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Theorem mulassnq 10116
 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 10054 . . . . . . 7 (((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)) = ((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶)))
2 mulasspi 10054 . . . . . . 7 (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶)) = ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))
31, 2opeq12i 4641 . . . . . 6 ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩ = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩
4 elpqn 10082 . . . . . . . . . 10 (𝐴Q𝐴 ∈ (N × N))
543ad2ant1 1124 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
6 elpqn 10082 . . . . . . . . . 10 (𝐵Q𝐵 ∈ (N × N))
763ad2ant2 1125 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
8 mulpipq2 10096 . . . . . . . . 9 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
95, 7, 8syl2anc 579 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
10 relxp 5373 . . . . . . . . 9 Rel (N × N)
11 elpqn 10082 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
12113ad2ant3 1126 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
13 1st2nd 7493 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐶 ∈ (N × N)) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
1410, 12, 13sylancr 581 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
159, 14oveq12d 6940 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ·pQ ⟨(1st𝐶), (2nd𝐶)⟩))
16 xp1st 7477 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
175, 16syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (1st𝐴) ∈ N)
18 xp1st 7477 . . . . . . . . . 10 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
197, 18syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (1st𝐵) ∈ N)
20 mulclpi 10050 . . . . . . . . 9 (((1st𝐴) ∈ N ∧ (1st𝐵) ∈ N) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
2117, 19, 20syl2anc 579 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐴) ·N (1st𝐵)) ∈ N)
22 xp2nd 7478 . . . . . . . . . 10 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
235, 22syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
24 xp2nd 7478 . . . . . . . . . 10 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
257, 24syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
26 mulclpi 10050 . . . . . . . . 9 (((2nd𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
2723, 25, 26syl2anc 579 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐴) ·N (2nd𝐵)) ∈ N)
28 xp1st 7477 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
2912, 28syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
30 xp2nd 7478 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
3112, 30syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
32 mulpipq 10097 . . . . . . . 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 829 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ·pQ ⟨(1st𝐶), (2nd𝐶)⟩) = ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
3415, 33eqtrd 2814 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = ⟨(((1st𝐴) ·N (1st𝐵)) ·N (1st𝐶)), (((2nd𝐴) ·N (2nd𝐵)) ·N (2nd𝐶))⟩)
35 1st2nd 7493 . . . . . . . . 9 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
3610, 5, 35sylancr 581 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
37 mulpipq2 10096 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
387, 12, 37syl2anc 579 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
3936, 38oveq12d 6940 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩))
40 mulclpi 10050 . . . . . . . . 9 (((1st𝐵) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
4119, 29, 40syl2anc 579 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
42 mulclpi 10050 . . . . . . . . 9 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
4325, 31, 42syl2anc 579 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
44 mulpipq 10097 . . . . . . . 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 829 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (⟨(1st𝐴), (2nd𝐴)⟩ ·pQ ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩) = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
4639, 45eqtrd 2814 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·pQ (𝐵 ·pQ 𝐶)) = ⟨((1st𝐴) ·N ((1st𝐵) ·N (1st𝐶))), ((2nd𝐴) ·N ((2nd𝐵) ·N (2nd𝐶)))⟩)
473, 34, 463eqtr4a 2840 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·pQ 𝐵) ·pQ 𝐶) = (𝐴 ·pQ (𝐵 ·pQ 𝐶)))
4847fveq2d 6450 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶)) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶))))
49 mulerpq 10114 . . . 4 (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = ([Q]‘((𝐴 ·pQ 𝐵) ·pQ 𝐶))
50 mulerpq 10114 . . . 4 (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))) = ([Q]‘(𝐴 ·pQ (𝐵 ·pQ 𝐶)))
5148, 49, 503eqtr4g 2839 . . 3 ((𝐴Q𝐵Q𝐶Q) → (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
52 mulpqnq 10098 . . . . 5 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
53523adant3 1123 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
54 nqerid 10090 . . . . . 6 (𝐶Q → ([Q]‘𝐶) = 𝐶)
5554eqcomd 2784 . . . . 5 (𝐶Q𝐶 = ([Q]‘𝐶))
56553ad2ant3 1126 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐶 = ([Q]‘𝐶))
5753, 56oveq12d 6940 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (([Q]‘(𝐴 ·pQ 𝐵)) ·Q ([Q]‘𝐶)))
58 nqerid 10090 . . . . . 6 (𝐴Q → ([Q]‘𝐴) = 𝐴)
5958eqcomd 2784 . . . . 5 (𝐴Q𝐴 = ([Q]‘𝐴))
60593ad2ant1 1124 . . . 4 ((𝐴Q𝐵Q𝐶Q) → 𝐴 = ([Q]‘𝐴))
61 mulpqnq 10098 . . . . 5 ((𝐵Q𝐶Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
62613adant1 1121 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (𝐵 ·Q 𝐶) = ([Q]‘(𝐵 ·pQ 𝐶)))
6360, 62oveq12d 6940 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 ·Q 𝐶)) = (([Q]‘𝐴) ·Q ([Q]‘(𝐵 ·pQ 𝐶))))
6451, 57, 633eqtr4d 2824 . 2 ((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
65 mulnqf 10106 . . . 4 ·Q :(Q × Q)⟶Q
6665fdmi 6301 . . 3 dom ·Q = (Q × Q)
67 0nnq 10081 . . 3 ¬ ∅ ∈ Q
6866, 67ndmovass 7099 . 2 (¬ (𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
6964, 68pm2.61i 177 1 ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ⟨cop 4404   × cxp 5353  Rel wrel 5360  ‘cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  Ncnpi 10001   ·N cmi 10003   ·pQ cmpq 10006  Qcnq 10009  [Q]cerq 10011   ·Q cmq 10013 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-omul 7848  df-er 8026  df-ni 10029  df-mi 10031  df-lti 10032  df-mpq 10066  df-enq 10068  df-nq 10069  df-erq 10070  df-mq 10072  df-1nq 10073 This theorem is referenced by:  recmulnq  10121  halfnq  10133  ltrnq  10136  addclprlem2  10174  mulclprlem  10176  mulasspr  10181  1idpr  10186  prlem934  10190  prlem936  10204  reclem3pr  10206
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