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Mirrors > Home > ILE Home > Th. List > peano2zm | GIF version |
Description: "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.) |
Ref | Expression |
---|---|
peano2zm | ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9172 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1cnd 7894 | . . . 4 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
3 | 1, 2 | negsubdid 8201 | . . 3 ⊢ (𝑁 ∈ ℤ → -(𝑁 − 1) = (-𝑁 + 1)) |
4 | znegcl 9198 | . . . 4 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
5 | peano2z 9203 | . . . 4 ⊢ (-𝑁 ∈ ℤ → (-𝑁 + 1) ∈ ℤ) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝑁 ∈ ℤ → (-𝑁 + 1) ∈ ℤ) |
7 | 3, 6 | eqeltrd 2234 | . 2 ⊢ (𝑁 ∈ ℤ → -(𝑁 − 1) ∈ ℤ) |
8 | 1, 2 | subcld 8186 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℂ) |
9 | znegclb 9200 | . . 3 ⊢ ((𝑁 − 1) ∈ ℂ → ((𝑁 − 1) ∈ ℤ ↔ -(𝑁 − 1) ∈ ℤ)) | |
10 | 8, 9 | syl 14 | . 2 ⊢ (𝑁 ∈ ℤ → ((𝑁 − 1) ∈ ℤ ↔ -(𝑁 − 1) ∈ ℤ)) |
11 | 7, 10 | mpbird 166 | 1 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 2128 (class class class)co 5824 ℂcc 7730 1c1 7733 + caddc 7735 − cmin 8046 -cneg 8047 ℤcz 9167 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-n0 9091 df-z 9168 |
This theorem is referenced by: zaddcllemneg 9206 zlem1lt 9223 zltlem1 9224 zextlt 9256 zeo 9269 eluzp1m1 9462 fz01en 9955 fzsuc2 9981 elfzm11 9993 uzdisj 9995 fzof 10043 fzoval 10047 elfzo 10048 fzodcel 10051 fzon 10065 fzoss2 10071 fzossrbm1 10072 fzosplitsnm1 10108 ubmelm1fzo 10125 elfzom1b 10128 fzosplitprm1 10133 fzoshftral 10137 fzofig 10331 uzsinds 10341 ser3mono 10377 iseqf1olemqcl 10385 iseqf1olemnab 10387 iseqf1olemab 10388 seq3f1olemqsumkj 10397 seq3f1olemqsum 10399 bcm1k 10634 bcn2 10638 bcp1m1 10639 bcpasc 10640 bccl 10641 zfz1isolemiso 10710 seq3coll 10713 resqrexlemcalc3 10916 resqrexlemnm 10918 fsumm1 11313 binomlem 11380 binom1dif 11384 isumsplit 11388 arisum2 11396 pwm1geoserap1 11405 mertenslemi1 11432 fprodm1 11495 fprodeq0 11514 zeo3 11759 oddm1even 11766 oddp1even 11767 zob 11782 nno 11797 isprm3 11995 prmdc 12007 phibnd 12092 hashdvds 12096 odzcllem 12117 odzdvds 12120 oddennn 12132 |
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