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Mirrors > Home > ILE Home > Th. List > zsubcl | GIF version |
Description: Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) |
Ref | Expression |
---|---|
zsubcl | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9187 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
2 | zcn 9187 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
3 | negsub 8137 | . . 3 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 + -𝑁) = (𝑀 − 𝑁)) | |
4 | 1, 2, 3 | syl2an 287 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + -𝑁) = (𝑀 − 𝑁)) |
5 | znegcl 9213 | . . 3 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
6 | zaddcl 9222 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 + -𝑁) ∈ ℤ) | |
7 | 5, 6 | sylan2 284 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + -𝑁) ∈ ℤ) |
8 | 4, 7 | eqeltrrd 2242 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 (class class class)co 5836 ℂcc 7742 + caddc 7747 − cmin 8060 -cneg 8061 ℤcz 9182 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 |
This theorem is referenced by: ztri3or 9225 zrevaddcl 9232 znnsub 9233 nzadd 9234 znn0sub 9247 zneo 9283 zsubcld 9309 eluzsubi 9484 fzen 9968 uzsubsubfz 9972 fzrev 10009 fzrev2 10010 fzrevral2 10031 fzshftral 10033 fz0fzdiffz0 10055 difelfzle 10059 difelfznle 10060 elfzomelpfzo 10156 zmodcl 10269 frecfzen2 10352 facndiv 10641 bccmpl 10656 bcpasc 10668 hashfz 10723 moddvds 11725 modmulconst 11749 dvds2sub 11752 dvdssub2 11760 dvdssubr 11764 fzocongeq 11781 odd2np1 11795 omoe 11818 omeo 11820 divalgb 11847 divalgmod 11849 ndvdsadd 11853 nn0seqcvgd 11952 congr 12011 cncongr1 12014 cncongr2 12015 prmdiv 12146 prmdiveq 12147 pythagtriplem4 12179 pythagtriplem8 12183 difsqpwdvds 12248 |
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