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Theorem mulpqf 9647
 Description: Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulpqf ·pQ :((N × N) × (N × N))⟶(N × N)

Proof of Theorem mulpqf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 7089 . . . . 5 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
2 xp1st 7089 . . . . 5 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
3 mulclpi 9594 . . . . 5 (((1st𝑥) ∈ N ∧ (1st𝑦) ∈ N) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
41, 2, 3syl2an 493 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
5 xp2nd 7090 . . . . 5 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
6 xp2nd 7090 . . . . 5 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
7 mulclpi 9594 . . . . 5 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
85, 6, 7syl2an 493 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
9 opelxpi 5072 . . . 4 ((((1st𝑥) ·N (1st𝑦)) ∈ N ∧ ((2nd𝑥) ·N (2nd𝑦)) ∈ N) → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
104, 8, 9syl2anc 691 . . 3 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
1110rgen2a 2960 . 2 𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N)
12 df-mpq 9610 . . 3 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
1312fmpt2 7126 . 2 (∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N) ↔ ·pQ :((N × N) × (N × N))⟶(N × N))
1411, 13mpbi 219 1 ·pQ :((N × N) × (N × N))⟶(N × N)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Ncnpi 9545   ·N cmi 9547   ·pQ cmpq 9550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452  df-ni 9573  df-mi 9575  df-mpq 9610 This theorem is referenced by:  mulclnq  9648  mulnqf  9650  mulcompq  9653  mulerpq  9658  distrnq  9662
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