Theorem addpinq1 7659

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 7546	. . . . 5 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
2	1	oveq2i 6018	. . . 4 ⊢ ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
3		1pi 7510	. . . . 5 ⊢ 1o ∈ N
4		addpipqqs 7565	. . . . . 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 439	. . . . 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 425	. . . 4 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
7	2, 6	eqtrid 2274	. . 3 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
8		mulidpi 7513	. . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
9	3, 8	ax-mp 5	. . . . . 6 ⊢ (1o ·N 1o) = 1o
10	9	oveq2i 6018	. . . . 5 ⊢ ((𝐴 ·N 1o) +N (1o ·N 1o)) = ((𝐴 ·N 1o) +N 1o)
11	10, 9	opeq12i 3862	. . . 4 ⊢ ⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨((𝐴 ·N 1o) +N 1o), 1o⟩
12		eceq1 6723	. . . 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 5	. . 3 ⊢ [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q = [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q
14	7, 13	eqtrdi 2278	. 2 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q )
15		mulidpi 7513	. . . . 5 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴)
16	15	oveq1d 6022	. . . 4 ⊢ (𝐴 ∈ N → ((𝐴 ·N 1o) +N 1o) = (𝐴 +N 1o))
17	16	opeq1d 3863	. . 3 ⊢ (𝐴 ∈ N → ⟨((𝐴 ·N 1o) +N 1o), 1o⟩ = ⟨(𝐴 +N 1o), 1o⟩)
18	17	eceq1d 6724	. 2 ⊢ (𝐴 ∈ N → [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q = [⟨(𝐴 +N 1o), 1o⟩] ~Q )
19	14, 18	eqtr2d 2263	1 ⊢ (𝐴 ∈ N → [⟨(𝐴 +N 1o), 1o⟩] ~Q = ([⟨𝐴, 1o⟩] ~Q +Q 1Q))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ⟨cop 3669  (class class class)co 6007  1_oc1o 6561  [cec 6686  Ncnpi 7467   +_N cpli 7468   ·_N cmi 7469   ~_Q ceq 7474  1_Qc1q 7476   +_Q cplq 7477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7499  df-pli 7500  df-mi 7501  df-plpq 7539  df-enq 7542  df-nqqs 7543  df-plqqs 7544  df-1nqqs 7546

This theorem is referenced by:  pitonnlem2  8042