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Theorem addpqf 10700
Description: Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addpqf +pQ :((N × N) × (N × N))⟶(N × N)

Proof of Theorem addpqf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 7863 . . . . . 6 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
2 xp2nd 7864 . . . . . 6 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
3 mulclpi 10649 . . . . . 6 (((1st𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
41, 2, 3syl2an 596 . . . . 5 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
5 xp1st 7863 . . . . . 6 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
6 xp2nd 7864 . . . . . 6 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
7 mulclpi 10649 . . . . . 6 (((1st𝑦) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
85, 6, 7syl2anr 597 . . . . 5 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
9 addclpi 10648 . . . . 5 ((((1st𝑥) ·N (2nd𝑦)) ∈ N ∧ ((1st𝑦) ·N (2nd𝑥)) ∈ N) → (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))) ∈ N)
104, 8, 9syl2anc 584 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))) ∈ N)
11 mulclpi 10649 . . . . 5 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
126, 2, 11syl2an 596 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
1310, 12opelxpd 5627 . . 3 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
1413rgen2 3120 . 2 𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N)
15 df-plpq 10664 . . 3 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
1615fmpo 7908 . 2 (∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N) ↔ +pQ :((N × N) × (N × N))⟶(N × N))
1714, 16mpbi 229 1 +pQ :((N × N) × (N × N))⟶(N × N)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wcel 2106  wral 3064  cop 4567   × cxp 5587  wf 6429  cfv 6433  (class class class)co 7275  1st c1st 7829  2nd c2nd 7830  Ncnpi 10600   +N cpli 10601   ·N cmi 10602   +pQ cplpq 10604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-oadd 8301  df-omul 8302  df-ni 10628  df-pli 10629  df-mi 10630  df-plpq 10664
This theorem is referenced by:  addclnq  10701  addnqf  10704  addcompq  10706  adderpq  10712  distrnq  10717
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