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Mirrors > Home > ILE Home > Th. List > m1modge3gt1 | GIF version |
Description: Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.) |
Ref | Expression |
---|---|
m1modge3gt1 | ⊢ (𝑀 ∈ (ℤ≥‘3) → 1 < (-1 mod 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1p1e2 9054 | . . . 4 ⊢ (1 + 1) = 2 | |
2 | 2p1e3 9070 | . . . . . 6 ⊢ (2 + 1) = 3 | |
3 | eluzle 9558 | . . . . . 6 ⊢ (𝑀 ∈ (ℤ≥‘3) → 3 ≤ 𝑀) | |
4 | 2, 3 | eqbrtrid 4053 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘3) → (2 + 1) ≤ 𝑀) |
5 | 2z 9299 | . . . . . 6 ⊢ 2 ∈ ℤ | |
6 | eluzelz 9555 | . . . . . 6 ⊢ (𝑀 ∈ (ℤ≥‘3) → 𝑀 ∈ ℤ) | |
7 | zltp1le 9325 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀)) | |
8 | 5, 6, 7 | sylancr 414 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘3) → (2 < 𝑀 ↔ (2 + 1) ≤ 𝑀)) |
9 | 4, 8 | mpbird 167 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘3) → 2 < 𝑀) |
10 | 1, 9 | eqbrtrid 4053 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘3) → (1 + 1) < 𝑀) |
11 | 1red 7990 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘3) → 1 ∈ ℝ) | |
12 | eluzelre 9556 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘3) → 𝑀 ∈ ℝ) | |
13 | 11, 11, 12 | ltaddsub2d 8521 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘3) → ((1 + 1) < 𝑀 ↔ 1 < (𝑀 − 1))) |
14 | 10, 13 | mpbid 147 | . 2 ⊢ (𝑀 ∈ (ℤ≥‘3) → 1 < (𝑀 − 1)) |
15 | eluzge3nn 9590 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘3) → 𝑀 ∈ ℕ) | |
16 | m1modnnsub1 10388 | . . 3 ⊢ (𝑀 ∈ ℕ → (-1 mod 𝑀) = (𝑀 − 1)) | |
17 | 15, 16 | syl 14 | . 2 ⊢ (𝑀 ∈ (ℤ≥‘3) → (-1 mod 𝑀) = (𝑀 − 1)) |
18 | 14, 17 | breqtrrd 4046 | 1 ⊢ (𝑀 ∈ (ℤ≥‘3) → 1 < (-1 mod 𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 ‘cfv 5231 (class class class)co 5891 1c1 7830 + caddc 7832 < clt 8010 ≤ cle 8011 − cmin 8146 -cneg 8147 ℕcn 8937 2c2 8988 3c3 8989 ℤcz 9271 ℤ≥cuz 9546 mod cmo 10340 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 ax-arch 7948 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-2 8996 df-3 8997 df-n0 9195 df-z 9272 df-uz 9547 df-q 9638 df-rp 9672 df-fl 10288 df-mod 10341 |
This theorem is referenced by: (None) |
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