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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 10601 and ax-mulass 10603. (Revised by Steven Nguyen, 24-Sep-2022.) |
Ref | Expression |
---|---|
nnmulcl | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7164 | . . . . 5 ⊢ (𝑥 = 1 → (𝐴 · 𝑥) = (𝐴 · 1)) | |
2 | 1 | eleq1d 2897 | . . . 4 ⊢ (𝑥 = 1 → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · 1) ∈ ℕ)) |
3 | 2 | imbi2d 343 | . . 3 ⊢ (𝑥 = 1 → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · 1) ∈ ℕ))) |
4 | oveq2 7164 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 · 𝑥) = (𝐴 · 𝑦)) | |
5 | 4 | eleq1d 2897 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · 𝑦) ∈ ℕ)) |
6 | 5 | imbi2d 343 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · 𝑦) ∈ ℕ))) |
7 | oveq2 7164 | . . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 · 𝑥) = (𝐴 · (𝑦 + 1))) | |
8 | 7 | eleq1d 2897 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · (𝑦 + 1)) ∈ ℕ)) |
9 | 8 | imbi2d 343 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · (𝑦 + 1)) ∈ ℕ))) |
10 | oveq2 7164 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 · 𝑥) = (𝐴 · 𝐵)) | |
11 | 10 | eleq1d 2897 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 · 𝑥) ∈ ℕ ↔ (𝐴 · 𝐵) ∈ ℕ)) |
12 | 11 | imbi2d 343 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ℕ → (𝐴 · 𝑥) ∈ ℕ) ↔ (𝐴 ∈ ℕ → (𝐴 · 𝐵) ∈ ℕ))) |
13 | nnre 11645 | . . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
14 | ax-1rid 10607 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
15 | 14 | eleq1d 2897 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝐴 · 1) ∈ ℕ ↔ 𝐴 ∈ ℕ)) |
16 | 15 | biimprd 250 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℕ → (𝐴 · 1) ∈ ℕ)) |
17 | 13, 16 | mpcom 38 | . . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) ∈ ℕ) |
18 | nnaddcl 11661 | . . . . . . . 8 ⊢ (((𝐴 · 𝑦) ∈ ℕ ∧ 𝐴 ∈ ℕ) → ((𝐴 · 𝑦) + 𝐴) ∈ ℕ) | |
19 | 18 | ancoms 461 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐴 · 𝑦) ∈ ℕ) → ((𝐴 · 𝑦) + 𝐴) ∈ ℕ) |
20 | nncn 11646 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
21 | nncn 11646 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
22 | ax-1cn 10595 | . . . . . . . . . . 11 ⊢ 1 ∈ ℂ | |
23 | adddi 10626 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + (𝐴 · 1))) | |
24 | 22, 23 | mp3an3 1446 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + (𝐴 · 1))) |
25 | 20, 21, 24 | syl2an 597 | . . . . . . . . 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 483 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝐴 · 1) = 𝐴) |
28 | 27 | oveq2d 7172 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 · 𝑦) + (𝐴 · 1)) = ((𝐴 · 𝑦) + 𝐴)) |
29 | 25, 28 | eqtrd 2856 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + 𝐴)) |
30 | 29 | eleq1d 2897 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 · (𝑦 + 1)) ∈ ℕ ↔ ((𝐴 · 𝑦) + 𝐴) ∈ ℕ)) |
31 | 19, 30 | syl5ibr 248 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝐴 ∈ ℕ ∧ (𝐴 · 𝑦) ∈ ℕ) → (𝐴 · (𝑦 + 1)) ∈ ℕ)) |
32 | 31 | exp4b 433 | . . . . 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 11656 | . 2 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ ℕ → (𝐴 · 𝐵) ∈ ℕ)) |
36 | 35 | impcom 410 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 ℝcr 10536 1c1 10538 + caddc 10540 · cmul 10542 ℕcn 11638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-addass 10602 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rrecex 10609 ax-cnre 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 |
This theorem is referenced by: nnmulcli 11663 nnmtmip 11664 nndivtr 11685 nnmulcld 11691 nn0mulcl 11934 qaddcl 12365 qmulcl 12367 modmulnn 13258 nnexpcl 13443 nnsqcl 13494 expmulnbnd 13597 faccl 13644 facdiv 13648 faclbnd3 13653 faclbnd4lem3 13656 faclbnd5 13659 bcrpcl 13669 trirecip 15218 fprodnncl 15309 nnrisefaccl 15373 lcmgcdlem 15950 lcmgcdnn 15955 pcmptcl 16227 prmreclem1 16252 prmreclem6 16257 4sqlem12 16292 vdwlem3 16319 vdwlem9 16325 vdwlem10 16326 mulgnnass 18262 ovolunlem1a 24097 ovolunlem1 24098 mbfi1fseqlem3 24318 mbfi1fseqlem4 24319 elqaalem2 24909 elqaalem3 24910 log2cnv 25522 log2tlbnd 25523 log2ublem2 25525 log2ub 25527 basellem1 25658 basellem2 25659 basellem3 25660 basellem4 25661 basellem5 25662 basellem6 25663 basellem7 25664 basellem8 25665 basellem9 25666 efnnfsumcl 25680 efchtdvds 25736 mumullem1 25756 mumullem2 25757 fsumdvdscom 25762 dvdsflf1o 25764 chtublem 25787 pcbcctr 25852 bclbnd 25856 bposlem1 25860 bposlem2 25861 bposlem3 25862 bposlem4 25863 bposlem5 25864 bposlem6 25865 lgseisenlem1 25951 lgseisenlem2 25952 lgseisenlem3 25953 lgseisenlem4 25954 lgsquadlem1 25956 lgsquadlem2 25957 chebbnd1lem1 26045 chebbnd1lem3 26047 dchrisumlem1 26065 mulogsum 26108 pntrsumo1 26141 pntrsumbnd 26142 ostth2lem1 26194 subfaclim 32435 jm2.17a 39606 jm2.17b 39607 jm2.17c 39608 acongrep 39626 acongeq 39629 jm2.27a 39651 jm2.27c 39653 |
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