Theorem nnanq0 7668

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 7659	. . 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 1328	. 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 7519	. . . . . . . 8 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
4		nnmcom 6652	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ ω) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
5	3, 4	sylan2 286	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
6	5	3adant2 1040	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
7	6	oveq1d 6028	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
8		nndi 6649	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
9	8	3coml 1234	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
10	3, 9	syl3an3 1306	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
11	7, 10	eqtr4d 2265	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = (𝐴 ·o (𝑁 +o 𝑀)))
12	11	opeq1d 3866	. . 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 6733	. 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 1023	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → 𝐴 ∈ N)
15		nnacl 6643	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +o 𝑀) ∈ ω)
16	15	3adant3 1041	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 +o 𝑀) ∈ ω)
17		mulcanenq0ec 7655	. . 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 1271	. 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 2267	1 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝑁 +o 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))

Syntax hints:   → wi 4   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ⟨cop 3670  ωcom 4686  (class class class)co 6013   +_o coa 6574   ·_o comu 6575  [cec 6695  Ncnpi 7482   ~_Q0 ceq0 7496   +_Q0 cplq0 7499

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-mi 7516  df-enq0 7634  df-nq0 7635  df-plq0 7637

This theorem is referenced by:  nq02m  7675  prarloclemcalc  7712