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Theorem nnanq0 6614
Description: Addition of non-negative 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 6605 . . 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 1202 . 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 6465 . . . . . . . 8 (𝐴N𝐴 ∈ ω)
4 nnmcom 6099 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝐴 ∈ ω) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
53, 4sylan2 274 . . . . . . 7 ((𝑁 ∈ ω ∧ 𝐴N) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
653adant2 934 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
76oveq1d 5555 . . . . 5 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
8 nndi 6096 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
983coml 1122 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
103, 9syl3an3 1181 . . . . 5 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
117, 10eqtr4d 2091 . . . 4 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)))
1211opeq1d 3583 . . 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 6173 . 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 917 . . 3 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → 𝐴N)
15 nnacl 6090 . . . 4 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +𝑜 𝑀) ∈ ω)
16153adant3 935 . . 3 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝑁 +𝑜 𝑀) ∈ ω)
17 mulcanenq0ec 6601 . . 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 1146 . 2 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → [⟨(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩] ~Q0 = [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 )
192, 13, 183eqtrrd 2093 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 896   = wceq 1259  wcel 1409  cop 3406  ωcom 4341  (class class class)co 5540   +𝑜 coa 6029   ·𝑜 comu 6030  [cec 6135  Ncnpi 6428   ~Q0 ceq0 6442   +Q0 cplq0 6445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-enq0 6580  df-nq0 6581  df-plq0 6583
This theorem is referenced by:  nq02m  6621  prarloclemcalc  6658
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