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Mirrors > Home > MPE Home > Th. List > modcld | Structured version Visualization version GIF version |
Description: Closure law for the modulo operation. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
modcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
modcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
modcld | ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | modcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | modcl 13244 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 (class class class)co 7159 ℝcr 10539 ℝ+crp 12392 mod cmo 13240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fl 13165 df-mod 13241 |
This theorem is referenced by: modmulmodr 13308 modaddmulmod 13309 digit1 13601 bitsmod 15788 bitsinv1lem 15793 eulerthlem2 16122 vfermltlALT 16142 4sqlem5 16281 4sqlem6 16282 4sqlem10 16286 lgsvalmod 25895 irrapxlem2 39426 irrapxlem3 39427 modabsdifz 39589 jm2.19 39596 sineq0ALT 41277 lefldiveq 41565 ltmod 41925 dirkertrigeq 42393 sqwvfoura 42520 sqwvfourb 42521 fouriersw 42523 m1mod0mod1 43536 fsummmodsndifre 43541 fpprwppr 43911 m1modmmod 44588 difmodm1lt 44589 dignn0flhalflem1 44682 |
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