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Mirrors > Home > ILE Home > Th. List > mulclpi | GIF version |
Description: Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
mulclpi | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulpiord 7118 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) | |
2 | pinn 7110 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
3 | pinn 7110 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
4 | nnmcl 6370 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω) | |
5 | 2, 3, 4 | syl2an 287 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ∈ ω) |
6 | elni2 7115 | . . . . . . 7 ⊢ (𝐵 ∈ N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) | |
7 | 6 | simprbi 273 | . . . . . 6 ⊢ (𝐵 ∈ N → ∅ ∈ 𝐵) |
8 | 7 | adantl 275 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐵) |
9 | 3 | adantl 275 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐵 ∈ ω) |
10 | 2 | adantr 274 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐴 ∈ ω) |
11 | elni2 7115 | . . . . . . . 8 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) | |
12 | 11 | simprbi 273 | . . . . . . 7 ⊢ (𝐴 ∈ N → ∅ ∈ 𝐴) |
13 | 12 | adantr 274 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ∅ ∈ 𝐴) |
14 | nnmordi 6405 | . . . . . 6 ⊢ (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵))) | |
15 | 9, 10, 13, 14 | syl21anc 1215 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (∅ ∈ 𝐵 → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵))) |
16 | 8, 15 | mpd 13 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)) |
17 | ne0i 3364 | . . . 4 ⊢ ((𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵) → (𝐴 ·o 𝐵) ≠ ∅) | |
18 | 16, 17 | syl 14 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ≠ ∅) |
19 | elni 7109 | . . 3 ⊢ ((𝐴 ·o 𝐵) ∈ N ↔ ((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐵) ≠ ∅)) | |
20 | 5, 18, 19 | sylanbrc 413 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) ∈ N) |
21 | 1, 20 | eqeltrd 2214 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ≠ wne 2306 ∅c0 3358 ωcom 4499 (class class class)co 5767 ·o comu 6304 Ncnpi 7073 ·N cmi 7075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-omul 6311 df-ni 7105 df-mi 7107 |
This theorem is referenced by: mulasspig 7133 distrpig 7134 ltmpig 7140 enqer 7159 enqdc 7162 addcmpblnq 7168 mulcmpblnq 7169 addpipqqslem 7170 mulpipq2 7172 mulpipqqs 7174 ordpipqqs 7175 addclnq 7176 mulclnq 7177 addcomnqg 7182 addassnqg 7183 mulassnqg 7185 mulcanenq 7186 distrnqg 7188 recexnq 7191 nqtri3or 7197 ltdcnq 7198 ltsonq 7199 ltanqg 7201 ltmnqg 7202 1lt2nq 7207 ltexnqq 7209 archnqq 7218 addcmpblnq0 7244 mulcmpblnq0 7245 mulcanenq0ec 7246 addclnq0 7252 mulclnq0 7253 nqpnq0nq 7254 nqnq0a 7255 nqnq0m 7256 nq0m0r 7257 distrnq0 7260 addassnq0lemcl 7262 |
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