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Theorem mulpipq 9747
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 5138 . . 3 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 opelxpi 5138 . . 3 ((𝐶N𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
3 mulpipq2 9746 . . 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 494 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
5 op1stg 7165 . . . 4 ((𝐴N𝐵N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
6 op1stg 7165 . . . 4 ((𝐶N𝐷N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
75, 6oveqan12d 6654 . . 3 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)) = (𝐴 ·N 𝐶))
8 op2ndg 7166 . . . 4 ((𝐴N𝐵N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9 op2ndg 7166 . . . 4 ((𝐶N𝐷N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
108, 9oveqan12d 6654 . . 3 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐵 ·N 𝐷))
117, 10opeq12d 4401 . 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 2654 1 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  cop 4174   × cxp 5102  cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  Ncnpi 9651   ·N cmi 9653   ·pQ cmpq 9656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-mpq 9716
This theorem is referenced by:  mulassnq  9766  distrnq  9768  mulidnq  9770  recmulnq  9771
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