Theorem addpinq1 7684

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 7571	. . . . 5 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
2	1	oveq2i 6029	. . . 4 ⊢ ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
3		1pi 7535	. . . . 5 ⊢ 1o ∈ N
4		addpipqqs 7590	. . . . . 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 2276	. . 3 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
8		mulidpi 7538	. . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
9	3, 8	ax-mp 5	. . . . . 6 ⊢ (1o ·N 1o) = 1o
10	9	oveq2i 6029	. . . . 5 ⊢ ((𝐴 ·N 1o) +N (1o ·N 1o)) = ((𝐴 ·N 1o) +N 1o)
11	10, 9	opeq12i 3867	. . . 4 ⊢ ⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨((𝐴 ·N 1o) +N 1o), 1o⟩
12		eceq1 6737	. . . 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 2280	. 2 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q )
15		mulidpi 7538	. . . . 5 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴)
16	15	oveq1d 6033	. . . 4 ⊢ (𝐴 ∈ N → ((𝐴 ·N 1o) +N 1o) = (𝐴 +N 1o))
17	16	opeq1d 3868	. . 3 ⊢ (𝐴 ∈ N → ⟨((𝐴 ·N 1o) +N 1o), 1o⟩ = ⟨(𝐴 +N 1o), 1o⟩)
18	17	eceq1d 6738	. 2 ⊢ (𝐴 ∈ N → [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q = [⟨(𝐴 +N 1o), 1o⟩] ~Q )
19	14, 18	eqtr2d 2265	1 ⊢ (𝐴 ∈ N → [⟨(𝐴 +N 1o), 1o⟩] ~Q = ([⟨𝐴, 1o⟩] ~Q +Q 1Q))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ⟨cop 3672  (class class class)co 6018  1_oc1o 6575  [cec 6700  Ncnpi 7492   +_N cpli 7493   ·_N cmi 7494   ~_Q ceq 7499  1_Qc1q 7501   +_Q cplq 7502

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-plpq 7564  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-1nqqs 7571

This theorem is referenced by:  pitonnlem2  8067