Theorem addpinq1 7590

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 7477	. . . . 5 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
2	1	oveq2i 5965	. . . 4 ⊢ ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = ([⟨𝐴, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
3		1pi 7441	. . . . 5 ⊢ 1o ∈ N
4		addpipqqs 7496	. . . . . 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 2251	. . 3 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
8		mulidpi 7444	. . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
9	3, 8	ax-mp 5	. . . . . 6 ⊢ (1o ·N 1o) = 1o
10	9	oveq2i 5965	. . . . 5 ⊢ ((𝐴 ·N 1o) +N (1o ·N 1o)) = ((𝐴 ·N 1o) +N 1o)
11	10, 9	opeq12i 3827	. . . 4 ⊢ ⟨((𝐴 ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨((𝐴 ·N 1o) +N 1o), 1o⟩
12		eceq1 6665	. . . 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 2255	. 2 ⊢ (𝐴 ∈ N → ([⟨𝐴, 1o⟩] ~Q +Q 1Q) = [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q )
15		mulidpi 7444	. . . . 5 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴)
16	15	oveq1d 5969	. . . 4 ⊢ (𝐴 ∈ N → ((𝐴 ·N 1o) +N 1o) = (𝐴 +N 1o))
17	16	opeq1d 3828	. . 3 ⊢ (𝐴 ∈ N → ⟨((𝐴 ·N 1o) +N 1o), 1o⟩ = ⟨(𝐴 +N 1o), 1o⟩)
18	17	eceq1d 6666	. 2 ⊢ (𝐴 ∈ N → [⟨((𝐴 ·N 1o) +N 1o), 1o⟩] ~Q = [⟨(𝐴 +N 1o), 1o⟩] ~Q )
19	14, 18	eqtr2d 2240	1 ⊢ (𝐴 ∈ N → [⟨(𝐴 +N 1o), 1o⟩] ~Q = ([⟨𝐴, 1o⟩] ~Q +Q 1Q))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ⟨cop 3638  (class class class)co 5954  1_oc1o 6505  [cec 6628  Ncnpi 7398   +_N cpli 7399   ·_N cmi 7400   ~_Q ceq 7405  1_Qc1q 7407   +_Q cplq 7408

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-irdg 6466  df-1o 6512  df-oadd 6516  df-omul 6517  df-er 6630  df-ec 6632  df-qs 6636  df-ni 7430  df-pli 7431  df-mi 7432  df-plpq 7470  df-enq 7473  df-nqqs 7474  df-plqqs 7475  df-1nqqs 7477

This theorem is referenced by:  pitonnlem2  7973