Theorem addpinq1 6654

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 6541	. . . . 5 ⊢ 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
2	1	oveq2i 5543	. . . 4 ⊢ ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q) = ([⟨𝐴, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q )
3		1pi 6505	. . . . 5 ⊢ 1𝑜 ∈ N
4		addpipqqs 6560	. . . . . 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 429	. . . . 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 415	. . . 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 2125	. . 3 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q )
8		mulidpi 6508	. . . . . . 7 ⊢ (1𝑜 ∈ N → (1𝑜 ·N 1𝑜) = 1𝑜)
9	3, 8	ax-mp 7	. . . . . 6 ⊢ (1𝑜 ·N 1𝑜) = 1𝑜
10	9	oveq2i 5543	. . . . 5 ⊢ ((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = ((𝐴 ·N 1𝑜) +N 1𝑜)
11	10, 9	opeq12i 3575	. . . 4 ⊢ ⟨((𝐴 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩ = ⟨((𝐴 ·N 1𝑜) +N 1𝑜), 1𝑜⟩
12		eceq1 6164	. . . 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 2129	. 2 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1𝑜) +N 1𝑜), 1𝑜⟩] ~Q )
15		mulidpi 6508	. . . . 5 ⊢ (𝐴 ∈ N → (𝐴 ·N 1𝑜) = 𝐴)
16	15	oveq1d 5547	. . . 4 ⊢ (𝐴 ∈ N → ((𝐴 ·N 1𝑜) +N 1𝑜) = (𝐴 +N 1𝑜))
17	16	opeq1d 3576	. . 3 ⊢ (𝐴 ∈ N → ⟨((𝐴 ·N 1𝑜) +N 1𝑜), 1𝑜⟩ = ⟨(𝐴 +N 1𝑜), 1𝑜⟩)
18	17	eceq1d 6165	. 2 ⊢ (𝐴 ∈ N → [⟨((𝐴 ·N 1𝑜) +N 1𝑜), 1𝑜⟩] ~Q = [⟨(𝐴 +N 1𝑜), 1𝑜⟩] ~Q )
19	14, 18	eqtr2d 2114	1 ⊢ (𝐴 ∈ N → [⟨(𝐴 +N 1𝑜), 1𝑜⟩] ~Q = ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ⟨cop 3401  (class class class)co 5532  1_𝑜c1o 6017  [cec 6127  Ncnpi 6462   +_N cpli 6463   ·_N cmi 6464   ~_Q ceq 6469  1_Qc1q 6471   +_Q cplq 6472

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-plpq 6534  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-1nqqs 6541

This theorem is referenced by:  pitonnlem2  7015