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Theorem mulpqf 9753
 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 7183 . . . . 5 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
2 xp1st 7183 . . . . 5 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
3 mulclpi 9700 . . . . 5 (((1st𝑥) ∈ N ∧ (1st𝑦) ∈ N) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
41, 2, 3syl2an 494 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
5 xp2nd 7184 . . . . 5 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
6 xp2nd 7184 . . . . 5 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
7 mulclpi 9700 . . . . 5 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
85, 6, 7syl2an 494 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
9 opelxpi 5138 . . . 4 ((((1st𝑥) ·N (1st𝑦)) ∈ N ∧ ((2nd𝑥) ·N (2nd𝑦)) ∈ N) → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
104, 8, 9syl2anc 692 . . 3 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
1110rgen2a 2974 . 2 𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N)
12 df-mpq 9716 . . 3 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
1312fmpt2 7222 . 2 (∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N) ↔ ·pQ :((N × N) × (N × N))⟶(N × N))
1411, 13mpbi 220 1 ·pQ :((N × N) × (N × N))⟶(N × N)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   ∈ wcel 1988  ∀wral 2909  ⟨cop 4174   × cxp 5102  ⟶wf 5872  ‘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-3or 1037  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-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549  df-omul 7550  df-ni 9679  df-mi 9681  df-mpq 9716 This theorem is referenced by:  mulclnq  9754  mulnqf  9756  mulcompq  9759  mulerpq  9764  distrnq  9768
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