![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > addpqf | Structured version Visualization version GIF version |
Description: Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addpqf | ⊢ +pQ :((N × N) × (N × N))⟶(N × N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xp1st 7703 | . . . . . 6 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N) | |
2 | xp2nd 7704 | . . . . . 6 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N) | |
3 | mulclpi 10304 | . . . . . 6 ⊢ (((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | |
4 | 1, 2, 3 | syl2an 598 | . . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) |
5 | xp1st 7703 | . . . . . 6 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N) | |
6 | xp2nd 7704 | . . . . . 6 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N) | |
7 | mulclpi 10304 | . . . . . 6 ⊢ (((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑥) ∈ N) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N) | |
8 | 5, 6, 7 | syl2anr 599 | . . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N) |
9 | addclpi 10303 | . . . . 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 587 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∈ N) |
11 | mulclpi 10304 | . . . . 5 ⊢ (((2nd ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | |
12 | 6, 2, 11 | syl2an 598 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) |
13 | 10, 12 | opelxpd 5557 | . . 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 3168 | . 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 10319 | . . 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 7748 | . 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 233 | 1 ⊢ +pQ :((N × N) × (N × N))⟶(N × N) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2111 ∀wral 3106 〈cop 4531 × cxp 5517 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 Ncnpi 10255 +N cpli 10256 ·N cmi 10257 +pQ cplpq 10259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 df-omul 8090 df-ni 10283 df-pli 10284 df-mi 10285 df-plpq 10319 |
This theorem is referenced by: addclnq 10356 addnqf 10359 addcompq 10361 adderpq 10367 distrnq 10372 |
Copyright terms: Public domain | W3C validator |