Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10589 and ax-mulass 10591. (Revised by Steven Nguyen, 24-Sep-2022.) |
Ref | Expression |
---|---|
nnmulcl | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7153 | . . . . 5 ⊢ (𝑥 = 1 → (𝐴 · 𝑥) = (𝐴 · 1)) | |
2 | 1 | eleq1d 2894 | . . . 4 ⊢ (𝑥 = 1 → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · 1) ∈ ℕ)) |
3 | 2 | imbi2d 342 | . . 3 ⊢ (𝑥 = 1 → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · 1) ∈ ℕ))) |
4 | oveq2 7153 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 · 𝑥) = (𝐴 · 𝑦)) | |
5 | 4 | eleq1d 2894 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · 𝑦) ∈ ℕ)) |
6 | 5 | imbi2d 342 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · 𝑦) ∈ ℕ))) |
7 | oveq2 7153 | . . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 · 𝑥) = (𝐴 · (𝑦 + 1))) | |
8 | 7 | eleq1d 2894 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · (𝑦 + 1)) ∈ ℕ)) |
9 | 8 | imbi2d 342 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · (𝑦 + 1)) ∈ ℕ))) |
10 | oveq2 7153 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 · 𝑥) = (𝐴 · 𝐵)) | |
11 | 10 | eleq1d 2894 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · 𝐵) ∈ ℕ)) |
12 | 11 | imbi2d 342 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · 𝐵) ∈ ℕ))) |
13 | nnre 11633 | . . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
14 | ax-1rid 10595 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
15 | 14 | eleq1d 2894 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝐴 · 1) ∈ ℕ ↔ 𝐴 ∈ ℕ)) |
16 | 15 | biimprd 249 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℕ → (𝐴 · 1) ∈ ℕ)) |
17 | 13, 16 | mpcom 38 | . . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) ∈ ℕ) |
18 | nnaddcl 11648 | . . . . . . . 8 ⊢ (((𝐴 · 𝑦) ∈ ℕ ∧ 𝐴 ∈ ℕ) → ((𝐴 · 𝑦) + 𝐴) ∈ ℕ) | |
19 | 18 | ancoms 459 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐴 · 𝑦) ∈ ℕ) → ((𝐴 · 𝑦) + 𝐴) ∈ ℕ) |
20 | nncn 11634 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
21 | nncn 11634 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
22 | ax-1cn 10583 | . . . . . . . . . . 11 ⊢ 1 ∈ ℂ | |
23 | adddi 10614 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + (𝐴 · 1))) | |
24 | 22, 23 | mp3an3 1441 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + (𝐴 · 1))) |
25 | 20, 21, 24 | syl2an 595 | . . . . . . . . 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 481 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝐴 · 1) = 𝐴) |
28 | 27 | oveq2d 7161 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 · 𝑦) + (𝐴 · 1)) = ((𝐴 · 𝑦) + 𝐴)) |
29 | 25, 28 | eqtrd 2853 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + 𝐴)) |
30 | 29 | eleq1d 2894 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 · (𝑦 + 1)) ∈ ℕ ↔ ((𝐴 · 𝑦) + 𝐴) ∈ ℕ)) |
31 | 19, 30 | syl5ibr 247 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 ∈ ℕ ∧ (𝐴 · 𝑦) ∈ ℕ) → (𝐴 · (𝑦 + 1)) ∈ ℕ)) |
32 | 31 | exp4b 431 | . . . . 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 11644 | . 2 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ ℕ → (𝐴 · 𝐵) ∈ ℕ)) |
36 | 35 | impcom 408 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 ℝcr 10524 1c1 10526 + caddc 10528 · cmul 10530 ℕcn 11626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-addass 10590 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rrecex 10597 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 |
This theorem is referenced by: nnmulcli 11650 nnmtmip 11651 nndivtr 11672 nnmulcld 11678 nn0mulcl 11921 qaddcl 12352 qmulcl 12354 modmulnn 13245 nnexpcl 13430 nnsqcl 13481 expmulnbnd 13584 faccl 13631 facdiv 13635 faclbnd3 13640 faclbnd4lem3 13643 faclbnd5 13646 bcrpcl 13656 trirecip 15206 fprodnncl 15297 nnrisefaccl 15361 lcmgcdlem 15938 lcmgcdnn 15943 pcmptcl 16215 prmreclem1 16240 prmreclem6 16245 4sqlem12 16280 vdwlem3 16307 vdwlem9 16313 vdwlem10 16314 mulgnnass 18200 ovolunlem1a 24024 ovolunlem1 24025 mbfi1fseqlem3 24245 mbfi1fseqlem4 24246 elqaalem2 24836 elqaalem3 24837 log2cnv 25449 log2tlbnd 25450 log2ublem2 25452 log2ub 25454 basellem1 25585 basellem2 25586 basellem3 25587 basellem4 25588 basellem5 25589 basellem6 25590 basellem7 25591 basellem8 25592 basellem9 25593 efnnfsumcl 25607 efchtdvds 25663 mumullem1 25683 mumullem2 25684 fsumdvdscom 25689 dvdsflf1o 25691 chtublem 25714 pcbcctr 25779 bclbnd 25783 bposlem1 25787 bposlem2 25788 bposlem3 25789 bposlem4 25790 bposlem5 25791 bposlem6 25792 lgseisenlem1 25878 lgseisenlem2 25879 lgseisenlem3 25880 lgseisenlem4 25881 lgsquadlem1 25883 lgsquadlem2 25884 chebbnd1lem1 25972 chebbnd1lem3 25974 dchrisumlem1 25992 mulogsum 26035 pntrsumo1 26068 pntrsumbnd 26069 ostth2lem1 26121 subfaclim 32332 jm2.17a 39435 jm2.17b 39436 jm2.17c 39437 acongrep 39455 acongeq 39458 jm2.27a 39480 jm2.27c 39482 |
Copyright terms: Public domain | W3C validator |