Theorem addpqf 10938

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.

Step	Hyp	Ref	Expression
1		xp1st 8003	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
2		xp2nd 8004	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
3		mulclpi 10887	. . . . . 6 ⊢ (((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
4	1, 2, 3	syl2an 595	. . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
5		xp1st 8003	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
6		xp2nd 8004	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
7		mulclpi 10887	. . . . . 6 ⊢ (((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑥) ∈ N) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
8	5, 6, 7	syl2anr 596	. . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
9		addclpi 10886	. . . . 5 ⊢ ((((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N ∧ ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∈ N)
10	4, 8, 9	syl2anc 583	. . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∈ N)
11		mulclpi 10887	. . . . 5 ⊢ (((2nd ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
12	6, 2, 11	syl2an 595	. . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
13	10, 12	opelxpd 5708	. . 3 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ (N × N))
14	13	rgen2 3191	. 2 ⊢ ∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ (N × N)
15		df-plpq 10902	. . 3 ⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
16	15	fmpo 8050	. 2 ⊢ (∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ (N × N) ↔ +pQ :((N × N) × (N × N))⟶(N × N))
17	14, 16	mpbi 229	1 ⊢ +pQ :((N × N) × (N × N))⟶(N × N)

Syntax hints:   ∧ wa 395   ∈ wcel 2098  ∀wral 3055  ⟨cop 4629   × cxp 5667  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404  1^st c1st 7969  2^nd c2nd 7970  Ncnpi 10838   +_N cpli 10839   ·_N cmi 10840   +_pQ cplpq 10842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-oadd 8468  df-omul 8469  df-ni 10866  df-pli 10867  df-mi 10868  df-plpq 10902

This theorem is referenced by:  addclnq  10939  addnqf  10942  addcompq  10944  adderpq  10950  distrnq  10955