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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 9083 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1cnd 7806 | . . . 4 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
3 | 1, 2 | negsubdid 8112 | . . 3 ⊢ (𝑁 ∈ ℤ → -(𝑁 − 1) = (-𝑁 + 1)) |
4 | znegcl 9109 | . . . 4 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
5 | peano2z 9114 | . . . 4 ⊢ (-𝑁 ∈ ℤ → (-𝑁 + 1) ∈ ℤ) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝑁 ∈ ℤ → (-𝑁 + 1) ∈ ℤ) |
7 | 3, 6 | eqeltrd 2217 | . 2 ⊢ (𝑁 ∈ ℤ → -(𝑁 − 1) ∈ ℤ) |
8 | 1, 2 | subcld 8097 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℂ) |
9 | znegclb 9111 | . . 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 1481 (class class class)co 5782 ℂcc 7642 1c1 7645 + caddc 7647 − cmin 7957 -cneg 7958 ℤcz 9078 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 |
This theorem is referenced by: zaddcllemneg 9117 zlem1lt 9134 zltlem1 9135 zextlt 9167 zeo 9180 eluzp1m1 9373 fz01en 9864 fzsuc2 9890 elfzm11 9902 uzdisj 9904 fzof 9952 fzoval 9956 elfzo 9957 fzodcel 9960 fzon 9974 fzoss2 9980 fzossrbm1 9981 fzosplitsnm1 10017 ubmelm1fzo 10034 elfzom1b 10037 fzosplitprm1 10042 fzoshftral 10046 fzofig 10236 uzsinds 10246 ser3mono 10282 iseqf1olemqcl 10290 iseqf1olemnab 10292 iseqf1olemab 10293 seq3f1olemqsumkj 10302 seq3f1olemqsum 10304 bcm1k 10538 bcn2 10542 bcp1m1 10543 bcpasc 10544 bccl 10545 zfz1isolemiso 10614 seq3coll 10617 resqrexlemcalc3 10820 resqrexlemnm 10822 fsumm1 11217 binomlem 11284 binom1dif 11288 isumsplit 11292 arisum2 11300 pwm1geoserap1 11309 mertenslemi1 11336 zeo3 11601 oddm1even 11608 oddp1even 11609 zob 11624 nno 11639 isprm3 11835 phibnd 11929 hashdvds 11933 oddennn 11941 |
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