Theorem mulpqf 10906

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.

Step	Hyp	Ref	Expression
1		xp1st 8004	. . . . 5 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
2		xp1st 8004	. . . . 5 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
3		mulclpi 10853	. . . . 5 ⊢ (((1st ‘𝑥) ∈ N ∧ (1st ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (1st ‘𝑦)) ∈ N)
4	1, 2, 3	syl2an 605	. . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑥) ·N (1st ‘𝑦)) ∈ N)
5		xp2nd 8005	. . . . 5 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
6		xp2nd 8005	. . . . 5 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
7		mulclpi 10853	. . . . 5 ⊢ (((2nd ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
8	5, 6, 7	syl2an 605	. . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
9	4, 8	opelxpd 5688	. . 3 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ (N × N))
10	9	rgen2 3204	. 2 ⊢ ∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ (N × N)
11		df-mpq 10869	. . 3 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
12	11	fmpo 8051	. 2 ⊢ (∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ (N × N) ↔ ·pQ :((N × N) × (N × N))⟶(N × N))
13	10, 12	mpbi 232	1 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N)

Syntax hints:   ∧ wa 399   ∈ wcel 2144  ∀wral 3078  ⟨cop 4590   × cxp 5647  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  1^st c1st 7970  2^nd c2nd 7971  Ncnpi 10804   ·_N cmi 10806   ·_pQ cmpq 10809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-oadd 8443  df-omul 8444  df-ni 10832  df-mi 10834  df-mpq 10869

This theorem is referenced by:  mulclnq  10907  mulnqf  10909  mulcompq  10912  mulerpq  10917  distrnq  10921