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Mirrors > Home > ILE Home > Th. List > zsubcld | GIF version |
Description: Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
zred.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
zaddcld.1 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
Ref | Expression |
---|---|
zsubcld | ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | zaddcld.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
3 | zsubcl 8852 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) ∈ ℤ) | |
4 | 1, 2, 3 | syl2anc 404 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 (class class class)co 5666 − cmin 7714 ℤcz 8811 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-addass 7508 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-ltadd 7522 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-inn 8484 df-n0 8735 df-z 8812 |
This theorem is referenced by: eluzsub 9109 uzm1 9110 uzsubsubfz 9522 fzm1 9575 eluzgtdifelfzo 9669 ubmelm1fzo 9698 intfracq 9788 modqsubdir 9861 modsumfzodifsn 9864 addmodlteq 9866 seq3f1olemqsumkj 9988 zesq 10133 ibcval5 10232 hashfz 10290 seq3shft 10333 resqrexlemnmsq 10511 resqrexlemcvg 10513 fzomaxdiflem 10606 fsum0diaglem 10895 fisum0diag 10896 mptfzshft 10897 fsumrev 10898 fsumshft 10899 fisum0diag2 10902 geo2sum 10969 cvgratnnlemabsle 10982 cvgratnnlemsumlt 10983 cvgratz 10987 mertenslemub 10989 mertenslemi1 10990 mertenslem2 10991 mertensabs 10992 eirraplem 11125 fzocongeq 11198 modremain 11268 bezoutlemnewy 11324 cncongr1 11424 prmind2 11441 pw2dvds 11483 hashdvds 11536 phiprmpw 11537 crth 11539 |
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