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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 10023 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) | |
2 | pinn 10016 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
3 | pinn 10016 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
4 | nnmcl 7960 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω) | |
5 | 2, 3, 4 | syl2an 591 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ∈ ω) |
6 | elni2 10015 | . . . . . . 7 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) | |
7 | 6 | simprbi 492 | . . . . . 6 ⊢ (𝐵 ∈ N → ∅ ∈ 𝐵) |
8 | 7 | adantl 475 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐵) |
9 | 3 | adantl 475 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐵 ∈ ω) |
10 | 2 | adantr 474 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐴 ∈ ω) |
11 | elni2 10015 | . . . . . . . 8 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) | |
12 | 11 | simprbi 492 | . . . . . . 7 ⊢ (𝐴 ∈ N → ∅ ∈ 𝐴) |
13 | 12 | adantr 474 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐴) |
14 | nnmordi 7979 | . . . . . 6 ⊢ (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵))) | |
15 | 9, 10, 13, 14 | syl21anc 873 | . . . . 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 4152 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ≠ ∅) |
18 | elni 10014 | . . 3 ⊢ ((𝐴 ·o 𝐵) ∈ N ↔ ((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐵) ≠ ∅)) | |
19 | 5, 17, 18 | sylanbrc 580 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ∈ N) |
20 | 1, 19 | eqeltrd 2907 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 ≠ wne 3000 ∅c0 4145 (class class class)co 6906 ωcom 7327 ·o comu 7825 Ncnpi 9982 ·N cmi 9984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-oadd 7831 df-omul 7832 df-ni 10010 df-mi 10012 |
This theorem is referenced by: mulasspi 10035 distrpi 10036 mulcanpi 10038 ltmpi 10042 enqer 10059 addpqf 10082 mulpqf 10084 adderpqlem 10092 mulerpqlem 10093 addassnq 10096 mulassnq 10097 mulcanenq 10098 distrnq 10099 recmulnq 10102 ltsonq 10107 lterpq 10108 ltanq 10109 ltmnq 10110 ltexnq 10113 archnq 10118 |
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