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Mirrors > Home > MPE Home > Th. List > mulmod0 | Structured version Visualization version GIF version |
Description: The product of an integer and a positive real number is 0 modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.) (Revised by AV, 5-Jul-2020.) |
Ref | Expression |
---|---|
mulmod0 | ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · 𝑀) mod 𝑀) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 11791 | . . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
2 | 1 | adantr 473 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → 𝐴 ∈ ℂ) |
3 | rpcn 12209 | . . . . 5 ⊢ (𝑀 ∈ ℝ+ → 𝑀 ∈ ℂ) | |
4 | 3 | adantl 474 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → 𝑀 ∈ ℂ) |
5 | rpne0 12215 | . . . . 5 ⊢ (𝑀 ∈ ℝ+ → 𝑀 ≠ 0) | |
6 | 5 | adantl 474 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → 𝑀 ≠ 0) |
7 | 2, 4, 6 | divcan4d 11215 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · 𝑀) / 𝑀) = 𝐴) |
8 | simpl 475 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → 𝐴 ∈ ℤ) | |
9 | 7, 8 | eqeltrd 2860 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · 𝑀) / 𝑀) ∈ ℤ) |
10 | zre 11790 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
11 | rpre 12205 | . . . 4 ⊢ (𝑀 ∈ ℝ+ → 𝑀 ∈ ℝ) | |
12 | remulcl 10412 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐴 · 𝑀) ∈ ℝ) | |
13 | 10, 11, 12 | syl2an 586 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (𝐴 · 𝑀) ∈ ℝ) |
14 | mod0 13052 | . . 3 ⊢ (((𝐴 · 𝑀) ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 · 𝑀) mod 𝑀) = 0 ↔ ((𝐴 · 𝑀) / 𝑀) ∈ ℤ)) | |
15 | 13, 14 | sylancom 579 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (((𝐴 · 𝑀) mod 𝑀) = 0 ↔ ((𝐴 · 𝑀) / 𝑀) ∈ ℤ)) |
16 | 9, 15 | mpbird 249 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · 𝑀) mod 𝑀) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 (class class class)co 6970 ℂcc 10325 ℝcr 10326 0cc0 10327 · cmul 10332 / cdiv 11090 ℤcz 11786 ℝ+crp 12197 mod cmo 13045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-n0 11701 df-z 11787 df-uz 12052 df-rp 12198 df-fl 12970 df-mod 13046 |
This theorem is referenced by: mulp1mod1 13088 mod2eq1n2dvds 15546 modprm0 15988 2lgslem3a1 25668 2lgslem3d1 25671 |
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