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Theorem mulpqf 10984
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 8045 . . . . 5 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
2 xp1st 8045 . . . . 5 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
3 mulclpi 10931 . . . . 5 (((1st𝑥) ∈ N ∧ (1st𝑦) ∈ N) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
41, 2, 3syl2an 596 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
5 xp2nd 8046 . . . . 5 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
6 xp2nd 8046 . . . . 5 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
7 mulclpi 10931 . . . . 5 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
85, 6, 7syl2an 596 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
94, 8opelxpd 5728 . . 3 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
109rgen2 3197 . 2 𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N)
11 df-mpq 10947 . . 3 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
1211fmpo 8092 . 2 (∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N) ↔ ·pQ :((N × N) × (N × N))⟶(N × N))
1310, 12mpbi 230 1 ·pQ :((N × N) × (N × N))⟶(N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2106  wral 3059  cop 4637   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Ncnpi 10882   ·N cmi 10884   ·pQ cmpq 10887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509  df-omul 8510  df-ni 10910  df-mi 10912  df-mpq 10947
This theorem is referenced by:  mulclnq  10985  mulnqf  10987  mulcompq  10990  mulerpq  10995  distrnq  10999
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