Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 1mod | Structured version Visualization version GIF version |
Description: Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
1mod | ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1 11156 | . . . . . 6 ⊢ 0 < 1 | |
2 | 0re 10637 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
3 | 1re 10635 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
4 | lttr 10711 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝑁) → 0 < 𝑁)) | |
5 | 2, 3, 4 | mp3an12 1447 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → ((0 < 1 ∧ 1 < 𝑁) → 0 < 𝑁)) |
6 | 1, 5 | mpani 694 | . . . . 5 ⊢ (𝑁 ∈ ℝ → (1 < 𝑁 → 0 < 𝑁)) |
7 | 6 | imdistani 571 | . . . 4 ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → (𝑁 ∈ ℝ ∧ 0 < 𝑁)) |
8 | elrp 12385 | . . . 4 ⊢ (𝑁 ∈ ℝ+ ↔ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) | |
9 | 7, 8 | sylibr 236 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → 𝑁 ∈ ℝ+) |
10 | 9, 3 | jctil 522 | . 2 ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → (1 ∈ ℝ ∧ 𝑁 ∈ ℝ+)) |
11 | simpr 487 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → 1 < 𝑁) | |
12 | 0le1 11157 | . . 3 ⊢ 0 ≤ 1 | |
13 | 11, 12 | jctil 522 | . 2 ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → (0 ≤ 1 ∧ 1 < 𝑁)) |
14 | modid 13258 | . 2 ⊢ (((1 ∈ ℝ ∧ 𝑁 ∈ ℝ+) ∧ (0 ≤ 1 ∧ 1 < 𝑁)) → (1 mod 𝑁) = 1) | |
15 | 10, 13, 14 | syl2anc 586 | 1 ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 < clt 10669 ≤ cle 10670 ℝ+crp 12383 mod cmo 13231 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 ax-pre-sup 10609 |
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-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 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-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fl 13156 df-mod 13232 |
This theorem is referenced by: mulp1mod1 13274 p1modz1 15608 modm1div 15613 mod2eq1n2dvds 15690 vfermltl 16132 pockthlem 16235 pockthi 16237 sylow3lem6 18751 wilthlem1 25639 lgsne0 25905 gausslemma2dlem0i 25934 gausslemma2dlem7 25943 gausslemma2d 25944 numclwwlk5 28161 numclwwlk7 28164 m1mod0mod1 43522 fmtnoprmfac1lem 43719 fmtnoprmfac2lem1 43721 sfprmdvdsmersenne 43761 modexp2m1d 43770 4fppr1 43893 digexp 44660 |
Copyright terms: Public domain | W3C validator |