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Theorem mulassnq 9637
 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 9575 . . . . . . 7 (((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)) = ((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶)))
2 mulasspi 9575 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
31, 2opeq12i 4339 . . . . . 6 ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩ = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩
4 elpqn 9603 . . . . . . . . . 10 (𝐴Q𝐴 ∈ (N × N))
543ad2ant1 1074 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
6 elpqn 9603 . . . . . . . . . 10 (𝐵Q𝐵 ∈ (N × N))
763ad2ant2 1075 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
8 mulpipq2 9617 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
95, 7, 8syl2anc 690 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
10 relxp 5139 . . . . . . . . 9 Rel (N × N)
11 elpqn 9603 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
12113ad2ant3 1076 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
13 1st2nd 7082 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
1410, 12, 13sylancr 693 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
159, 14oveq12d 6545 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ·pQ ⟨(1st𝐶), (2nd𝐶)⟩))
16 xp1st 7066 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
175, 16syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
18 xp1st 7066 . . . . . . . . . 10 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
197, 18syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
20 mulclpi 9571 . . . . . . . . 9 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
2117, 19, 20syl2anc 690 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
22 xp2nd 7067 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
235, 22syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
24 xp2nd 7067 . . . . . . . . . 10 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
257, 24syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
26 mulclpi 9571 . . . . . . . . 9 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
2723, 25, 26syl2anc 690 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
28 xp1st 7066 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
2912, 28syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
30 xp2nd 7067 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
3112, 30syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
32 mulpipq 9618 . . . . . . . 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 1318 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ·pQ ⟨(1st𝐶), (2nd𝐶)⟩) = ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
3415, 33eqtrd 2643 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
35 1st2nd 7082 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
3610, 5, 35sylancr 693 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
37 mulpipq2 9617 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
387, 12, 37syl2anc 690 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
3936, 38oveq12d 6545 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩))
40 mulclpi 9571 . . . . . . . . 9 (((1st𝐵) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
4119, 29, 40syl2anc 690 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
42 mulclpi 9571 . . . . . . . . 9 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
4325, 31, 42syl2anc 690 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
44 mulpipq 9618 . . . . . . . 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 1318 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩) = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
4639, 45eqtrd 2643 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
473, 34, 463eqtr4a 2669 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (𝐴 ·pQ (𝐵 ·pQ 𝐶)))
4847fveq2d 6092 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶)) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶))))
49 mulerpq 9635 . . . 4 (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶))
50 mulerpq 9635 . . . 4 (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶)))
5148, 49, 503eqtr4g 2668 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
52 mulpqnq 9619 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
53523adant3 1073 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
54 nqerid 9611 . . . . . 6 (𝐶Q → ([Q]‘𝐶) = 𝐶)
5554eqcomd 2615 . . . . 5 (𝐶Q𝐶 = ([Q]‘𝐶))
56553ad2ant3 1076 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ([Q]‘𝐶))
5753, 56oveq12d 6545 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)))
58 nqerid 9611 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
5958eqcomd 2615 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
60593ad2ant1 1074 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
61 mulpqnq 9619 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
62613adant1 1071 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
6360, 62oveq12d 6545 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 ·Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
6451, 57, 633eqtr4d 2653 . 2 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
65 mulnqf 9627 . . . 4 ·Q :(Q × Q)⟶Q
6665fdmi 5951 . . 3 dom ·Q = (Q × Q)
67 0nnq 9602 . . 3 ¬ ∅ ∈ Q
6866, 67ndmovass 6697 . 2 (¬ (𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
6964, 68pm2.61i 174 1 ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∧ w3a 1030   = wceq 1474   ∈ wcel 1976  ⟨cop 4130   × cxp 5026  Rel wrel 5033  ‘cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  Ncnpi 9522   ·N cmi 9524   ·pQ cmpq 9527  Qcnq 9530  [Q]cerq 9532   ·Q cmq 9534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ni 9550  df-mi 9552  df-lti 9553  df-mpq 9587  df-enq 9589  df-nq 9590  df-erq 9591  df-mq 9593  df-1nq 9594 This theorem is referenced by:  recmulnq  9642  halfnq  9654  ltrnq  9657  addclprlem2  9695  mulclprlem  9697  mulasspr  9702  1idpr  9707  prlem934  9711  prlem936  9725  reclem3pr  9727
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