Theorem nnanq0 6648

Description: Addition of non-negative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.)

Assertion

Ref	Expression
nnanq0	⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))

Proof of Theorem nnanq0

Step	Hyp	Ref	Expression
1		addnnnq0 6639	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝐴 ∈ N) ∧ (𝑀 ∈ ω ∧ 𝐴 ∈ N)) → ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ) = [⟨((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩] ~Q0 )
2	1	3impdir 1225	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ) = [⟨((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩] ~Q0 )
3		pinn 6499	. . . . . . . 8 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
4		nnmcom 6091	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ ω) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
5	3, 4	sylan2 280	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
6	5	3adant2 957	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
7	6	oveq1d 5547	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
8		nndi 6088	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
9	8	3coml 1145	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
10	3, 9	syl3an3 1204	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
11	7, 10	eqtr4d 2116	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)))
12	11	opeq1d 3576	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ⟨((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩ = ⟨(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩)
13	12	eceq1d 6165	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩] ~Q0 = [⟨(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩] ~Q0 )
14		simp3 940	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → 𝐴 ∈ N)
15		nnacl 6082	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +𝑜 𝑀) ∈ ω)
16	15	3adant3 958	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 +𝑜 𝑀) ∈ ω)
17		mulcanenq0ec 6635	. . 3 ⊢ ((𝐴 ∈ N ∧ (𝑁 +𝑜 𝑀) ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩] ~Q0 = [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 )
18	14, 16, 14, 17	syl3anc 1169	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩] ~Q0 = [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 )
19	2, 13, 18	3eqtrrd 2118	1 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))

Syntax hints:   → wi 4   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ⟨cop 3401  ωcom 4331  (class class class)co 5532   +_𝑜 coa 6021   ·_𝑜 comu 6022  [cec 6127  Ncnpi 6462   ~_Q0 ceq0 6476   +_Q0 cplq0 6479

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-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-mi 6496  df-enq0 6614  df-nq0 6615  df-plq0 6617

This theorem is referenced by:  nq02m  6655  prarloclemcalc  6692