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Theorem nnanq0 7286
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 ∧ 𝐴N) → [⟨(𝑁 +o 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))

Proof of Theorem nnanq0
StepHypRef Expression
1 addnnnq0 7277 . . 3 (((𝑁 ∈ ω ∧ 𝐴N) ∧ (𝑀 ∈ ω ∧ 𝐴N)) → ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ) = [⟨((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)), (𝐴 ·o 𝐴)⟩] ~Q0 )
213impdir 1273 . 2 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ) = [⟨((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)), (𝐴 ·o 𝐴)⟩] ~Q0 )
3 pinn 7137 . . . . . . . 8 (𝐴N𝐴 ∈ ω)
4 nnmcom 6389 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝐴 ∈ ω) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
53, 4sylan2 284 . . . . . . 7 ((𝑁 ∈ ω ∧ 𝐴N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
653adant2 1001 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
76oveq1d 5793 . . . . 5 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
8 nndi 6386 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
983coml 1189 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
103, 9syl3an3 1252 . . . . 5 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
117, 10eqtr4d 2176 . . . 4 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = (𝐴 ·o (𝑁 +o 𝑀)))
1211opeq1d 3715 . . 3 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ⟨((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)), (𝐴 ·o 𝐴)⟩ = ⟨(𝐴 ·o (𝑁 +o 𝑀)), (𝐴 ·o 𝐴)⟩)
1312eceq1d 6469 . 2 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → [⟨((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)), (𝐴 ·o 𝐴)⟩] ~Q0 = [⟨(𝐴 ·o (𝑁 +o 𝑀)), (𝐴 ·o 𝐴)⟩] ~Q0 )
14 simp3 984 . . 3 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → 𝐴N)
15 nnacl 6380 . . . 4 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +o 𝑀) ∈ ω)
16153adant3 1002 . . 3 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝑁 +o 𝑀) ∈ ω)
17 mulcanenq0ec 7273 . . 3 ((𝐴N ∧ (𝑁 +o 𝑀) ∈ ω ∧ 𝐴N) → [⟨(𝐴 ·o (𝑁 +o 𝑀)), (𝐴 ·o 𝐴)⟩] ~Q0 = [⟨(𝑁 +o 𝑀), 𝐴⟩] ~Q0 )
1814, 16, 14, 17syl3anc 1217 . 2 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → [⟨(𝐴 ·o (𝑁 +o 𝑀)), (𝐴 ·o 𝐴)⟩] ~Q0 = [⟨(𝑁 +o 𝑀), 𝐴⟩] ~Q0 )
192, 13, 183eqtrrd 2178 1 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → [⟨(𝑁 +o 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 963   = wceq 1332  wcel 1481  cop 3531  ωcom 4508  (class class class)co 5778   +o coa 6314   ·o comu 6315  [cec 6431  Ncnpi 7100   ~Q0 ceq0 7114   +Q0 cplq0 7117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-mi 7134  df-enq0 7252  df-nq0 7253  df-plq0 7255
This theorem is referenced by:  nq02m  7293  prarloclemcalc  7330
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