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Theorem mulpipqqs 7402
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 7357 . . . 4 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) ∈ N)
2 mulclpi 7357 . . . 4 ((𝐵N𝐷N) → (𝐵 ·N 𝐷) ∈ N)
3 opelxpi 4676 . . . 4 (((𝐴 ·N 𝐶) ∈ N ∧ (𝐵 ·N 𝐷) ∈ N) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
41, 2, 3syl2an 289 . . 3 (((𝐴N𝐶N) ∧ (𝐵N𝐷N)) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
54an4s 588 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
6 mulclpi 7357 . . . 4 ((𝑎N𝑔N) → (𝑎 ·N 𝑔) ∈ N)
7 mulclpi 7357 . . . 4 ((𝑏NN) → (𝑏 ·N ) ∈ N)
8 opelxpi 4676 . . . 4 (((𝑎 ·N 𝑔) ∈ N ∧ (𝑏 ·N ) ∈ N) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ∈ (N × N))
96, 7, 8syl2an 289 . . 3 (((𝑎N𝑔N) ∧ (𝑏NN)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ∈ (N × N))
109an4s 588 . 2 (((𝑎N𝑏N) ∧ (𝑔NN)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ∈ (N × N))
11 mulclpi 7357 . . . 4 ((𝑐N𝑡N) → (𝑐 ·N 𝑡) ∈ N)
12 mulclpi 7357 . . . 4 ((𝑑N𝑠N) → (𝑑 ·N 𝑠) ∈ N)
13 opelxpi 4676 . . . 4 (((𝑐 ·N 𝑡) ∈ N ∧ (𝑑 ·N 𝑠) ∈ N) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
1411, 12, 13syl2an 289 . . 3 (((𝑐N𝑡N) ∧ (𝑑N𝑠N)) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
1514an4s 588 . 2 (((𝑐N𝑑N) ∧ (𝑡N𝑠N)) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
16 enqex 7389 . 2 ~Q ∈ V
17 enqer 7387 . 2 ~Q Er (N × N)
18 df-enq 7376 . 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 527 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑧 = 𝑎)
20 simprr 531 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑢 = 𝑑)
2119, 20oveq12d 5914 . . 3 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
22 simplr 528 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑤 = 𝑏)
23 simprl 529 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑣 = 𝑐)
2422, 23oveq12d 5914 . . 3 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
2521, 24eqeq12d 2204 . 2 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
26 simpll 527 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑧 = 𝑔)
27 simprr 531 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑢 = 𝑠)
2826, 27oveq12d 5914 . . 3 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
29 simplr 528 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑤 = )
30 simprl 529 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑣 = 𝑡)
3129, 30oveq12d 5914 . . 3 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝑤 ·N 𝑣) = ( ·N 𝑡))
3228, 31eqeq12d 2204 . 2 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = ( ·N 𝑡)))
33 dfmpq2 7384 . 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 527 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑤 = 𝑎)
35 simprl 529 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑢 = 𝑔)
3634, 35oveq12d 5914 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → (𝑤 ·N 𝑢) = (𝑎 ·N 𝑔))
37 simplr 528 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑣 = 𝑏)
38 simprr 531 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑓 = )
3937, 38oveq12d 5914 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → (𝑣 ·N 𝑓) = (𝑏 ·N ))
4036, 39opeq12d 3801 . 2 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩)
41 simpll 527 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑤 = 𝑐)
42 simprl 529 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑢 = 𝑡)
4341, 42oveq12d 5914 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → (𝑤 ·N 𝑢) = (𝑐 ·N 𝑡))
44 simplr 528 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑣 = 𝑑)
45 simprr 531 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑓 = 𝑠)
4644, 45oveq12d 5914 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
4743, 46opeq12d 3801 . 2 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩)
48 simpll 527 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑤 = 𝐴)
49 simprl 529 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑢 = 𝐶)
5048, 49oveq12d 5914 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑤 ·N 𝑢) = (𝐴 ·N 𝐶))
51 simplr 528 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑣 = 𝐵)
52 simprr 531 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑓 = 𝐷)
5351, 52oveq12d 5914 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
5450, 53opeq12d 3801 . 2 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
55 df-mqqs 7379 . 2 ·Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑎𝑏𝑐𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ~Q𝑦 = [⟨𝑐, 𝑑⟩] ~Q ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ ·pQ𝑐, 𝑑⟩)] ~Q ))}
56 df-nqqs 7377 . 2 Q = ((N × N) / ~Q )
57 mulcmpblnq 7397 . 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 6667 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 104   = wceq 1364  wcel 2160  cop 3610   × cxp 4642  (class class class)co 5896  [cec 6557  Ncnpi 7301   ·N cmi 7303   ·pQ cmpq 7306   ~Q ceq 7308  Qcnq 7309   ·Q cmq 7312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-mi 7335  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-mqqs 7379
This theorem is referenced by:  mulclnq  7405  mulcomnqg  7412  mulassnqg  7413  distrnqg  7416  mulidnq  7418  recexnq  7419  ltmnqg  7430  nqnq0m  7484
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