
4847fveq2d 6233 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶)) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶))))
49 mulerpq 9817 . . . 4 (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶))
50 mulerpq 9817 . . . 4 (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶)))
5148, 49, 503eqtr4g 2710 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
52 mulpqnq 9801 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
53523adant3 1101 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
54 nqerid 9793 . . . . . 6 (𝐶Q → ([Q]‘𝐶) = 𝐶)
5554eqcomd 2657 . . . . 5 (𝐶Q𝐶 = ([Q]‘𝐶))
56553ad2ant3 1104 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ([Q]‘𝐶))
5753, 56oveq12d 6708 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)))
58 nqerid 9793 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
5958eqcomd 2657 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
60593ad2ant1 1102 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
61 mulpqnq 9801 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
62613adant1 1099 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
6360, 62oveq12d 6708 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 ·Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
6451, 57, 633eqtr4d 2695 . 2 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
65 mulnqf 9809 . . . 4 ·Q :(Q × Q)⟶Q
6665fdmi 6090 . . 3 dom ·Q = (Q × Q)
67 0nnq 9784 . . 3 ¬ ∅ ∈ Q
6866, 67ndmovass 6864 . 2 (¬ (𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
6964, 68pm2.61i 176 1 ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶))
Colors of variables: wff setvar class
Syntax hints:  w3a 1054   = wceq 1523  wcel 2030  cop 4216   × cxp 5141  Rel wrel 5148  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Ncnpi 9704   ·N cmi 9706   ·pQ cmpq 9709  Qcnq 9712  [Q]cerq 9714   ·Q cmq 9716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-mi 9734  df-lti 9735  df-mpq 9769  df-enq 9771  df-nq 9772  df-erq 9773  df-mq 9775  df-1nq 9776
This theorem is referenced by:  recmulnq  9824  halfnq  9836  ltrnq  9839  addclprlem2  9877  mulclprlem  9879  mulasspr  9884  1idpr  9889  prlem934  9893  prlem936  9907  reclem3pr  9909
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