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Theorem addpinq1 7240
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))

Proof of Theorem addpinq1
StepHypRef Expression
1 df-1nqqs 7127 . . . . 5 1Q = [⟨1o, 1o⟩] ~Q
21oveq2i 5753 . . . 4 ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
3 1pi 7091 . . . . 5 1oN
4 addpipqqs 7146 . . . . . 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 435 . . . . 5 ((𝐴N ∧ 1oN) → ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
63, 5mpan2 421 . . . 4 (𝐴N → ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
72, 6syl5eq 2162 . . 3 (𝐴N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
8 mulidpi 7094 . . . . . . 7 (1oN → (1o ·N 1o) = 1o)
93, 8ax-mp 5 . . . . . 6 (1o ·N 1o) = 1o
109oveq2i 5753 . . . . 5 ((𝐴 ·N 1o) +N (1o ·N 1o)) = ((𝐴 ·N 1o) +N 1o)
1110, 9opeq12i 3680 . . . 4 ⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨((𝐴 ·N 1o) +N 1o), 1o
12 eceq1 6432 . . . 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, 13syl6eq 2166 . 2 (𝐴N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q )
15 mulidpi 7094 . . . . 5 (𝐴N → (𝐴 ·N 1o) = 𝐴)
1615oveq1d 5757 . . . 4 (𝐴N → ((𝐴 ·N 1o) +N 1o) = (𝐴 +N 1o))
1716opeq1d 3681 . . 3 (𝐴N → ⟨((𝐴 ·N 1o) +N 1o), 1o⟩ = ⟨(𝐴 +N 1o), 1o⟩)
1817eceq1d 6433 . 2 (𝐴N → [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q = [⟨(𝐴 +N 1o), 1o⟩] ~Q )
1914, 18eqtr2d 2151 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 1316  wcel 1465  cop 3500  (class class class)co 5742  1oc1o 6274  [cec 6395  Ncnpi 7048   +N cpli 7049   ·N cmi 7050   ~Q ceq 7055  1Qc1q 7057   +Q cplq 7058
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-plpq 7120  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-1nqqs 7127
This theorem is referenced by:  pitonnlem2  7623
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