![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9258 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1cnd 7973 | . . . 4 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
3 | 1, 2 | negsubdid 8283 | . . 3 ⊢ (𝑁 ∈ ℤ → -(𝑁 − 1) = (-𝑁 + 1)) |
4 | znegcl 9284 | . . . 4 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
5 | peano2z 9289 | . . . 4 ⊢ (-𝑁 ∈ ℤ → (-𝑁 + 1) ∈ ℤ) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝑁 ∈ ℤ → (-𝑁 + 1) ∈ ℤ) |
7 | 3, 6 | eqeltrd 2254 | . 2 ⊢ (𝑁 ∈ ℤ → -(𝑁 − 1) ∈ ℤ) |
8 | 1, 2 | subcld 8268 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℂ) |
9 | znegclb 9286 | . . 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 167 | 1 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2148 (class class class)co 5875 ℂcc 7809 1c1 7812 + caddc 7814 − cmin 8128 -cneg 8129 ℤcz 9253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-ltadd 7927 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-iota 5179 df-fun 5219 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-inn 8920 df-n0 9177 df-z 9254 |
This theorem is referenced by: zaddcllemneg 9292 zlem1lt 9309 zltlem1 9310 zextlt 9345 zeo 9358 eluzp1m1 9551 fz01en 10053 fzsuc2 10079 elfzm11 10091 uzdisj 10093 fzof 10144 fzoval 10148 elfzo 10149 fzodcel 10152 fzon 10166 fzoss2 10172 fzossrbm1 10173 fzosplitsnm1 10209 ubmelm1fzo 10226 elfzom1b 10229 fzosplitprm1 10234 fzoshftral 10238 fzofig 10432 uzsinds 10442 ser3mono 10478 iseqf1olemqcl 10486 iseqf1olemnab 10488 iseqf1olemab 10489 seq3f1olemqsumkj 10498 seq3f1olemqsum 10500 bcm1k 10740 bcn2 10744 bcp1m1 10745 bcpasc 10746 bccl 10747 zfz1isolemiso 10819 seq3coll 10822 resqrexlemcalc3 11025 resqrexlemnm 11027 fsumm1 11424 binomlem 11491 binom1dif 11495 isumsplit 11499 arisum2 11507 pwm1geoserap1 11516 mertenslemi1 11543 fprodm1 11606 fprodeq0 11625 zeo3 11873 oddm1even 11880 oddp1even 11881 zob 11896 nno 11911 isprm3 12118 prmdc 12130 isprm5 12142 phibnd 12217 hashdvds 12221 odzcllem 12242 odzdvds 12245 fldivp1 12346 pockthlem 12354 oddennn 12393 lgslem1 14404 lgsval2lem 14414 lgseisenlem1 14453 lgseisenlem2 14454 2sqlem8 14473 |
Copyright terms: Public domain | W3C validator |