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 Description: Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
Assertion
Ref Expression
addpinq1 (𝐴N → [⟨(𝐴 +N 1o), 1o⟩] ~Q = ([⟨𝐴, 1o⟩] ~Q +Q 1Q))

StepHypRef Expression
1 df-1nqqs 7254 . . . . 5 1Q = [⟨1o, 1o⟩] ~Q
21oveq2i 5829 . . . 4 ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
3 1pi 7218 . . . . 5 1oN
4 addpipqqs 7273 . . . . . 6 (((𝐴N ∧ 1oN) ∧ (1oN ∧ 1oN)) → ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
53, 3, 4mpanr12 436 . . . . 5 ((𝐴N ∧ 1oN) → ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
63, 5mpan2 422 . . . 4 (𝐴N → ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
72, 6syl5eq 2202 . . 3 (𝐴N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
8 mulidpi 7221 . . . . . . 7 (1oN → (1o ·N 1o) = 1o)
93, 8ax-mp 5 . . . . . 6 (1o ·N 1o) = 1o
109oveq2i 5829 . . . . 5 ((𝐴 ·N 1o) +N (1o ·N 1o)) = ((𝐴 ·N 1o) +N 1o)
1110, 9opeq12i 3746 . . . 4 ⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨((𝐴 ·N 1o) +N 1o), 1o
12 eceq1 6508 . . . 4 (⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨((𝐴 ·N 1o) +N 1o), 1o⟩ → [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q = [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q )
1311, 12ax-mp 5 . . 3 [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q = [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q
147, 13eqtrdi 2206 . 2 (𝐴N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q )
15 mulidpi 7221 . . . . 5 (𝐴N → (𝐴 ·N 1o) = 𝐴)
1615oveq1d 5833 . . . 4 (𝐴N → ((𝐴 ·N 1o) +N 1o) = (𝐴 +N 1o))
1716opeq1d 3747 . . 3 (𝐴N → ⟨((𝐴 ·N 1o) +N 1o), 1o⟩ = ⟨(𝐴 +N 1o), 1o⟩)
1817eceq1d 6509 . 2 (𝐴N → [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q = [⟨(𝐴 +N 1o), 1o⟩] ~Q )
1914, 18eqtr2d 2191 1 (𝐴N → [⟨(𝐴 +N 1o), 1o⟩] ~Q = ([⟨𝐴, 1o⟩] ~Q +Q 1Q))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∈ wcel 2128  ⟨cop 3563  (class class class)co 5818  1oc1o 6350  [cec 6471  Ncnpi 7175   +N cpli 7176   ·N cmi 7177   ~Q ceq 7182  1Qc1q 7184   +Q cplq 7185 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-1o 6357  df-oadd 6361  df-omul 6362  df-er 6473  df-ec 6475  df-qs 6479  df-ni 7207  df-pli 7208  df-mi 7209  df-plpq 7247  df-enq 7250  df-nqqs 7251  df-plqqs 7252  df-1nqqs 7254 This theorem is referenced by:  pitonnlem2  7750
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