Theorem nnanq0 7570

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 7561	. . 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 1306	. 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 7421	. . . . . . . 8 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
4		nnmcom 6574	. . . . . . . 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 1018	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
7	6	oveq1d 5958	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
8		nndi 6571	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
9	8	3coml 1212	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
10	3, 9	syl3an3 1284	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
11	7, 10	eqtr4d 2240	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = (𝐴 ·o (𝑁 +o 𝑀)))
12	11	opeq1d 3824	. . 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 6655	. 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 1001	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → 𝐴 ∈ N)
15		nnacl 6565	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +o 𝑀) ∈ ω)
16	15	3adant3 1019	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → (𝑁 +o 𝑀) ∈ ω)
17		mulcanenq0ec 7557	. . 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 1249	. 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 2242	1 ⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝑁 +o 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))

Syntax hints:   → wi 4   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  ⟨cop 3635  ωcom 4637  (class class class)co 5943   +_o coa 6498   ·_o comu 6499  [cec 6617  Ncnpi 7384   ~_Q0 ceq0 7398   +_Q0 cplq0 7401

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-mi 7418  df-enq0 7536  df-nq0 7537  df-plq0 7539

This theorem is referenced by:  nq02m  7577  prarloclemcalc  7614