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Theorem mulpipqqs 6471
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
Assertion
Ref Expression
mulpipqqs (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q ·Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩] ~Q )

Proof of Theorem mulpipqqs
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑡 𝑠 𝑓 𝑔 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpi 6426 . . . 4 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) ∈ N)
2 mulclpi 6426 . . . 4 ((𝐵N𝐷N) → (𝐵 ·N 𝐷) ∈ N)
3 opelxpi 4376 . . . 4 (((𝐴 ·N 𝐶) ∈ N ∧ (𝐵 ·N 𝐷) ∈ N) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
41, 2, 3syl2an 273 . . 3 (((𝐴N𝐶N) ∧ (𝐵N𝐷N)) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
54an4s 522 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
6 mulclpi 6426 . . . 4 ((𝑎N𝑔N) → (𝑎 ·N 𝑔) ∈ N)
7 mulclpi 6426 . . . 4 ((𝑏NN) → (𝑏 ·N ) ∈ N)
8 opelxpi 4376 . . . 4 (((𝑎 ·N 𝑔) ∈ N ∧ (𝑏 ·N ) ∈ N) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ∈ (N × N))
96, 7, 8syl2an 273 . . 3 (((𝑎N𝑔N) ∧ (𝑏NN)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ∈ (N × N))
109an4s 522 . 2 (((𝑎N𝑏N) ∧ (𝑔NN)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ∈ (N × N))
11 mulclpi 6426 . . . 4 ((𝑐N𝑡N) → (𝑐 ·N 𝑡) ∈ N)
12 mulclpi 6426 . . . 4 ((𝑑N𝑠N) → (𝑑 ·N 𝑠) ∈ N)
13 opelxpi 4376 . . . 4 (((𝑐 ·N 𝑡) ∈ N ∧ (𝑑 ·N 𝑠) ∈ N) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
1411, 12, 13syl2an 273 . . 3 (((𝑐N𝑡N) ∧ (𝑑N𝑠N)) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
1514an4s 522 . 2 (((𝑐N𝑑N) ∧ (𝑡N𝑠N)) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
16 enqex 6458 . 2 ~Q ∈ V
17 enqer 6456 . 2 ~Q Er (N × N)
18 df-enq 6445 . 2 ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
19 simpll 481 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑧 = 𝑎)
20 simprr 484 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑢 = 𝑑)
2119, 20oveq12d 5530 . . 3 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
22 simplr 482 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑤 = 𝑏)
23 simprl 483 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑣 = 𝑐)
2422, 23oveq12d 5530 . . 3 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
2521, 24eqeq12d 2054 . 2 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
26 simpll 481 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑧 = 𝑔)
27 simprr 484 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑢 = 𝑠)
2826, 27oveq12d 5530 . . 3 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
29 simplr 482 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑤 = )
30 simprl 483 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑣 = 𝑡)
3129, 30oveq12d 5530 . . 3 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝑤 ·N 𝑣) = ( ·N 𝑡))
3228, 31eqeq12d 2054 . 2 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = ( ·N 𝑡)))
33 dfmpq2 6453 . 2 ·pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))}
34 simpll 481 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑤 = 𝑎)
35 simprl 483 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑢 = 𝑔)
3634, 35oveq12d 5530 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → (𝑤 ·N 𝑢) = (𝑎 ·N 𝑔))
37 simplr 482 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑣 = 𝑏)
38 simprr 484 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑓 = )
3937, 38oveq12d 5530 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → (𝑣 ·N 𝑓) = (𝑏 ·N ))
4036, 39opeq12d 3557 . 2 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩)
41 simpll 481 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑤 = 𝑐)
42 simprl 483 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑢 = 𝑡)
4341, 42oveq12d 5530 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → (𝑤 ·N 𝑢) = (𝑐 ·N 𝑡))
44 simplr 482 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑣 = 𝑑)
45 simprr 484 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑓 = 𝑠)
4644, 45oveq12d 5530 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
4743, 46opeq12d 3557 . 2 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩)
48 simpll 481 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑤 = 𝐴)
49 simprl 483 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑢 = 𝐶)
5048, 49oveq12d 5530 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑤 ·N 𝑢) = (𝐴 ·N 𝐶))
51 simplr 482 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑣 = 𝐵)
52 simprr 484 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑓 = 𝐷)
5351, 52oveq12d 5530 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
5450, 53opeq12d 3557 . 2 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
55 df-mqqs 6448 . 2 ·Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑎𝑏𝑐𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ~Q𝑦 = [⟨𝑐, 𝑑⟩] ~Q ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ ·pQ𝑐, 𝑑⟩)] ~Q ))}
56 df-nqqs 6446 . 2 Q = ((N × N) / ~Q )
57 mulcmpblnq 6466 . 2 ((((𝑎N𝑏N) ∧ (𝑐N𝑑N)) ∧ ((𝑔NN) ∧ (𝑡N𝑠N))) → (((𝑎 ·N 𝑑) = (𝑏 ·N 𝑐) ∧ (𝑔 ·N 𝑠) = ( ·N 𝑡)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ~Q ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩))
585, 10, 15, 16, 17, 18, 25, 32, 33, 40, 47, 54, 55, 56, 57oviec 6212 1 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q ·Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩] ~Q )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  cop 3378   × cxp 4343  (class class class)co 5512  [cec 6104  Ncnpi 6370   ·N cmi 6372   ·pQ cmpq 6375   ~Q ceq 6377  Qcnq 6378   ·Q cmq 6381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448
This theorem is referenced by:  mulclnq  6474  mulcomnqg  6481  mulassnqg  6482  distrnqg  6485  mulidnq  6487  recexnq  6488  ltmnqg  6499  nqnq0m  6553
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