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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 7089 | . . . . . 6 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N) | |
2 | xp2nd 7090 | . . . . . 6 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N) | |
3 | mulclpi 9594 | . . . . . 6 ⊢ (((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | |
4 | 1, 2, 3 | syl2an 493 | . . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) |
5 | xp1st 7089 | . . . . . 6 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N) | |
6 | xp2nd 7090 | . . . . . 6 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N) | |
7 | mulclpi 9594 | . . . . . 6 ⊢ (((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑥) ∈ N) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N) | |
8 | 5, 6, 7 | syl2anr 494 | . . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N) |
9 | addclpi 9593 | . . . . 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 691 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∈ N) |
11 | mulclpi 9594 | . . . . 5 ⊢ (((2nd ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | |
12 | 6, 2, 11 | syl2an 493 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) |
13 | opelxpi 5072 | . . . 4 ⊢ (((((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∈ N ∧ ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) → 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ (N × N)) | |
14 | 10, 12, 13 | syl2anc 691 | . . 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)) |
15 | 14 | rgen2a 2960 | . 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) |
16 | df-plpq 9609 | . . 3 ⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
17 | 16 | fmpt2 7126 | . 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)) |
18 | 15, 17 | mpbi 219 | 1 ⊢ +pQ :((N × N) × (N × N))⟶(N × N) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 1977 ∀wral 2896 〈cop 4131 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 Ncnpi 9545 +N cpli 9546 ·N cmi 9547 +pQ cplpq 9549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-omul 7452 df-ni 9573 df-pli 9574 df-mi 9575 df-plpq 9609 |
This theorem is referenced by: addclnq 9646 addnqf 9649 addcompq 9651 adderpq 9657 distrnq 9662 |
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