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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 9256 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1cnd 7972 | . . . 4 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
3 | 1, 2 | negsubdid 8281 | . . 3 ⊢ (𝑁 ∈ ℤ → -(𝑁 − 1) = (-𝑁 + 1)) |
4 | znegcl 9282 | . . . 4 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
5 | peano2z 9287 | . . . 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 8266 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℂ) |
9 | znegclb 9284 | . . 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 5874 ℂcc 7808 1c1 7811 + caddc 7813 − cmin 8126 -cneg 8127 ℤcz 9251 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-n0 9175 df-z 9252 |
This theorem is referenced by: zaddcllemneg 9290 zlem1lt 9307 zltlem1 9308 zextlt 9343 zeo 9356 eluzp1m1 9549 fz01en 10050 fzsuc2 10076 elfzm11 10088 uzdisj 10090 fzof 10141 fzoval 10145 elfzo 10146 fzodcel 10149 fzon 10163 fzoss2 10169 fzossrbm1 10170 fzosplitsnm1 10206 ubmelm1fzo 10223 elfzom1b 10226 fzosplitprm1 10231 fzoshftral 10235 fzofig 10429 uzsinds 10439 ser3mono 10475 iseqf1olemqcl 10483 iseqf1olemnab 10485 iseqf1olemab 10486 seq3f1olemqsumkj 10495 seq3f1olemqsum 10497 bcm1k 10735 bcn2 10739 bcp1m1 10740 bcpasc 10741 bccl 10742 zfz1isolemiso 10814 seq3coll 10817 resqrexlemcalc3 11020 resqrexlemnm 11022 fsumm1 11419 binomlem 11486 binom1dif 11490 isumsplit 11494 arisum2 11502 pwm1geoserap1 11511 mertenslemi1 11538 fprodm1 11601 fprodeq0 11620 zeo3 11867 oddm1even 11874 oddp1even 11875 zob 11890 nno 11905 isprm3 12112 prmdc 12124 isprm5 12136 phibnd 12211 hashdvds 12215 odzcllem 12236 odzdvds 12239 fldivp1 12340 pockthlem 12348 oddennn 12387 lgslem1 14294 lgsval2lem 14304 2sqlem8 14352 |
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