Theorem nnanq0 7815

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 7806	. . 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 1335	. 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 7666	. . . . . . . 8 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
4		nnmcom 6752	. . . . . . . 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 1047	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
7	6	oveq1d 6090	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
8		nndi 6749	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
9	8	3coml 1241	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
10	3, 9	syl3an3 1313	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
11	7, 10	eqtr4d 2274	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = (𝐴 ·o (𝑁 +o 𝑀)))
12	11	opeq1d 3905	. . 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 6833	. 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 1030	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → 𝐴 ∈ N)
15		nnacl 6743	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +o 𝑀) ∈ ω)
16	15	3adant3 1048	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 +o 𝑀) ∈ ω)
17		mulcanenq0ec 7802	. . 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 1278	. 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 2276	1 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝑁 +o 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))

Syntax hints:   → wi 4   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ⟨cop 3708  ωcom 4732  (class class class)co 6075   +_o coa 6674   ·_o comu 6675  [cec 6795  Ncnpi 7629   ~_Q0 ceq0 7643   +_Q0 cplq0 7646

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-mi 7663  df-enq0 7781  df-nq0 7782  df-plq0 7784

This theorem is referenced by:  nq02m  7822  prarloclemcalc  7859