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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 7710 | . . . . 5 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N) | |
2 | xp1st 7710 | . . . . 5 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N) | |
3 | mulclpi 10303 | . . . . 5 ⊢ (((1st ‘𝑥) ∈ N ∧ (1st ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (1st ‘𝑦)) ∈ N) | |
4 | 1, 2, 3 | syl2an 595 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑥) ·N (1st ‘𝑦)) ∈ N) |
5 | xp2nd 7711 | . . . . 5 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N) | |
6 | xp2nd 7711 | . . . . 5 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N) | |
7 | mulclpi 10303 | . . . . 5 ⊢ (((2nd ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | |
8 | 5, 6, 7 | syl2an 595 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) |
9 | 4, 8 | opelxpd 5586 | . . 3 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ (N × N)) |
10 | 9 | rgen2 3200 | . 2 ⊢ ∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ (N × N) |
11 | df-mpq 10319 | . . 3 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
12 | 11 | fmpo 7755 | . 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 231 | 1 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2105 ∀wral 3135 〈cop 4563 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 Ncnpi 10254 ·N cmi 10256 ·pQ cmpq 10259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-oadd 8095 df-omul 8096 df-ni 10282 df-mi 10284 df-mpq 10319 |
This theorem is referenced by: mulclnq 10357 mulnqf 10359 mulcompq 10362 mulerpq 10367 distrnq 10371 |
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