![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7347 | . . . . 5 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N) | |
2 | xp1st 7347 | . . . . 5 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N) | |
3 | mulclpi 9917 | . . . . 5 ⊢ (((1st ‘𝑥) ∈ N ∧ (1st ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (1st ‘𝑦)) ∈ N) | |
4 | 1, 2, 3 | syl2an 583 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((1st ‘𝑥) ·N (1st ‘𝑦)) ∈ N) |
5 | xp2nd 7348 | . . . . 5 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N) | |
6 | xp2nd 7348 | . . . . 5 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N) | |
7 | mulclpi 9917 | . . . . 5 ⊢ (((2nd ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) | |
8 | 5, 6, 7 | syl2an 583 | . . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) |
9 | opelxpi 5288 | . . . 4 ⊢ ((((1st ‘𝑥) ·N (1st ‘𝑦)) ∈ N ∧ ((2nd ‘𝑥) ·N (2nd ‘𝑦)) ∈ N) → 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ (N × N)) | |
10 | 4, 8, 9 | syl2anc 573 | . . 3 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ (N × N)) |
11 | 10 | rgen2a 3126 | . 2 ⊢ ∀𝑥 ∈ (N × N)∀𝑦 ∈ (N × N)〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉 ∈ (N × N) |
12 | df-mpq 9933 | . . 3 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ 〈((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))〉) | |
13 | 12 | fmpt2 7387 | . 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)) |
14 | 11, 13 | mpbi 220 | 1 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 ∈ wcel 2145 ∀wral 3061 〈cop 4322 × cxp 5247 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 1st c1st 7313 2nd c2nd 7314 Ncnpi 9868 ·N cmi 9870 ·pQ cmpq 9873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-oadd 7717 df-omul 7718 df-ni 9896 df-mi 9898 df-mpq 9933 |
This theorem is referenced by: mulclnq 9971 mulnqf 9973 mulcompq 9976 mulerpq 9981 distrnq 9985 |
Copyright terms: Public domain | W3C validator |