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Mirrors > Home > ILE Home > Th. List > zltp1le | GIF version |
Description: Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
zltp1le | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1 8945 | . . . 4 ⊢ ((𝑁 − 𝑀) ∈ ℕ → 1 ≤ (𝑁 − 𝑀)) | |
2 | 1 | a1i 9 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 − 𝑀) ∈ ℕ → 1 ≤ (𝑁 − 𝑀))) |
3 | znnsub 9307 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ)) | |
4 | zre 9260 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 9260 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | 1re 7959 | . . . . 5 ⊢ 1 ∈ ℝ | |
7 | leaddsub2 8399 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 1) ≤ 𝑁 ↔ 1 ≤ (𝑁 − 𝑀))) | |
8 | 6, 7 | mp3an2 1325 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 1) ≤ 𝑁 ↔ 1 ≤ (𝑁 − 𝑀))) |
9 | 4, 5, 8 | syl2an 289 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 + 1) ≤ 𝑁 ↔ 1 ≤ (𝑁 − 𝑀))) |
10 | 2, 3, 9 | 3imtr4d 203 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 → (𝑀 + 1) ≤ 𝑁)) |
11 | 4 | adantr 276 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ) |
12 | 11 | ltp1d 8890 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 < (𝑀 + 1)) |
13 | peano2re 8096 | . . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ) | |
14 | 11, 13 | syl 14 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 1) ∈ ℝ) |
15 | 5 | adantl 277 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ) |
16 | ltletr 8050 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 < (𝑀 + 1) ∧ (𝑀 + 1) ≤ 𝑁) → 𝑀 < 𝑁)) | |
17 | 11, 14, 15, 16 | syl3anc 1238 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 < (𝑀 + 1) ∧ (𝑀 + 1) ≤ 𝑁) → 𝑀 < 𝑁)) |
18 | 12, 17 | mpand 429 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 + 1) ≤ 𝑁 → 𝑀 < 𝑁)) |
19 | 10, 18 | impbid 129 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 class class class wbr 4005 (class class class)co 5878 ℝcr 7813 1c1 7815 + caddc 7817 < clt 7995 ≤ cle 7996 − cmin 8131 ℕcn 8922 ℤcz 9256 |
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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-ltadd 7930 |
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 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-inn 8923 df-n0 9180 df-z 9257 |
This theorem is referenced by: zleltp1 9311 zlem1lt 9312 zgt0ge1 9314 nnltp1le 9316 nn0ltp1le 9318 btwnnz 9350 uzind2 9368 fzind 9371 btwnapz 9386 eluzp1l 9555 eluz2b1 9604 zltaddlt1le 10010 fzsplit2 10053 m1modge3gt1 10374 seq3f1olemqsumkj 10501 seq3f1olemqsumk 10502 bcval5 10746 seq3coll 10825 cvgratnnlemseq 11537 nn0o1gt2 11913 divalglemnqt 11928 zsupcllemstep 11949 infssuzex 11953 suprzubdc 11956 isprm3 12121 dvdsnprmd 12128 prmgt1 12135 oddprmge3 12138 znege1 12181 hashdvds 12224 lgsdilem2 14625 |
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