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Theorem nnanq0 6556
 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 6547 . . 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 1191 . 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 6407 . . . . . . . 8 (𝐴N𝐴 ∈ ω)
4 nnmcom 6068 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝐴 ∈ ω) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
53, 4sylan2 270 . . . . . . 7 ((𝑁 ∈ ω ∧ 𝐴N) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
653adant2 923 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝑁 ·𝑜 𝐴) = (𝐴 ·𝑜 𝑁))
76oveq1d 5527 . . . . 5 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
8 nndi 6065 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
983coml 1111 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
103, 9syl3an3 1170 . . . . 5 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)) = ((𝐴 ·𝑜 𝑁) +𝑜 (𝐴 ·𝑜 𝑀)))
117, 10eqtr4d 2075 . . . 4 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ((𝑁 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 𝑀)) = (𝐴 ·𝑜 (𝑁 +𝑜 𝑀)))
1211opeq1d 3555 . . 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 6142 . 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 906 . . 3 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → 𝐴N)
15 nnacl 6059 . . . 4 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +𝑜 𝑀) ∈ ω)
16153adant3 924 . . 3 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝑁 +𝑜 𝑀) ∈ ω)
17 mulcanenq0ec 6543 . . 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 1135 . 2 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → [⟨(𝐴 ·𝑜 (𝑁 +𝑜 𝑀)), (𝐴 ·𝑜 𝐴)⟩] ~Q0 = [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 )
192, 13, 183eqtrrd 2077 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 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  ωcom 4313  (class class class)co 5512   +𝑜 coa 5998   ·𝑜 comu 5999  [cec 6104  Ncnpi 6370   ~Q0 ceq0 6384   +Q0 cplq0 6387 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-enq0 6522  df-nq0 6523  df-plq0 6525 This theorem is referenced by:  nq02m  6563  prarloclemcalc  6600
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