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Theorem addpipqqs 6468
Description: Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
Assertion
Ref Expression
addpipqqs (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q +Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩] ~Q )

Proof of Theorem addpipqqs
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 addpipqqslem 6467 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
2 addpipqqslem 6467 . 2 (((𝑎N𝑏N) ∧ (𝑔NN)) → ⟨((𝑎 ·N ) +N (𝑏 ·N 𝑔)), (𝑏 ·N )⟩ ∈ (N × N))
3 addpipqqslem 6467 . 2 (((𝑐N𝑑N) ∧ (𝑡N𝑠N)) → ⟨((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
4 enqex 6458 . 2 ~Q ∈ V
5 enqer 6456 . 2 ~Q Er (N × N)
6 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)))}
7 oveq12 5521 . . . 4 ((𝑧 = 𝑎𝑢 = 𝑑) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
8 oveq12 5521 . . . 4 ((𝑤 = 𝑏𝑣 = 𝑐) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
97, 8eqeqan12d 2055 . . 3 (((𝑧 = 𝑎𝑢 = 𝑑) ∧ (𝑤 = 𝑏𝑣 = 𝑐)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
109an42s 523 . 2 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
11 oveq12 5521 . . . 4 ((𝑧 = 𝑔𝑢 = 𝑠) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
12 oveq12 5521 . . . 4 ((𝑤 = 𝑣 = 𝑡) → (𝑤 ·N 𝑣) = ( ·N 𝑡))
1311, 12eqeqan12d 2055 . . 3 (((𝑧 = 𝑔𝑢 = 𝑠) ∧ (𝑤 = 𝑣 = 𝑡)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = ( ·N 𝑡)))
1413an42s 523 . 2 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = ( ·N 𝑡)))
15 dfplpq2 6452 . 2 +pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))}
16 oveq12 5521 . . . . 5 ((𝑤 = 𝑎𝑓 = ) → (𝑤 ·N 𝑓) = (𝑎 ·N ))
17 oveq12 5521 . . . . 5 ((𝑣 = 𝑏𝑢 = 𝑔) → (𝑣 ·N 𝑢) = (𝑏 ·N 𝑔))
1816, 17oveqan12d 5531 . . . 4 (((𝑤 = 𝑎𝑓 = ) ∧ (𝑣 = 𝑏𝑢 = 𝑔)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝑎 ·N ) +N (𝑏 ·N 𝑔)))
1918an42s 523 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝑎 ·N ) +N (𝑏 ·N 𝑔)))
20 oveq12 5521 . . . 4 ((𝑣 = 𝑏𝑓 = ) → (𝑣 ·N 𝑓) = (𝑏 ·N ))
2120ad2ant2l 477 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → (𝑣 ·N 𝑓) = (𝑏 ·N ))
2219, 21opeq12d 3557 . 2 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩ = ⟨((𝑎 ·N ) +N (𝑏 ·N 𝑔)), (𝑏 ·N )⟩)
23 oveq12 5521 . . . . 5 ((𝑤 = 𝑐𝑓 = 𝑠) → (𝑤 ·N 𝑓) = (𝑐 ·N 𝑠))
24 oveq12 5521 . . . . 5 ((𝑣 = 𝑑𝑢 = 𝑡) → (𝑣 ·N 𝑢) = (𝑑 ·N 𝑡))
2523, 24oveqan12d 5531 . . . 4 (((𝑤 = 𝑐𝑓 = 𝑠) ∧ (𝑣 = 𝑑𝑢 = 𝑡)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)))
2625an42s 523 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)))
27 oveq12 5521 . . . 4 ((𝑣 = 𝑑𝑓 = 𝑠) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
2827ad2ant2l 477 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
2926, 28opeq12d 3557 . 2 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩ = ⟨((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)), (𝑑 ·N 𝑠)⟩)
30 oveq12 5521 . . . . 5 ((𝑤 = 𝐴𝑓 = 𝐷) → (𝑤 ·N 𝑓) = (𝐴 ·N 𝐷))
31 oveq12 5521 . . . . 5 ((𝑣 = 𝐵𝑢 = 𝐶) → (𝑣 ·N 𝑢) = (𝐵 ·N 𝐶))
3230, 31oveqan12d 5531 . . . 4 (((𝑤 = 𝐴𝑓 = 𝐷) ∧ (𝑣 = 𝐵𝑢 = 𝐶)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)))
3332an42s 523 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)))
34 oveq12 5521 . . . 4 ((𝑣 = 𝐵𝑓 = 𝐷) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
3534ad2ant2l 477 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
3633, 35opeq12d 3557 . 2 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩ = ⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩)
37 df-plqqs 6447 . 2 +Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑎𝑏𝑐𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ~Q𝑦 = [⟨𝑐, 𝑑⟩] ~Q ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ +pQ𝑐, 𝑑⟩)] ~Q ))}
38 df-nqqs 6446 . 2 Q = ((N × N) / ~Q )
39 addcmpblnq 6465 . 2 ((((𝑎N𝑏N) ∧ (𝑐N𝑑N)) ∧ ((𝑔NN) ∧ (𝑡N𝑠N))) → (((𝑎 ·N 𝑑) = (𝑏 ·N 𝑐) ∧ (𝑔 ·N 𝑠) = ( ·N 𝑡)) → ⟨((𝑎 ·N ) +N (𝑏 ·N 𝑔)), (𝑏 ·N )⟩ ~Q ⟨((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)), (𝑑 ·N 𝑠)⟩))
401, 2, 3, 4, 5, 6, 10, 14, 15, 22, 29, 36, 37, 38, 39oviec 6212 1 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q +Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩] ~Q )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  cop 3378  (class class class)co 5512  [cec 6104  Ncnpi 6370   +N cpli 6371   ·N cmi 6372   +pQ cplpq 6374   ~Q ceq 6377  Qcnq 6378   +Q cplq 6380
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-pli 6403  df-mi 6404  df-plpq 6442  df-enq 6445  df-nqqs 6446  df-plqqs 6447
This theorem is referenced by:  addclnq  6473  addcomnqg  6479  addassnqg  6480  distrnqg  6485  ltanqg  6498  1lt2nq  6504  ltexnqq  6506  nqnq0a  6552  addpinq1  6562
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