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Theorem mulpqf 10356
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 7710 . . . . 5 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
2 xp1st 7710 . . . . 5 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
3 mulclpi 10303 . . . . 5 (((1st𝑥) ∈ N ∧ (1st𝑦) ∈ N) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
41, 2, 3syl2an 595 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st𝑥) ·N (1st𝑦)) ∈ N)
5 xp2nd 7711 . . . . 5 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
6 xp2nd 7711 . . . . 5 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
7 mulclpi 10303 . . . . 5 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
85, 6, 7syl2an 595 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
94, 8opelxpd 5586 . . 3 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
109rgen2 3200 . 2 𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N)
11 df-mpq 10319 . . 3 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
1211fmpo 7755 . 2 (∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N) ↔ ·pQ :((N × N) × (N × N))⟶(N × N))
1310, 12mpbi 231 1 ·pQ :((N × N) × (N × N))⟶(N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wcel 2105  wral 3135  cop 4563   × cxp 5546  wf 6344  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  Ncnpi 10254   ·N cmi 10256   ·pQ cmpq 10259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-omul 8096  df-ni 10282  df-mi 10284  df-mpq 10319
This theorem is referenced by:  mulclnq  10357  mulnqf  10359  mulcompq  10362  mulerpq  10367  distrnq  10371
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