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Theorem mulpipqqs 6911
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 6866 . . . 4 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) ∈ N)
2 mulclpi 6866 . . . 4 ((𝐵N𝐷N) → (𝐵 ·N 𝐷) ∈ N)
3 opelxpi 4459 . . . 4 (((𝐴 ·N 𝐶) ∈ N ∧ (𝐵 ·N 𝐷) ∈ N) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
41, 2, 3syl2an 283 . . 3 (((𝐴N𝐶N) ∧ (𝐵N𝐷N)) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
54an4s 555 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
6 mulclpi 6866 . . . 4 ((𝑎N𝑔N) → (𝑎 ·N 𝑔) ∈ N)
7 mulclpi 6866 . . . 4 ((𝑏NN) → (𝑏 ·N ) ∈ N)
8 opelxpi 4459 . . . 4 (((𝑎 ·N 𝑔) ∈ N ∧ (𝑏 ·N ) ∈ N) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ∈ (N × N))
96, 7, 8syl2an 283 . . 3 (((𝑎N𝑔N) ∧ (𝑏NN)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ∈ (N × N))
109an4s 555 . 2 (((𝑎N𝑏N) ∧ (𝑔NN)) → ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩ ∈ (N × N))
11 mulclpi 6866 . . . 4 ((𝑐N𝑡N) → (𝑐 ·N 𝑡) ∈ N)
12 mulclpi 6866 . . . 4 ((𝑑N𝑠N) → (𝑑 ·N 𝑠) ∈ N)
13 opelxpi 4459 . . . 4 (((𝑐 ·N 𝑡) ∈ N ∧ (𝑑 ·N 𝑠) ∈ N) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
1411, 12, 13syl2an 283 . . 3 (((𝑐N𝑡N) ∧ (𝑑N𝑠N)) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
1514an4s 555 . 2 (((𝑐N𝑑N) ∧ (𝑡N𝑠N)) → ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
16 enqex 6898 . 2 ~Q ∈ V
17 enqer 6896 . 2 ~Q Er (N × N)
18 df-enq 6885 . 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 496 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑧 = 𝑎)
20 simprr 499 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑢 = 𝑑)
2119, 20oveq12d 5652 . . 3 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
22 simplr 497 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑤 = 𝑏)
23 simprl 498 . . . 4 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → 𝑣 = 𝑐)
2422, 23oveq12d 5652 . . 3 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
2521, 24eqeq12d 2102 . 2 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
26 simpll 496 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑧 = 𝑔)
27 simprr 499 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑢 = 𝑠)
2826, 27oveq12d 5652 . . 3 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
29 simplr 497 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑤 = )
30 simprl 498 . . . 4 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → 𝑣 = 𝑡)
3129, 30oveq12d 5652 . . 3 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝑤 ·N 𝑣) = ( ·N 𝑡))
3228, 31eqeq12d 2102 . 2 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = ( ·N 𝑡)))
33 dfmpq2 6893 . 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 496 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑤 = 𝑎)
35 simprl 498 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑢 = 𝑔)
3634, 35oveq12d 5652 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → (𝑤 ·N 𝑢) = (𝑎 ·N 𝑔))
37 simplr 497 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑣 = 𝑏)
38 simprr 499 . . . 4 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝑓 = )
3937, 38oveq12d 5652 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → (𝑣 ·N 𝑓) = (𝑏 ·N ))
4036, 39opeq12d 3625 . 2 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝑎 ·N 𝑔), (𝑏 ·N )⟩)
41 simpll 496 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑤 = 𝑐)
42 simprl 498 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑢 = 𝑡)
4341, 42oveq12d 5652 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → (𝑤 ·N 𝑢) = (𝑐 ·N 𝑡))
44 simplr 497 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑣 = 𝑑)
45 simprr 499 . . . 4 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝑓 = 𝑠)
4644, 45oveq12d 5652 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
4743, 46opeq12d 3625 . 2 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝑐 ·N 𝑡), (𝑑 ·N 𝑠)⟩)
48 simpll 496 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑤 = 𝐴)
49 simprl 498 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑢 = 𝐶)
5048, 49oveq12d 5652 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑤 ·N 𝑢) = (𝐴 ·N 𝐶))
51 simplr 497 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑣 = 𝐵)
52 simprr 499 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑓 = 𝐷)
5351, 52oveq12d 5652 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
5450, 53opeq12d 3625 . 2 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
55 df-mqqs 6888 . 2 ·Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑎𝑏𝑐𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ~Q𝑦 = [⟨𝑐, 𝑑⟩] ~Q ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ ·pQ𝑐, 𝑑⟩)] ~Q ))}
56 df-nqqs 6886 . 2 Q = ((N × N) / ~Q )
57 mulcmpblnq 6906 . 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 6378 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 102   = wceq 1289  wcel 1438  cop 3444   × cxp 4426  (class class class)co 5634  [cec 6270  Ncnpi 6810   ·N cmi 6812   ·pQ cmpq 6815   ~Q ceq 6817  Qcnq 6818   ·Q cmq 6821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-mi 6844  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-mqqs 6888
This theorem is referenced by:  mulclnq  6914  mulcomnqg  6921  mulassnqg  6922  distrnqg  6925  mulidnq  6927  recexnq  6928  ltmnqg  6939  nqnq0m  6993
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