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| Mirrors > Home > MPE Home > Th. List > mulpqf | Structured version Visualization version GIF version | ||
| Description: Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| mulpqf | ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xp1st 8046 | . . . . 5 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N) | |
| 2 | xp1st 8046 | . . . . 5 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N) | |
| 3 | mulclpi 10933 | . . . . 5 ⊢ (((1st ‘𝑥) ∈ N ∧ (1st ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (1st ‘𝑦)) ∈ N) | |
| 4 | 1, 2, 3 | syl2an 596 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑥) ·N (1st ‘𝑦)) ∈ N) | 
| 5 | xp2nd 8047 | . . . . 5 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N) | |
| 6 | xp2nd 8047 | . . . . 5 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N) | |
| 7 | mulclpi 10933 | . . . . 5 ⊢ (((2nd ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | |
| 8 | 5, 6, 7 | syl2an 596 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | 
| 9 | 4, 8 | opelxpd 5724 | . . 3 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ (N × N)) | 
| 10 | 9 | rgen2 3199 | . 2 ⊢ ∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ (N × N) | 
| 11 | df-mpq 10949 | . . 3 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
| 12 | 11 | fmpo 8093 | . 2 ⊢ (∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ (N × N) ↔ ·pQ :((N × N) × (N × N))⟶(N × N)) | 
| 13 | 10, 12 | mpbi 230 | 1 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∈ wcel 2108 ∀wral 3061 〈cop 4632 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 Ncnpi 10884 ·N cmi 10886 ·pQ cmpq 10889 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-oadd 8510 df-omul 8511 df-ni 10912 df-mi 10914 df-mpq 10949 | 
| This theorem is referenced by: mulclnq 10987 mulnqf 10989 mulcompq 10992 mulerpq 10997 distrnq 11001 | 
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