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Mirrors > Home > MPE Home > Th. List > mulclpi | Structured version Visualization version GIF version |
Description: Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulclpi | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulpiord 10296 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) | |
2 | pinn 10289 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
3 | pinn 10289 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
4 | nnmcl 8221 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω) | |
5 | 2, 3, 4 | syl2an 598 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ∈ ω) |
6 | elni2 10288 | . . . . . . 7 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) | |
7 | 6 | simprbi 500 | . . . . . 6 ⊢ (𝐵 ∈ N → ∅ ∈ 𝐵) |
8 | 7 | adantl 485 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐵) |
9 | 3 | adantl 485 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐵 ∈ ω) |
10 | 2 | adantr 484 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐴 ∈ ω) |
11 | elni2 10288 | . . . . . . . 8 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) | |
12 | 11 | simprbi 500 | . . . . . . 7 ⊢ (𝐴 ∈ N → ∅ ∈ 𝐴) |
13 | 12 | adantr 484 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐴) |
14 | nnmordi 8240 | . . . . . 6 ⊢ (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵))) | |
15 | 9, 10, 13, 14 | syl21anc 836 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (∅ ∈ 𝐵 → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵))) |
16 | 8, 15 | mpd 15 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)) |
17 | 16 | ne0d 4251 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ≠ ∅) |
18 | elni 10287 | . . 3 ⊢ ((𝐴 ·o 𝐵) ∈ N ↔ ((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐵) ≠ ∅)) | |
19 | 5, 17, 18 | sylanbrc 586 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ∈ N) |
20 | 1, 19 | eqeltrd 2890 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 (class class class)co 7135 ωcom 7560 ·o comu 8083 Ncnpi 10255 ·N cmi 10257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 df-omul 8090 df-ni 10283 df-mi 10285 |
This theorem is referenced by: mulasspi 10308 distrpi 10309 mulcanpi 10311 ltmpi 10315 enqer 10332 addpqf 10355 mulpqf 10357 adderpqlem 10365 mulerpqlem 10366 addassnq 10369 mulassnq 10370 mulcanenq 10371 distrnq 10372 recmulnq 10375 ltsonq 10380 lterpq 10381 ltanq 10382 ltmnq 10383 ltexnq 10386 archnq 10391 |
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