Theorem addpinq1 6562

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	⊢ (𝐴 ∈ N → [⟨(𝐴 +N 1𝑜), 1𝑜⟩] ~Q = ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q))

Proof of Theorem addpinq1

Step	Hyp	Ref	Expression
1		df-1nqqs 6449	. . . . 5 ⊢ 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
2	1	oveq2i 5523	. . . 4 ⊢ ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q) = ([⟨𝐴, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q )
3		1pi 6413	. . . . 5 ⊢ 1𝑜 ∈ N
4		addpipqqs 6468	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 1𝑜 ∈ N) ∧ (1𝑜 ∈ N ∧ 1𝑜 ∈ N)) → ([⟨𝐴, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q ) = [⟨((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q )
5	3, 3, 4	mpanr12 415	. . . . 5 ⊢ ((𝐴 ∈ N ∧ 1𝑜 ∈ N) → ([⟨𝐴, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q ) = [⟨((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q )
6	3, 5	mpan2 401	. . . 4 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q ) = [⟨((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q )
7	2, 6	syl5eq 2084	. . 3 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q )
8		mulidpi 6416	. . . . . . 7 ⊢ (1𝑜 ∈ N → (1𝑜 ·N 1𝑜) = 1𝑜)
9	3, 8	ax-mp 7	. . . . . 6 ⊢ (1𝑜 ·N 1𝑜) = 1𝑜
10	9	oveq2i 5523	. . . . 5 ⊢ ((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = ((𝐴 ·N 1𝑜) +N 1𝑜)
11	10, 9	opeq12i 3554	. . . 4 ⊢ ⟨((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩ = ⟨((𝐴 ·N 1𝑜) +N 1𝑜), 1𝑜⟩
12		eceq1 6141	. . . 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 )
13	11, 12	ax-mp 7	. . 3 ⊢ [⟨((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q = [⟨((𝐴 ·N 1𝑜) +N 1𝑜), 1𝑜⟩] ~Q
14	7, 13	syl6eq 2088	. 2 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1𝑜) +N 1𝑜), 1𝑜⟩] ~Q )
15		mulidpi 6416	. . . . 5 ⊢ (𝐴 ∈ N → (𝐴 ·N 1𝑜) = 𝐴)
16	15	oveq1d 5527	. . . 4 ⊢ (𝐴 ∈ N → ((𝐴 ·N 1𝑜) +N 1𝑜) = (𝐴 +N 1𝑜))
17	16	opeq1d 3555	. . 3 ⊢ (𝐴 ∈ N → ⟨((𝐴 ·N 1𝑜) +N 1𝑜), 1𝑜⟩ = ⟨(𝐴 +N 1𝑜), 1𝑜⟩)
18	17	eceq1d 6142	. 2 ⊢ (𝐴 ∈ N → [⟨((𝐴 ·N 1𝑜) +N 1𝑜), 1𝑜⟩] ~Q = [⟨(𝐴 +N 1𝑜), 1𝑜⟩] ~Q )
19	14, 18	eqtr2d 2073	1 ⊢ (𝐴 ∈ N → [⟨(𝐴 +N 1𝑜), 1𝑜⟩] ~Q = ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   +_N cpli 6371   ·_N cmi 6372   ~_Q ceq 6377  1_Qc1q 6379   +_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-1o 6001  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  df-1nqqs 6449

This theorem is referenced by:  pitonnlem2  6923