Theorem nnanq0 7015

Description: Addition of nonnegative 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 7006	. . 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 1230	. 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 6866	. . . . . . . 8 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
4		nnmcom 6250	. . . . . . . 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 962	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
7	6	oveq1d 5667	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
8		nndi 6247	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
9	8	3coml 1150	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
10	3, 9	syl3an3 1209	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
11	7, 10	eqtr4d 2123	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)))
12	11	opeq1d 3628	. . 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 6326	. 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 945	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → 𝐴 ∈ N)
15		nnacl 6241	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +𝑜 𝑀) ∈ ω)
16	15	3adant3 963	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 +𝑜 𝑀) ∈ ω)
17		mulcanenq0ec 7002	. . 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 1174	. 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 2125	1 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))

Syntax hints:   → wi 4   ∧ w3a 924   = wceq 1289   ∈ wcel 1438  ⟨cop 3449  ωcom 4405  (class class class)co 5652   +_𝑜 coa 6178   ·_𝑜 comu 6179  [cec 6288  Ncnpi 6829   ~_Q0 ceq0 6843   +_Q0 cplq0 6846

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-mi 6863  df-enq0 6981  df-nq0 6982  df-plq0 6984

This theorem is referenced by:  nq02m  7022  prarloclemcalc  7059