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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 10637 | . . . . 5 ⊢ (𝑀 ∈ ℝ+ → 0 ∈ ℝ) | |
2 | rpxr 12392 | . . . . 5 ⊢ (𝑀 ∈ ℝ+ → 𝑀 ∈ ℝ*) | |
3 | elico2 12794 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑀 ∈ ℝ*) → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) | |
4 | 1, 2, 3 | syl2anc 586 | . . . 4 ⊢ (𝑀 ∈ ℝ+ → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) |
5 | 4 | adantl 484 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (𝐴 ∈ (0[,)𝑀) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀))) |
6 | zcn 11980 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
7 | rpcn 12393 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℝ+ → 𝑀 ∈ ℂ) | |
8 | mulcl 10614 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 · 𝑀) ∈ ℂ) | |
9 | 6, 7, 8 | syl2an 597 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (𝑁 · 𝑀) ∈ ℂ) |
10 | 9 | adantr 483 | . . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (𝑁 · 𝑀) ∈ ℂ) |
11 | recn 10620 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
12 | 11 | 3ad2ant1 1128 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀) → 𝐴 ∈ ℂ) |
13 | 12 | adantl 484 | . . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → 𝐴 ∈ ℂ) |
14 | 10, 13 | addcomd 10835 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → ((𝑁 · 𝑀) + 𝐴) = (𝐴 + (𝑁 · 𝑀))) |
15 | 14 | oveq1d 7164 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = ((𝐴 + (𝑁 · 𝑀)) mod 𝑀)) |
16 | simp1 1131 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀) → 𝐴 ∈ ℝ) | |
17 | 16 | adantl 484 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → 𝐴 ∈ ℝ) |
18 | simpr 487 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → 𝑀 ∈ ℝ+) | |
19 | 18 | adantr 483 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → 𝑀 ∈ ℝ+) |
20 | simpll 765 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → 𝑁 ∈ ℤ) | |
21 | modcyc 13271 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → ((𝐴 + (𝑁 · 𝑀)) mod 𝑀) = (𝐴 mod 𝑀)) | |
22 | 17, 19, 20, 21 | syl3anc 1366 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → ((𝐴 + (𝑁 · 𝑀)) mod 𝑀) = (𝐴 mod 𝑀)) |
23 | 18, 16 | anim12ci 615 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+)) |
24 | 3simpc 1145 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀) → (0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) | |
25 | 24 | adantl 484 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) |
26 | modid 13261 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (𝐴 mod 𝑀) = 𝐴) | |
27 | 23, 25, 26 | syl2anc 586 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (𝐴 mod 𝑀) = 𝐴) |
28 | 15, 22, 27 | 3eqtrd 2859 | . . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀)) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴) |
29 | 28 | ex 415 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < 𝑀) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴)) |
30 | 5, 29 | sylbid 242 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (𝐴 ∈ (0[,)𝑀) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴)) |
31 | 30 | 3impia 1112 | 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 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 class class class wbr 5059 (class class class)co 7149 ℂcc 10528 ℝcr 10529 0cc0 10530 + caddc 10533 · cmul 10535 ℝ*cxr 10667 < clt 10668 ≤ cle 10669 ℤcz 11975 ℝ+crp 12383 [,)cico 12734 mod cmo 13234 |
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 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-fl 13159 df-mod 13235 |
This theorem is referenced by: modmuladd 13278 addmodid 13284 mod42tp1mod8 43841 |
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