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| 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 10926 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) | |
| 2 | pinn 10919 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
| 3 | pinn 10919 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
| 4 | nnmcl 8651 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω) | |
| 5 | 2, 3, 4 | syl2an 596 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ∈ ω) | 
| 6 | elni2 10918 | . . . . . . 7 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) | |
| 7 | 6 | simprbi 496 | . . . . . 6 ⊢ (𝐵 ∈ N → ∅ ∈ 𝐵) | 
| 8 | 7 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐵) | 
| 9 | 3 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐵 ∈ ω) | 
| 10 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐴 ∈ ω) | 
| 11 | elni2 10918 | . . . . . . . 8 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) | |
| 12 | 11 | simprbi 496 | . . . . . . 7 ⊢ (𝐴 ∈ N → ∅ ∈ 𝐴) | 
| 13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐴) | 
| 14 | nnmordi 8670 | . . . . . 6 ⊢ (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵))) | |
| 15 | 9, 10, 13, 14 | syl21anc 837 | . . . . 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 4341 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ≠ ∅) | 
| 18 | elni 10917 | . . 3 ⊢ ((𝐴 ·o 𝐵) ∈ N ↔ ((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐵) ≠ ∅)) | |
| 19 | 5, 17, 18 | sylanbrc 583 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ∈ N) | 
| 20 | 1, 19 | eqeltrd 2840 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 (class class class)co 7432 ωcom 7888 ·o comu 8505 Ncnpi 10885 ·N cmi 10887 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-oadd 8511 df-omul 8512 df-ni 10913 df-mi 10915 | 
| This theorem is referenced by: mulasspi 10938 distrpi 10939 mulcanpi 10941 ltmpi 10945 enqer 10962 addpqf 10985 mulpqf 10987 adderpqlem 10995 mulerpqlem 10996 addassnq 10999 mulassnq 11000 mulcanenq 11001 distrnq 11002 recmulnq 11005 ltsonq 11010 lterpq 11011 ltanq 11012 ltmnq 11013 ltexnq 11016 archnq 11021 | 
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