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Theorem mulpqf 9970
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 7347 . . . . 5 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
2 xp1st 7347 . . . . 5 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
3 mulclpi 9917 . . . . 5 (((1st𝑥) ∈ N ∧ (1st𝑦) ∈ N) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
41, 2, 3syl2an 583 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
5 xp2nd 7348 . . . . 5 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
6 xp2nd 7348 . . . . 5 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
7 mulclpi 9917 . . . . 5 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
85, 6, 7syl2an 583 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
9 opelxpi 5288 . . . 4 ((((1st𝑥) ·N (1st𝑦)) ∈ N ∧ ((2nd𝑥) ·N (2nd𝑦)) ∈ N) → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
104, 8, 9syl2anc 573 . . 3 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
1110rgen2a 3126 . 2 𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N)
12 df-mpq 9933 . . 3 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
1312fmpt2 7387 . 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 382  wcel 2145  wral 3061  cop 4322   × cxp 5247  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  Ncnpi 9868   ·N cmi 9870   ·pQ cmpq 9873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-omul 7718  df-ni 9896  df-mi 9898  df-mpq 9933
This theorem is referenced by:  mulclnq  9971  mulnqf  9973  mulcompq  9976  mulerpq  9981  distrnq  9985
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