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Theorem mulpipq 6356
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.)
Assertion
Ref Expression
mulpipq (((A N B N) (𝐶 N 𝐷 N)) → (⟨A, B⟩ ·pQ𝐶, 𝐷⟩) = ⟨(A ·N 𝐶), (B ·N 𝐷)⟩)

Proof of Theorem mulpipq
StepHypRef Expression
1 opelxpi 4319 . . 3 ((A N B N) → ⟨A, B (N × N))
2 opelxpi 4319 . . 3 ((𝐶 N 𝐷 N) → ⟨𝐶, 𝐷 (N × N))
3 mulpipq2 6355 . . 3 ((⟨A, B (N × N) 𝐶, 𝐷 (N × N)) → (⟨A, B⟩ ·pQ𝐶, 𝐷⟩) = ⟨((1st ‘⟨A, B⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)), ((2nd ‘⟨A, B⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
41, 2, 3syl2an 273 . 2 (((A N B N) (𝐶 N 𝐷 N)) → (⟨A, B⟩ ·pQ𝐶, 𝐷⟩) = ⟨((1st ‘⟨A, B⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)), ((2nd ‘⟨A, B⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
5 op1stg 5719 . . . 4 ((A N B N) → (1st ‘⟨A, B⟩) = A)
6 op1stg 5719 . . . 4 ((𝐶 N 𝐷 N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
75, 6oveqan12d 5474 . . 3 (((A N B N) (𝐶 N 𝐷 N)) → ((1st ‘⟨A, B⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)) = (A ·N 𝐶))
8 op2ndg 5720 . . . 4 ((A N B N) → (2nd ‘⟨A, B⟩) = B)
9 op2ndg 5720 . . . 4 ((𝐶 N 𝐷 N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
108, 9oveqan12d 5474 . . 3 (((A N B N) (𝐶 N 𝐷 N)) → ((2nd ‘⟨A, B⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (B ·N 𝐷))
117, 10opeq12d 3548 . 2 (((A N B N) (𝐶 N 𝐷 N)) → ⟨((1st ‘⟨A, B⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)), ((2nd ‘⟨A, B⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩ = ⟨(A ·N 𝐶), (B ·N 𝐷)⟩)
124, 11eqtrd 2069 1 (((A N B N) (𝐶 N 𝐷 N)) → (⟨A, B⟩ ·pQ𝐶, 𝐷⟩) = ⟨(A ·N 𝐶), (B ·N 𝐷)⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cop 3370   × cxp 4286  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Ncnpi 6256   ·N cmi 6258   ·pQ cmpq 6261
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-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-ni 6288  df-mi 6290  df-mpq 6329
This theorem is referenced by: (None)
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