Theorem mulpipq 6562

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⟩ ·pQ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)

Proof of Theorem mulpipq

Step	Hyp	Ref	Expression
1		opelxpi 4394	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		opelxpi 4394	. . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
3		mulpipq2 6561	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (⟨𝐴, 𝐵⟩ ·pQ ⟨𝐶, 𝐷⟩) = ⟨((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
4	1, 2, 3	syl2an 283	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ ·pQ ⟨𝐶, 𝐷⟩) = ⟨((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
5		op1stg 5797	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
6		op1stg 5797	. . . 4 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
7	5, 6	oveqan12d 5551	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)) = (𝐴 ·N 𝐶))
8		op2ndg 5798	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9		op2ndg 5798	. . . 4 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
10	8, 9	oveqan12d 5551	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐵 ·N 𝐷))
11	7, 10	opeq12d 3578	. 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)⟩)
12	4, 11	eqtrd 2113	1 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ ·pQ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ⟨cop 3401   × cxp 4361  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  2^nd c2nd 5786  Ncnpi 6462   ·_N cmi 6464   ·_pQ cmpq 6467

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-ni 6494  df-mi 6496  df-mpq 6535

This theorem is referenced by: (None)