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Mirrors > Home > ILE Home > Th. List > nnm1nn0 | GIF version |
Description: A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nnm1nn0 | ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn1m1nn 8937 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) | |
2 | oveq1 5882 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
3 | 1m1e0 8988 | . . . . . 6 ⊢ (1 − 1) = 0 | |
4 | 2, 3 | eqtrdi 2226 | . . . . 5 ⊢ (𝑁 = 1 → (𝑁 − 1) = 0) |
5 | 4 | orim1i 760 | . . . 4 ⊢ ((𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ) → ((𝑁 − 1) = 0 ∨ (𝑁 − 1) ∈ ℕ)) |
6 | 1, 5 | syl 14 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) = 0 ∨ (𝑁 − 1) ∈ ℕ)) |
7 | 6 | orcomd 729 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) = 0)) |
8 | elnn0 9178 | . 2 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ ((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) = 0)) | |
9 | 7, 8 | sylibr 134 | 1 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 708 = wceq 1353 ∈ wcel 2148 (class class class)co 5875 0cc0 7811 1c1 7812 − cmin 8128 ℕcn 8919 ℕ0cn0 9176 |
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-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 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-0id 7919 ax-rnegex 7920 ax-cnre 7922 |
This theorem depends on definitions: df-bi 117 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-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-sub 8130 df-inn 8920 df-n0 9177 |
This theorem is referenced by: elnn0nn 9218 nnaddm1cl 9314 nn0n0n1ge2 9323 fseq1m1p1 10095 nn0ennn 10433 expm1t 10548 expgt1 10558 nn0ltexp2 10689 bcn1 10738 bcm1k 10740 bcn2m1 10749 resqrexlemnm 11027 resqrexlemcvg 11028 resqrexlemga 11032 binomlem 11491 arisum 11506 arisum2 11507 cvgratnnlemnexp 11532 cvgratnnlemfm 11537 mertenslem2 11544 iddvdsexp 11822 dvdsfac 11866 oexpneg 11882 phibnd 12217 phiprmpw 12222 prmdiv 12235 oddprm 12259 fldivp1 12346 prmpwdvds 12353 dvexp 14178 lgslem1 14404 m1lgs 14455 |
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