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Mirrors > Home > MPE Home > Th. List > nnmulcl | Structured version Visualization version GIF version |
Description: Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) Remove dependency on ax-mulcom 10336 and ax-mulass 10338. (Revised by Steven Nguyen, 24-Sep-2022.) |
Ref | Expression |
---|---|
nnmulcl | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6930 | . . . . 5 ⊢ (𝑥 = 1 → (𝐴 · 𝑥) = (𝐴 · 1)) | |
2 | 1 | eleq1d 2844 | . . . 4 ⊢ (𝑥 = 1 → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · 1) ∈ ℕ)) |
3 | 2 | imbi2d 332 | . . 3 ⊢ (𝑥 = 1 → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · 1) ∈ ℕ))) |
4 | oveq2 6930 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 · 𝑥) = (𝐴 · 𝑦)) | |
5 | 4 | eleq1d 2844 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · 𝑦) ∈ ℕ)) |
6 | 5 | imbi2d 332 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · 𝑦) ∈ ℕ))) |
7 | oveq2 6930 | . . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 · 𝑥) = (𝐴 · (𝑦 + 1))) | |
8 | 7 | eleq1d 2844 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · (𝑦 + 1)) ∈ ℕ)) |
9 | 8 | imbi2d 332 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · (𝑦 + 1)) ∈ ℕ))) |
10 | oveq2 6930 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 · 𝑥) = (𝐴 · 𝐵)) | |
11 | 10 | eleq1d 2844 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · 𝐵) ∈ ℕ)) |
12 | 11 | imbi2d 332 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · 𝐵) ∈ ℕ))) |
13 | nnre 11382 | . . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
14 | ax-1rid 10342 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
15 | 14 | eleq1d 2844 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝐴 · 1) ∈ ℕ ↔ 𝐴 ∈ ℕ)) |
16 | 15 | biimprd 240 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℕ → (𝐴 · 1) ∈ ℕ)) |
17 | 13, 16 | mpcom 38 | . . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) ∈ ℕ) |
18 | nnaddcl 11398 | . . . . . . . 8 ⊢ (((𝐴 · 𝑦) ∈ ℕ ∧ 𝐴 ∈ ℕ) → ((𝐴 · 𝑦) + 𝐴) ∈ ℕ) | |
19 | 18 | ancoms 452 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐴 · 𝑦) ∈ ℕ) → ((𝐴 · 𝑦) + 𝐴) ∈ ℕ) |
20 | nncn 11383 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
21 | nncn 11383 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
22 | ax-1cn 10330 | . . . . . . . . . . 11 ⊢ 1 ∈ ℂ | |
23 | adddi 10361 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + (𝐴 · 1))) | |
24 | 22, 23 | mp3an3 1523 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + (𝐴 · 1))) |
25 | 20, 21, 24 | syl2an 589 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + (𝐴 · 1))) |
26 | 13, 14 | syl 17 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) = 𝐴) |
27 | 26 | adantr 474 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝐴 · 1) = 𝐴) |
28 | 27 | oveq2d 6938 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 · 𝑦) + (𝐴 · 1)) = ((𝐴 · 𝑦) + 𝐴)) |
29 | 25, 28 | eqtrd 2814 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + 𝐴)) |
30 | 29 | eleq1d 2844 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 · (𝑦 + 1)) ∈ ℕ ↔ ((𝐴 · 𝑦) + 𝐴) ∈ ℕ)) |
31 | 19, 30 | syl5ibr 238 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 ∈ ℕ ∧ (𝐴 · 𝑦) ∈ ℕ) → (𝐴 · (𝑦 + 1)) ∈ ℕ)) |
32 | 31 | exp4b 423 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (𝑦 ∈ ℕ → (𝐴 ∈ ℕ → ((𝐴 · 𝑦) ∈ ℕ → (𝐴 · (𝑦 + 1)) ∈ ℕ)))) |
33 | 32 | pm2.43b 55 | . . . 4 ⊢ (𝑦 ∈ ℕ → (𝐴 ∈ ℕ → ((𝐴 · 𝑦) ∈ ℕ → (𝐴 · (𝑦 + 1)) ∈ ℕ))) |
34 | 33 | a2d 29 | . . 3 ⊢ (𝑦 ∈ ℕ → ((𝐴 ∈ ℕ → (𝐴 · 𝑦) ∈ ℕ) → (𝐴 ∈ ℕ → (𝐴 · (𝑦 + 1)) ∈ ℕ))) |
35 | 3, 6, 9, 12, 17, 34 | nnind 11394 | . 2 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ ℕ → (𝐴 · 𝐵) ∈ ℕ)) |
36 | 35 | impcom 398 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 ℝcr 10271 1c1 10273 + caddc 10275 · cmul 10277 ℕcn 11374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-addass 10337 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rrecex 10344 ax-cnre 10345 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-nn 11375 |
This theorem is referenced by: nnmulcli 11401 nnmtmip 11402 nndivtr 11422 nnmulcld 11428 nn0mulcl 11680 qaddcl 12112 qmulcl 12114 modmulnn 13007 nnexpcl 13191 nnsqcl 13252 expmulnbnd 13315 faccl 13388 facdiv 13392 faclbnd3 13397 faclbnd4lem3 13400 faclbnd5 13403 bcrpcl 13413 trirecip 14999 fprodnncl 15088 nnrisefaccl 15152 lcmgcdlem 15725 lcmgcdnn 15730 pcmptcl 15999 prmreclem1 16024 prmreclem6 16029 4sqlem12 16064 vdwlem3 16091 vdwlem9 16097 vdwlem10 16098 mulgnnass 17961 ovolunlem1a 23700 ovolunlem1 23701 mbfi1fseqlem3 23921 mbfi1fseqlem4 23922 elqaalem2 24512 elqaalem3 24513 log2cnv 25123 log2tlbnd 25124 log2ublem2 25126 log2ub 25128 basellem1 25259 basellem2 25260 basellem3 25261 basellem4 25262 basellem5 25263 basellem6 25264 basellem7 25265 basellem8 25266 basellem9 25267 efnnfsumcl 25281 efchtdvds 25337 mumullem1 25357 mumullem2 25358 fsumdvdscom 25363 dvdsflf1o 25365 chtublem 25388 pcbcctr 25453 bclbnd 25457 bposlem1 25461 bposlem2 25462 bposlem3 25463 bposlem4 25464 bposlem5 25465 bposlem6 25466 lgseisenlem1 25552 lgseisenlem2 25553 lgseisenlem3 25554 lgseisenlem4 25555 lgsquadlem1 25557 lgsquadlem2 25558 chebbnd1lem1 25610 chebbnd1lem3 25612 dchrisumlem1 25630 mulogsum 25673 pntrsumo1 25706 pntrsumbnd 25707 ostth2lem1 25759 subfaclim 31769 jm2.17a 38490 jm2.17b 38491 jm2.17c 38492 acongrep 38510 acongeq 38513 jm2.27a 38535 jm2.27c 38537 |
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