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Mirrors > Home > MPE Home > Th. List > uz2m1nn | Structured version Visualization version GIF version |
Description: One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.) |
Ref | Expression |
---|---|
uz2m1nn | ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2b1 12313 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁)) | |
2 | 1z 12006 | . . . 4 ⊢ 1 ∈ ℤ | |
3 | znnsub 12022 | . . . 4 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 < 𝑁 ↔ (𝑁 − 1) ∈ ℕ)) | |
4 | 2, 3 | mpan 688 | . . 3 ⊢ (𝑁 ∈ ℤ → (1 < 𝑁 ↔ (𝑁 − 1) ∈ ℕ)) |
5 | 4 | biimpa 479 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (𝑁 − 1) ∈ ℕ) |
6 | 1, 5 | sylbi 219 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 1c1 10532 < clt 10669 − cmin 10864 ℕcn 11632 2c2 11686 ℤcz 11975 ℤ≥cuz 12237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 |
This theorem is referenced by: nn0ge2m1nnALT 12336 bernneq3 13586 pfxtrcfv0 14050 climcndslem1 15198 exprmfct 16042 oddprm 16141 pockthg 16236 vdwlem5 16315 vdwlem8 16318 efgs1b 18856 efgredlema 18860 wilthlem3 25641 ppiprm 25722 ppinprm 25723 chtprm 25724 chtnprm 25725 lgsval2lem 25877 lgsqrlem2 25917 lgseisenlem1 25945 lgseisenlem3 25947 lgsquadlem3 25952 rplogsumlem1 26054 rplogsumlem2 26055 rpvmasumlem 26057 clwwisshclwwslemlem 27785 umgr2cwwk2dif 27837 psgnfzto1stlem 30737 ballotlemic 31759 ballotlem1c 31760 signstfveq0 31842 fltnltalem 39267 fltnlta 39268 jm3.1lem1 39607 jm3.1lem2 39608 trclfvdecomr 40066 itgsinexp 42233 stirlinglem12 42364 fourierdlem54 42439 fourierdlem102 42487 fourierdlem114 42499 blennngt2o2 44646 |
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