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Theorem addpipqqs 7171
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 7170 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
2 addpipqqslem 7170 . 2 (((𝑎N𝑏N) ∧ (𝑔NN)) → ⟨((𝑎 ·N ) +N (𝑏 ·N 𝑔)), (𝑏 ·N )⟩ ∈ (N × N))
3 addpipqqslem 7170 . 2 (((𝑐N𝑑N) ∧ (𝑡N𝑠N)) → ⟨((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)), (𝑑 ·N 𝑠)⟩ ∈ (N × N))
4 enqex 7161 . 2 ~Q ∈ V
5 enqer 7159 . 2 ~Q Er (N × N)
6 df-enq 7148 . 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 5776 . . . 4 ((𝑧 = 𝑎𝑢 = 𝑑) → (𝑧 ·N 𝑢) = (𝑎 ·N 𝑑))
8 oveq12 5776 . . . 4 ((𝑤 = 𝑏𝑣 = 𝑐) → (𝑤 ·N 𝑣) = (𝑏 ·N 𝑐))
97, 8eqeqan12d 2153 . . 3 (((𝑧 = 𝑎𝑢 = 𝑑) ∧ (𝑤 = 𝑏𝑣 = 𝑐)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
109an42s 578 . 2 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑎 ·N 𝑑) = (𝑏 ·N 𝑐)))
11 oveq12 5776 . . . 4 ((𝑧 = 𝑔𝑢 = 𝑠) → (𝑧 ·N 𝑢) = (𝑔 ·N 𝑠))
12 oveq12 5776 . . . 4 ((𝑤 = 𝑣 = 𝑡) → (𝑤 ·N 𝑣) = ( ·N 𝑡))
1311, 12eqeqan12d 2153 . . 3 (((𝑧 = 𝑔𝑢 = 𝑠) ∧ (𝑤 = 𝑣 = 𝑡)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = ( ·N 𝑡)))
1413an42s 578 . 2 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑔 ·N 𝑠) = ( ·N 𝑡)))
15 dfplpq2 7155 . 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 5776 . . . . 5 ((𝑤 = 𝑎𝑓 = ) → (𝑤 ·N 𝑓) = (𝑎 ·N ))
17 oveq12 5776 . . . . 5 ((𝑣 = 𝑏𝑢 = 𝑔) → (𝑣 ·N 𝑢) = (𝑏 ·N 𝑔))
1816, 17oveqan12d 5786 . . . 4 (((𝑤 = 𝑎𝑓 = ) ∧ (𝑣 = 𝑏𝑢 = 𝑔)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝑎 ·N ) +N (𝑏 ·N 𝑔)))
1918an42s 578 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝑎 ·N ) +N (𝑏 ·N 𝑔)))
20 oveq12 5776 . . . 4 ((𝑣 = 𝑏𝑓 = ) → (𝑣 ·N 𝑓) = (𝑏 ·N ))
2120ad2ant2l 499 . . 3 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → (𝑣 ·N 𝑓) = (𝑏 ·N ))
2219, 21opeq12d 3708 . 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 5776 . . . . 5 ((𝑤 = 𝑐𝑓 = 𝑠) → (𝑤 ·N 𝑓) = (𝑐 ·N 𝑠))
24 oveq12 5776 . . . . 5 ((𝑣 = 𝑑𝑢 = 𝑡) → (𝑣 ·N 𝑢) = (𝑑 ·N 𝑡))
2523, 24oveqan12d 5786 . . . 4 (((𝑤 = 𝑐𝑓 = 𝑠) ∧ (𝑣 = 𝑑𝑢 = 𝑡)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)))
2625an42s 578 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝑐 ·N 𝑠) +N (𝑑 ·N 𝑡)))
27 oveq12 5776 . . . 4 ((𝑣 = 𝑑𝑓 = 𝑠) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
2827ad2ant2l 499 . . 3 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → (𝑣 ·N 𝑓) = (𝑑 ·N 𝑠))
2926, 28opeq12d 3708 . 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 5776 . . . . 5 ((𝑤 = 𝐴𝑓 = 𝐷) → (𝑤 ·N 𝑓) = (𝐴 ·N 𝐷))
31 oveq12 5776 . . . . 5 ((𝑣 = 𝐵𝑢 = 𝐶) → (𝑣 ·N 𝑢) = (𝐵 ·N 𝐶))
3230, 31oveqan12d 5786 . . . 4 (((𝑤 = 𝐴𝑓 = 𝐷) ∧ (𝑣 = 𝐵𝑢 = 𝐶)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)))
3332an42s 578 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)) = ((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)))
34 oveq12 5776 . . . 4 ((𝑣 = 𝐵𝑓 = 𝐷) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
3534ad2ant2l 499 . . 3 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑣 ·N 𝑓) = (𝐵 ·N 𝐷))
3633, 35opeq12d 3708 . 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 7150 . 2 +Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑎𝑏𝑐𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ~Q𝑦 = [⟨𝑐, 𝑑⟩] ~Q ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ +pQ𝑐, 𝑑⟩)] ~Q ))}
38 df-nqqs 7149 . 2 Q = ((N × N) / ~Q )
39 addcmpblnq 7168 . 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 6528 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 103  wb 104   = wceq 1331  wcel 1480  cop 3525  (class class class)co 5767  [cec 6420  Ncnpi 7073   +N cpli 7074   ·N cmi 7075   +pQ cplpq 7077   ~Q ceq 7080  Qcnq 7081   +Q cplq 7083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-plpq 7145  df-enq 7148  df-nqqs 7149  df-plqqs 7150
This theorem is referenced by:  addclnq  7176  addcomnqg  7182  addassnqg  7183  distrnqg  7188  ltanqg  7201  1lt2nq  7207  ltexnqq  7209  nqnq0a  7255  addpinq1  7265
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