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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 7958 | . . . . . 6 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N) | |
2 | xp2nd 7959 | . . . . . 6 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N) | |
3 | mulclpi 10836 | . . . . . 6 ⊢ (((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | |
4 | 1, 2, 3 | syl2an 597 | . . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) |
5 | xp1st 7958 | . . . . . 6 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N) | |
6 | xp2nd 7959 | . . . . . 6 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N) | |
7 | mulclpi 10836 | . . . . . 6 ⊢ (((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑥) ∈ N) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N) | |
8 | 5, 6, 7 | syl2anr 598 | . . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N) |
9 | addclpi 10835 | . . . . 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 585 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∈ N) |
11 | mulclpi 10836 | . . . . 5 ⊢ (((2nd ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | |
12 | 6, 2, 11 | syl2an 597 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) |
13 | 10, 12 | opelxpd 5676 | . . 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 3195 | . 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 10851 | . . 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 8005 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 ∀wral 3065 ⟨cop 4597 × cxp 5636 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 1st c1st 7924 2nd c2nd 7925 Ncnpi 10787 +N cpli 10788 ·N cmi 10789 +pQ cplpq 10791 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-oadd 8421 df-omul 8422 df-ni 10815 df-pli 10816 df-mi 10817 df-plpq 10851 |
This theorem is referenced by: addclnq 10888 addnqf 10891 addcompq 10893 adderpq 10899 distrnq 10904 |
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