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Theorem mulpipq 10350
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulpipq (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)

Proof of Theorem mulpipq
StepHypRef Expression
1 opelxpi 5585 . . 3 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 opelxpi 5585 . . 3 ((𝐶N𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
3 mulpipq2 10349 . . 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 595 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
5 op1stg 7690 . . . 4 ((𝐴N𝐵N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
6 op1stg 7690 . . . 4 ((𝐶N𝐷N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
75, 6oveqan12d 7164 . . 3 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)) = (𝐴 ·N 𝐶))
8 op2ndg 7691 . . . 4 ((𝐴N𝐵N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9 op2ndg 7691 . . . 4 ((𝐶N𝐷N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
108, 9oveqan12d 7164 . . 3 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐵 ·N 𝐷))
117, 10opeq12d 4803 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ⟨((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩ = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
124, 11eqtrd 2853 1 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cop 4563   × cxp 5546  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  Ncnpi 10254   ·N cmi 10256   ·pQ cmpq 10259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-mpq 10319
This theorem is referenced by:  mulassnq  10369  distrnq  10371  mulidnq  10373  recmulnq  10374
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