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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 9066 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
2 | zcn 9066 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
3 | negsub 8017 | . . 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 9092 | . . 3 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
6 | zaddcl 9101 | . . 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 2217 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7625 + caddc 7630 − cmin 7940 -cneg 7941 ℤcz 9061 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-addcom 7727 ax-addass 7729 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-0id 7735 ax-rnegex 7736 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-ltadd 7743 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-inn 8728 df-n0 8985 df-z 9062 |
This theorem is referenced by: ztri3or 9104 zrevaddcl 9111 znnsub 9112 nzadd 9113 znn0sub 9126 zneo 9159 zsubcld 9185 eluzsubi 9360 fzen 9830 uzsubsubfz 9834 fzrev 9871 fzrev2 9872 fzrevral2 9893 fzshftral 9895 fz0fzdiffz0 9914 difelfzle 9918 difelfznle 9919 elfzomelpfzo 10015 zmodcl 10124 frecfzen2 10207 facndiv 10492 bccmpl 10507 bcpasc 10519 hashfz 10574 moddvds 11509 modmulconst 11532 dvds2sub 11535 dvdssub2 11542 dvdssubr 11546 fzocongeq 11563 odd2np1 11577 omoe 11600 omeo 11602 divalgb 11629 divalgmod 11631 ndvdsadd 11635 nn0seqcvgd 11729 congr 11788 cncongr1 11791 cncongr2 11792 |
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