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Theorem nnanq0 7488
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 7479 . . 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 1305 . 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 7339 . . . . . . . 8 (𝐴N𝐴 ∈ ω)
4 nnmcom 6515 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝐴 ∈ ω) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
53, 4sylan2 286 . . . . . . 7 ((𝑁 ∈ ω ∧ 𝐴N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
653adant2 1018 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝑁 ·o 𝐴) = (𝐴 ·o 𝑁))
76oveq1d 5912 . . . . 5 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
8 nndi 6512 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
983coml 1212 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
103, 9syl3an3 1284 . . . . 5 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝐴 ·o (𝑁 +o 𝑀)) = ((𝐴 ·o 𝑁) +o (𝐴 ·o 𝑀)))
117, 10eqtr4d 2225 . . . 4 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → ((𝑁 ·o 𝐴) +o (𝐴 ·o 𝑀)) = (𝐴 ·o (𝑁 +o 𝑀)))
1211opeq1d 3799 . . 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 6596 . 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 1001 . . 3 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → 𝐴N)
15 nnacl 6506 . . . 4 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω) → (𝑁 +o 𝑀) ∈ ω)
16153adant3 1019 . . 3 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → (𝑁 +o 𝑀) ∈ ω)
17 mulcanenq0ec 7475 . . 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 1249 . 2 ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴N) → [⟨(𝐴 ·o (𝑁 +o 𝑀)), (𝐴 ·o 𝐴)⟩] ~Q0 = [⟨(𝑁 +o 𝑀), 𝐴⟩] ~Q0 )
192, 13, 183eqtrrd 2227 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 980   = wceq 1364  wcel 2160  cop 3610  ωcom 4607  (class class class)co 5897   +o coa 6439   ·o comu 6440  [cec 6558  Ncnpi 7302   ~Q0 ceq0 7316   +Q0 cplq0 7319
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-mi 7336  df-enq0 7454  df-nq0 7455  df-plq0 7457
This theorem is referenced by:  nq02m  7495  prarloclemcalc  7532
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