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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 8936 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) | |
2 | oveq1 5881 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
3 | 1m1e0 8987 | . . . . . 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 9177 | . 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 5874 0cc0 7810 1c1 7811 − cmin 8127 ℕcn 8918 ℕ0cn0 9175 |
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 4121 ax-pow 4174 ax-pr 4209 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-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
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 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-sub 8129 df-inn 8919 df-n0 9176 |
This theorem is referenced by: elnn0nn 9217 nnaddm1cl 9313 nn0n0n1ge2 9322 fseq1m1p1 10094 nn0ennn 10432 expm1t 10547 expgt1 10557 nn0ltexp2 10688 bcn1 10737 bcm1k 10739 bcn2m1 10748 resqrexlemnm 11026 resqrexlemcvg 11027 resqrexlemga 11031 binomlem 11490 arisum 11505 arisum2 11506 cvgratnnlemnexp 11531 cvgratnnlemfm 11536 mertenslem2 11543 iddvdsexp 11821 dvdsfac 11865 oexpneg 11881 phibnd 12216 phiprmpw 12221 prmdiv 12234 oddprm 12258 fldivp1 12345 prmpwdvds 12352 dvexp 14145 lgslem1 14371 m1lgs 14422 |
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