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 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 7479 . . . . . 6 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
2 xp2nd 7480 . . . . . 6 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
3 mulclpi 10052 . . . . . 6 (((1st𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
41, 2, 3syl2an 589 . . . . 5 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
5 xp1st 7479 . . . . . 6 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
6 xp2nd 7480 . . . . . 6 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
7 mulclpi 10052 . . . . . 6 (((1st𝑦) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
85, 6, 7syl2anr 590 . . . . 5 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
9 addclpi 10051 . . . . 5 ((((1st𝑥) ·N (2nd𝑦)) ∈ N ∧ ((1st𝑦) ·N (2nd𝑥)) ∈ N) → (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))) ∈ N)
104, 8, 9syl2anc 579 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))) ∈ N)
11 mulclpi 10052 . . . . 5 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
126, 2, 11syl2an 589 . . . 4 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd𝑥) ·N (2nd𝑦)) ∈ N)
1310, 12opelxpd 5395 . . 3 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N))
1413rgen2a 3159 . 2 𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ (N × N)
15 df-plpq 10067 . . 3 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
1615fmpt2 7519 . 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 222 1 +pQ :((N × N) × (N × N))⟶(N × N)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 386   ∈ wcel 2107  ∀wral 3090  ⟨cop 4404   × cxp 5355  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924  1st c1st 7445  2nd c2nd 7446  Ncnpi 10003   +N cpli 10004   ·N cmi 10005   +pQ cplpq 10007 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-oadd 7849  df-omul 7850  df-ni 10031  df-pli 10032  df-mi 10033  df-plpq 10067 This theorem is referenced by:  addclnq  10104  addnqf  10107  addcompq  10109  adderpq  10115  distrnq  10120
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