ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnanq0 Structured version   GIF version

Theorem nnanq0 6440
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 6431 . . 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 1190 . 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 6293 . . . . . . . 8 (A NA 𝜔)
4 nnmcom 6007 . . . . . . . 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 922 . . . . . 6 ((𝑁 𝜔 𝑀 𝜔 A N) → (𝑁 ·𝑜 A) = (A ·𝑜 𝑁))
76oveq1d 5470 . . . . 5 ((𝑁 𝜔 𝑀 𝜔 A N) → ((𝑁 ·𝑜 A) +𝑜 (A ·𝑜 𝑀)) = ((A ·𝑜 𝑁) +𝑜 (A ·𝑜 𝑀)))
8 nndi 6004 . . . . . . 7 ((A 𝜔 𝑁 𝜔 𝑀 𝜔) → (A ·𝑜 (𝑁 +𝑜 𝑀)) = ((A ·𝑜 𝑁) +𝑜 (A ·𝑜 𝑀)))
983coml 1110 . . . . . 6 ((𝑁 𝜔 𝑀 𝜔 A 𝜔) → (A ·𝑜 (𝑁 +𝑜 𝑀)) = ((A ·𝑜 𝑁) +𝑜 (A ·𝑜 𝑀)))
103, 9syl3an3 1169 . . . . 5 ((𝑁 𝜔 𝑀 𝜔 A N) → (A ·𝑜 (𝑁 +𝑜 𝑀)) = ((A ·𝑜 𝑁) +𝑜 (A ·𝑜 𝑀)))
117, 10eqtr4d 2072 . . . 4 ((𝑁 𝜔 𝑀 𝜔 A N) → ((𝑁 ·𝑜 A) +𝑜 (A ·𝑜 𝑀)) = (A ·𝑜 (𝑁 +𝑜 𝑀)))
1211opeq1d 3546 . . 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 6078 . 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 905 . . 3 ((𝑁 𝜔 𝑀 𝜔 A N) → A N)
15 nnacl 5998 . . . 4 ((𝑁 𝜔 𝑀 𝜔) → (𝑁 +𝑜 𝑀) 𝜔)
16153adant3 923 . . 3 ((𝑁 𝜔 𝑀 𝜔 A N) → (𝑁 +𝑜 𝑀) 𝜔)
17 mulcanenq0ec 6427 . . 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 1134 . 2 ((𝑁 𝜔 𝑀 𝜔 A N) → [⟨(A ·𝑜 (𝑁 +𝑜 𝑀)), (A ·𝑜 A)⟩] ~Q0 = [⟨(𝑁 +𝑜 𝑀), A⟩] ~Q0 )
192, 13, 183eqtrrd 2074 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 884   = wceq 1242   wcel 1390  cop 3370  𝜔com 4256  (class class class)co 5455   +𝑜 coa 5937   ·𝑜 comu 5938  [cec 6040  Ncnpi 6256   ~Q0 ceq0 6270   +Q0 cplq0 6273
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-enq0 6406  df-nq0 6407  df-plq0 6409
This theorem is referenced by:  nq02m  6447  prarloclemcalc  6484
  Copyright terms: Public domain W3C validator