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Theorem nnanq0 6313
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 A N) → [⟨(𝑁 +𝑜 𝑀), A⟩] ~Q0 = ([⟨𝑁, A⟩] ~Q0 +Q0 [⟨𝑀, A⟩] ~Q0 ))

Proof of Theorem nnanq0
StepHypRef Expression
1 addnnnq0 6304 . . 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 )
213impdir 1177 . 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 6169 . . . . . . . 8 (A NA 𝜔)
4 nnmcom 5983 . . . . . . . 8 ((𝑁 𝜔 A 𝜔) → (𝑁 ·𝑜 A) = (A ·𝑜 𝑁))
53, 4sylan2 270 . . . . . . 7 ((𝑁 𝜔 A N) → (𝑁 ·𝑜 A) = (A ·𝑜 𝑁))
653adant2 911 . . . . . 6 ((𝑁 𝜔 𝑀 𝜔 A N) → (𝑁 ·𝑜 A) = (A ·𝑜 𝑁))
76oveq1d 5451 . . . . 5 ((𝑁 𝜔 𝑀 𝜔 A N) → ((𝑁 ·𝑜 A) +𝑜 (A ·𝑜 𝑀)) = ((A ·𝑜 𝑁) +𝑜 (A ·𝑜 𝑀)))
8 nndi 5980 . . . . . . 7 ((A 𝜔 𝑁 𝜔 𝑀 𝜔) → (A ·𝑜 (𝑁 +𝑜 𝑀)) = ((A ·𝑜 𝑁) +𝑜 (A ·𝑜 𝑀)))
983coml 1097 . . . . . 6 ((𝑁 𝜔 𝑀 𝜔 A 𝜔) → (A ·𝑜 (𝑁 +𝑜 𝑀)) = ((A ·𝑜 𝑁) +𝑜 (A ·𝑜 𝑀)))
103, 9syl3an3 1156 . . . . 5 ((𝑁 𝜔 𝑀 𝜔 A N) → (A ·𝑜 (𝑁 +𝑜 𝑀)) = ((A ·𝑜 𝑁) +𝑜 (A ·𝑜 𝑀)))
117, 10eqtr4d 2057 . . . 4 ((𝑁 𝜔 𝑀 𝜔 A N) → ((𝑁 ·𝑜 A) +𝑜 (A ·𝑜 𝑀)) = (A ·𝑜 (𝑁 +𝑜 𝑀)))
1211opeq1d 3529 . . 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)⟩)
1312eceq1d 6053 . 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 894 . . 3 ((𝑁 𝜔 𝑀 𝜔 A N) → A N)
15 nnacl 5974 . . . 4 ((𝑁 𝜔 𝑀 𝜔) → (𝑁 +𝑜 𝑀) 𝜔)
16153adant3 912 . . 3 ((𝑁 𝜔 𝑀 𝜔 A N) → (𝑁 +𝑜 𝑀) 𝜔)
17 mulcanenq0ec 6300 . . 3 ((A N (𝑁 +𝑜 𝑀) 𝜔 A N) → [⟨(A ·𝑜 (𝑁 +𝑜 𝑀)), (A ·𝑜 A)⟩] ~Q0 = [⟨(𝑁 +𝑜 𝑀), A⟩] ~Q0 )
1814, 16, 14, 17syl3anc 1121 . 2 ((𝑁 𝜔 𝑀 𝜔 A N) → [⟨(A ·𝑜 (𝑁 +𝑜 𝑀)), (A ·𝑜 A)⟩] ~Q0 = [⟨(𝑁 +𝑜 𝑀), A⟩] ~Q0 )
192, 13, 183eqtrrd 2059 1 ((𝑁 𝜔 𝑀 𝜔 A N) → [⟨(𝑁 +𝑜 𝑀), A⟩] ~Q0 = ([⟨𝑁, A⟩] ~Q0 +Q0 [⟨𝑀, A⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4   w3a 873   = wceq 1228   wcel 1374  cop 3353  𝜔com 4240  (class class class)co 5436   +𝑜 coa 5913   ·𝑜 comu 5914  [cec 6015  Ncnpi 6130   ~Q0 ceq0 6144   +Q0 cplq0 6147
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-dc 734  df-3or 874  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921  df-er 6017  df-ec 6019  df-qs 6023  df-ni 6164  df-mi 6166  df-enq0 6279  df-nq0 6280  df-plq0 6282
This theorem is referenced by:  nq02m  6319  prarloclemcalc  6356
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