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Mirrors > Home > MPE Home > Th. List > ndvdsi | Structured version Visualization version GIF version |
Description: A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
ndvdsi.1 | ⊢ 𝐴 ∈ ℕ |
ndvdsi.2 | ⊢ 𝑄 ∈ ℕ0 |
ndvdsi.3 | ⊢ 𝑅 ∈ ℕ |
ndvdsi.4 | ⊢ ((𝐴 · 𝑄) + 𝑅) = 𝐵 |
ndvdsi.5 | ⊢ 𝑅 < 𝐴 |
Ref | Expression |
---|---|
ndvdsi | ⊢ ¬ 𝐴 ∥ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndvdsi.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ | |
2 | 1 | nnzi 12581 | . . . 4 ⊢ 𝐴 ∈ ℤ |
3 | ndvdsi.2 | . . . . 5 ⊢ 𝑄 ∈ ℕ0 | |
4 | 3 | nn0zi 12582 | . . . 4 ⊢ 𝑄 ∈ ℤ |
5 | dvdsmul1 16216 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝐴 ∥ (𝐴 · 𝑄)) | |
6 | 2, 4, 5 | mp2an 691 | . . 3 ⊢ 𝐴 ∥ (𝐴 · 𝑄) |
7 | zmulcl 12606 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (𝐴 · 𝑄) ∈ ℤ) | |
8 | 2, 4, 7 | mp2an 691 | . . . 4 ⊢ (𝐴 · 𝑄) ∈ ℤ |
9 | ndvdsi.3 | . . . . 5 ⊢ 𝑅 ∈ ℕ | |
10 | ndvdsi.5 | . . . . 5 ⊢ 𝑅 < 𝐴 | |
11 | 9, 10 | pm3.2i 472 | . . . 4 ⊢ (𝑅 ∈ ℕ ∧ 𝑅 < 𝐴) |
12 | ndvdsadd 16348 | . . . 4 ⊢ (((𝐴 · 𝑄) ∈ ℤ ∧ 𝐴 ∈ ℕ ∧ (𝑅 ∈ ℕ ∧ 𝑅 < 𝐴)) → (𝐴 ∥ (𝐴 · 𝑄) → ¬ 𝐴 ∥ ((𝐴 · 𝑄) + 𝑅))) | |
13 | 8, 1, 11, 12 | mp3an 1462 | . . 3 ⊢ (𝐴 ∥ (𝐴 · 𝑄) → ¬ 𝐴 ∥ ((𝐴 · 𝑄) + 𝑅)) |
14 | 6, 13 | ax-mp 5 | . 2 ⊢ ¬ 𝐴 ∥ ((𝐴 · 𝑄) + 𝑅) |
15 | ndvdsi.4 | . . 3 ⊢ ((𝐴 · 𝑄) + 𝑅) = 𝐵 | |
16 | 15 | breq2i 5154 | . 2 ⊢ (𝐴 ∥ ((𝐴 · 𝑄) + 𝑅) ↔ 𝐴 ∥ 𝐵) |
17 | 14, 16 | mtbi 322 | 1 ⊢ ¬ 𝐴 ∥ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5146 (class class class)co 7403 + caddc 11108 · cmul 11110 < clt 11243 ℕcn 12207 ℕ0cn0 12467 ℤcz 12553 ∥ cdvds 16192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-inf 9433 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-n0 12468 df-z 12554 df-uz 12818 df-rp 12970 df-fz 13480 df-seq 13962 df-exp 14023 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-dvds 16193 |
This theorem is referenced by: dec5dvds 16992 5prm 17037 7prm 17039 11prm 17043 13prm 17044 17prm 17045 19prm 17046 23prm 17047 37prm 17049 43prm 17050 83prm 17051 139prm 17052 163prm 17053 317prm 17054 631prm 17055 1259lem5 17063 2503lem3 17067 4001lem4 17072 257prm 46163 fmtno4nprmfac193 46176 3ndvds4 46197 139prmALT 46198 127prm 46201 |
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