Theorem mulpipq 7432

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 4691	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		opelxpi 4691	. . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
3		mulpipq2 7431	. . 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 289	. 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 6203	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
6		op1stg 6203	. . . 4 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
7	5, 6	oveqan12d 5937	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘⟨𝐴, 𝐵⟩) ·N (1st ‘⟨𝐶, 𝐷⟩)) = (𝐴 ·N 𝐶))
8		op2ndg 6204	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9		op2ndg 6204	. . . 4 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
10	8, 9	oveqan12d 5937	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐵 ·N 𝐷))
11	7, 10	opeq12d 3812	. 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 2226	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 104   = wceq 1364   ∈ wcel 2164  ⟨cop 3621   × cxp 4657  ‘cfv 5254  (class class class)co 5918  1^st c1st 6191  2^nd c2nd 6192  Ncnpi 7332   ·_N cmi 7334   ·_pQ cmpq 7337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473  df-omul 6474  df-ni 7364  df-mi 7366  df-mpq 7405

This theorem is referenced by: (None)