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Mirrors > Home > MPE Home > Th. List > zmulcl | Structured version Visualization version GIF version |
Description: Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.) |
Ref | Expression |
---|---|
zmulcl | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elznn0 11997 | . 2 ⊢ (𝑀 ∈ ℤ ↔ (𝑀 ∈ ℝ ∧ (𝑀 ∈ ℕ0 ∨ -𝑀 ∈ ℕ0))) | |
2 | elznn0 11997 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) | |
3 | nn0mulcl 11934 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0) | |
4 | 3 | orcd 869 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0)) |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0))) |
6 | remulcl 10622 | . . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 · 𝑁) ∈ ℝ) | |
7 | 5, 6 | jctild 528 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝑀 · 𝑁) ∈ ℝ ∧ ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0)))) |
8 | nn0mulcl 11934 | . . . . . . . . 9 ⊢ ((-𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (-𝑀 · 𝑁) ∈ ℕ0) | |
9 | recn 10627 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → 𝑀 ∈ ℂ) | |
10 | recn 10627 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ) | |
11 | mulneg1 11076 | . . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (-𝑀 · 𝑁) = -(𝑀 · 𝑁)) | |
12 | 9, 10, 11 | syl2an 597 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (-𝑀 · 𝑁) = -(𝑀 · 𝑁)) |
13 | 12 | eleq1d 2897 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((-𝑀 · 𝑁) ∈ ℕ0 ↔ -(𝑀 · 𝑁) ∈ ℕ0)) |
14 | 8, 13 | syl5ib 246 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((-𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → -(𝑀 · 𝑁) ∈ ℕ0)) |
15 | olc 864 | . . . . . . . 8 ⊢ (-(𝑀 · 𝑁) ∈ ℕ0 → ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0)) | |
16 | 14, 15 | syl6 35 | . . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((-𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0))) |
17 | 16, 6 | jctild 528 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((-𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝑀 · 𝑁) ∈ ℝ ∧ ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0)))) |
18 | nn0mulcl 11934 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ0 ∧ -𝑁 ∈ ℕ0) → (𝑀 · -𝑁) ∈ ℕ0) | |
19 | mulneg2 11077 | . . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 · -𝑁) = -(𝑀 · 𝑁)) | |
20 | 9, 10, 19 | syl2an 597 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 · -𝑁) = -(𝑀 · 𝑁)) |
21 | 20 | eleq1d 2897 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 · -𝑁) ∈ ℕ0 ↔ -(𝑀 · 𝑁) ∈ ℕ0)) |
22 | 18, 21 | syl5ib 246 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 ∈ ℕ0 ∧ -𝑁 ∈ ℕ0) → -(𝑀 · 𝑁) ∈ ℕ0)) |
23 | 22, 15 | syl6 35 | . . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 ∈ ℕ0 ∧ -𝑁 ∈ ℕ0) → ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0))) |
24 | 23, 6 | jctild 528 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 ∈ ℕ0 ∧ -𝑁 ∈ ℕ0) → ((𝑀 · 𝑁) ∈ ℝ ∧ ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0)))) |
25 | nn0mulcl 11934 | . . . . . . . . 9 ⊢ ((-𝑀 ∈ ℕ0 ∧ -𝑁 ∈ ℕ0) → (-𝑀 · -𝑁) ∈ ℕ0) | |
26 | mul2neg 11079 | . . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (-𝑀 · -𝑁) = (𝑀 · 𝑁)) | |
27 | 9, 10, 26 | syl2an 597 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (-𝑀 · -𝑁) = (𝑀 · 𝑁)) |
28 | 27 | eleq1d 2897 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((-𝑀 · -𝑁) ∈ ℕ0 ↔ (𝑀 · 𝑁) ∈ ℕ0)) |
29 | 25, 28 | syl5ib 246 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((-𝑀 ∈ ℕ0 ∧ -𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0)) |
30 | orc 863 | . . . . . . . 8 ⊢ ((𝑀 · 𝑁) ∈ ℕ0 → ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0)) | |
31 | 29, 30 | syl6 35 | . . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((-𝑀 ∈ ℕ0 ∧ -𝑁 ∈ ℕ0) → ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0))) |
32 | 31, 6 | jctild 528 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((-𝑀 ∈ ℕ0 ∧ -𝑁 ∈ ℕ0) → ((𝑀 · 𝑁) ∈ ℝ ∧ ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0)))) |
33 | 7, 17, 24, 32 | ccased 1033 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝑀 ∈ ℕ0 ∨ -𝑀 ∈ ℕ0) ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)) → ((𝑀 · 𝑁) ∈ ℝ ∧ ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0)))) |
34 | elznn0 11997 | . . . . 5 ⊢ ((𝑀 · 𝑁) ∈ ℤ ↔ ((𝑀 · 𝑁) ∈ ℝ ∧ ((𝑀 · 𝑁) ∈ ℕ0 ∨ -(𝑀 · 𝑁) ∈ ℕ0))) | |
35 | 33, 34 | syl6ibr 254 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝑀 ∈ ℕ0 ∨ -𝑀 ∈ ℕ0) ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)) → (𝑀 · 𝑁) ∈ ℤ)) |
36 | 35 | imp 409 | . . 3 ⊢ (((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) ∧ ((𝑀 ∈ ℕ0 ∨ -𝑀 ∈ ℕ0) ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) → (𝑀 · 𝑁) ∈ ℤ) |
37 | 36 | an4s 658 | . 2 ⊢ (((𝑀 ∈ ℝ ∧ (𝑀 ∈ ℕ0 ∨ -𝑀 ∈ ℕ0)) ∧ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) → (𝑀 · 𝑁) ∈ ℤ) |
38 | 1, 2, 37 | syl2anb 599 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 ℝcr 10536 · cmul 10542 -cneg 10871 ℕ0cn0 11898 ℤcz 11982 |
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-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
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-nel 3124 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-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 |
This theorem is referenced by: zdivmul 12055 msqznn 12065 zmulcld 12094 uz2mulcl 12327 qaddcl 12365 qmulcl 12367 qreccl 12369 fzctr 13020 flmulnn0 13198 zexpcl 13445 iexpcyc 13570 zesq 13588 cshweqrep 14183 fprodzcl 15308 zrisefaccl 15374 zfallfaccl 15375 dvdsmul1 15631 dvdsmul2 15632 muldvds1 15634 muldvds2 15635 dvdscmul 15636 dvdsmulc 15637 dvdscmulr 15638 dvdsmulcr 15639 dvds2ln 15642 dvdstr 15646 dvdsmultr1 15647 dvdsmultr2 15649 3dvdsdec 15681 3dvds2dec 15682 oexpneg 15694 mulsucdiv2z 15702 divalglem0 15744 divalglem2 15746 divalglem4 15747 divalglem8 15751 divalgb 15755 divalgmod 15757 ndvdsi 15763 gcdaddmlem 15872 absmulgcd 15897 gcdmultipleOLD 15900 gcdmultiplezOLD 15901 dvdsmulgcd 15905 rpmulgcd 15906 lcmcllem 15940 rpmul 16003 cncongr1 16011 cncongr2 16012 eulerthlem2 16119 modprminv 16136 modprminveq 16137 modprm0 16142 pythagtriplem4 16156 pcpremul 16180 pcmul 16188 gzmulcl 16274 pgpfac1lem2 19197 zsubrg 20598 dvdsrzring 20630 mulgrhm 20645 domnchr 20679 znfld 20707 znunit 20710 mbfi1fseqlem5 24320 dvexp3 24575 basellem2 25659 basellem5 25662 dvdsflf1o 25764 chtub 25788 bposlem1 25860 bposlem5 25864 bposlem6 25865 lgslem3 25875 lgsval4a 25895 lgsneg 25897 lgsdir2 25906 lgsdchr 25931 lgseisenlem1 25951 lgseisenlem2 25952 lgseisenlem3 25953 lgsquadlem1 25956 lgsquad2lem2 25961 2lgsoddprmlem2 25985 chebbnd1lem1 26045 chebbnd1lem3 26047 knoppndvlem2 33852 fzmul 35031 mzpclall 39373 mzpindd 39392 acongrep 39626 acongeq 39629 jm2.18 39634 jm2.21 39640 jm2.26a 39646 jm2.26 39648 jm2.16nn0 39650 jm2.27a 39651 jm2.27c 39653 jm3.1lem3 39665 fourierswlem 42564 oexpnegALTV 43891 oexpnegnz 43892 tgblthelfgott 44029 2zrngmmgm 44266 zlmodzxzequa 44600 zlmodzxzequap 44603 |
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