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Mirrors > Home > MPE Home > Th. List > 0mod | Structured version Visualization version GIF version |
Description: Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014.) |
Ref | Expression |
---|---|
0mod | ⊢ (𝑁 ∈ ℝ+ → (0 mod 𝑁) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10330 | . . 3 ⊢ 0 ∈ ℝ | |
2 | 1 | jctl 520 | . 2 ⊢ (𝑁 ∈ ℝ+ → (0 ∈ ℝ ∧ 𝑁 ∈ ℝ+)) |
3 | rpgt0 12088 | . . 3 ⊢ (𝑁 ∈ ℝ+ → 0 < 𝑁) | |
4 | 0le0 11421 | . . 3 ⊢ 0 ≤ 0 | |
5 | 3, 4 | jctil 516 | . 2 ⊢ (𝑁 ∈ ℝ+ → (0 ≤ 0 ∧ 0 < 𝑁)) |
6 | modid 12950 | . 2 ⊢ (((0 ∈ ℝ ∧ 𝑁 ∈ ℝ+) ∧ (0 ≤ 0 ∧ 0 < 𝑁)) → (0 mod 𝑁) = 0) | |
7 | 2, 5, 6 | syl2anc 580 | 1 ⊢ (𝑁 ∈ ℝ+ → (0 mod 𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 class class class wbr 4843 (class class class)co 6878 ℝcr 10223 0cc0 10224 < clt 10363 ≤ cle 10364 ℝ+crp 12074 mod cmo 12923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-inf 8591 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-rp 12075 df-fl 12848 df-mod 12924 |
This theorem is referenced by: addmodlteq 13000 cshw0 13879 dvdsmodexp 15327 moddvds 15330 modprm0 15843 nnnn0modprm0 15844 elqaalem2 24416 lgsne0 25412 gausslemma2dlem0i 25441 pellexlem6 38184 m1mod0mod1 42179 dignnld 43196 dig0 43199 digexp 43200 |
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