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Mirrors > Home > MPE Home > Th. List > muladdmodid | Structured version Visualization version GIF version |
Description: The sum of a positive real number less than an upper bound and the product of an integer and the upper bound is the positive real number modulo the upper bound. (Contributed by AV, 5-Jul-2020.) |
Ref | Expression |
---|---|
muladdmodid | ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ∧ 𝐴 ∈ (0[,)𝑀)) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10633 | . . . . 5 ⊢ (𝑀 ∈ ℝ+ → 0 ∈ ℝ) | |
2 | rpxr 12386 | . . . . 5 ⊢ (𝑀 ∈ ℝ+ → 𝑀 ∈ ℝ*) | |
3 | elico2 12789 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑀 ∈ ℝ*) → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) | |
4 | 1, 2, 3 | syl2anc 587 | . . . 4 ⊢ (𝑀 ∈ ℝ+ → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) |
5 | 4 | adantl 485 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) |
6 | zcn 11974 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
7 | rpcn 12387 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℝ+ → 𝑀 ∈ ℂ) | |
8 | mulcl 10610 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 · 𝑀) ∈ ℂ) | |
9 | 6, 7, 8 | syl2an 598 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (𝑁 · 𝑀) ∈ ℂ) |
10 | 9 | adantr 484 | . . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (𝑁 · 𝑀) ∈ ℂ) |
11 | recn 10616 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
12 | 11 | 3ad2ant1 1130 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀) → 𝐴 ∈ ℂ) |
13 | 12 | adantl 485 | . . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → 𝐴 ∈ ℂ) |
14 | 10, 13 | addcomd 10831 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → ((𝑁 · 𝑀) + 𝐴) = (𝐴 + (𝑁 · 𝑀))) |
15 | 14 | oveq1d 7150 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = ((𝐴 + (𝑁 · 𝑀)) mod 𝑀)) |
16 | simp1 1133 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀) → 𝐴 ∈ ℝ) | |
17 | 16 | adantl 485 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → 𝐴 ∈ ℝ) |
18 | simpr 488 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → 𝑀 ∈ ℝ+) | |
19 | 18 | adantr 484 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → 𝑀 ∈ ℝ+) |
20 | simpll 766 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → 𝑁 ∈ ℤ) | |
21 | modcyc 13269 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → ((𝐴 + (𝑁 · 𝑀)) mod 𝑀) = (𝐴 mod 𝑀)) | |
22 | 17, 19, 20, 21 | syl3anc 1368 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → ((𝐴 + (𝑁 · 𝑀)) mod 𝑀) = (𝐴 mod 𝑀)) |
23 | 18, 16 | anim12ci 616 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+)) |
24 | 3simpc 1147 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀) → (0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) | |
25 | 24 | adantl 485 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) |
26 | modid 13259 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (𝐴 mod 𝑀) = 𝐴) | |
27 | 23, 25, 26 | syl2anc 587 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (𝐴 mod 𝑀) = 𝐴) |
28 | 15, 22, 27 | 3eqtrd 2837 | . . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴) |
29 | 28 | ex 416 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴)) |
30 | 5, 29 | sylbid 243 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (𝐴 ∈ (0[,)𝑀) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴)) |
31 | 30 | 3impia 1114 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ∧ 𝐴 ∈ (0[,)𝑀)) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 ℤcz 11969 ℝ+crp 12377 [,)cico 12728 mod cmo 13232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fl 13157 df-mod 13233 |
This theorem is referenced by: modmuladd 13276 addmodid 13282 mod42tp1mod8 44120 |
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