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Theorem mulpipq 7148
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 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)

Proof of Theorem mulpipq
StepHypRef Expression
1 opelxpi 4541 . . 3 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 opelxpi 4541 . . 3 ((𝐶N𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
3 mulpipq2 7147 . . 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 287 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
5 op1stg 6016 . . . 4 ((𝐴N𝐵N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
6 op1stg 6016 . . . 4 ((𝐶N𝐷N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
75, 6oveqan12d 5761 . . 3 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)) = (𝐴 ·N 𝐶))
8 op2ndg 6017 . . . 4 ((𝐴N𝐵N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9 op2ndg 6017 . . . 4 ((𝐶N𝐷N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
108, 9oveqan12d 5761 . . 3 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐵 ·N 𝐷))
117, 10opeq12d 3683 . 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 2150 1 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  cop 3500   × cxp 4507  cfv 5093  (class class class)co 5742  1st c1st 6004  2nd c2nd 6005  Ncnpi 7048   ·N cmi 7050   ·pQ cmpq 7053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285  df-omul 6286  df-ni 7080  df-mi 7082  df-mpq 7121
This theorem is referenced by: (None)
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