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Mirrors > Home > MPE Home > Th. List > fzo0dvdseq | Structured version Visualization version GIF version |
Description: Zero is the only one of the first 𝐴 nonnegative integers that is divisible by 𝐴. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
fzo0dvdseq | ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 ↔ 𝐵 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzolt2 13510 | . . . . . . 7 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 < 𝐴) | |
2 | elfzoelz 13501 | . . . . . . . . 9 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℤ) | |
3 | 2 | zred 12540 | . . . . . . . 8 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℝ) |
4 | elfzoel2 13500 | . . . . . . . . 9 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∈ ℤ) | |
5 | 4 | zred 12540 | . . . . . . . 8 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∈ ℝ) |
6 | 3, 5 | ltnled 11236 | . . . . . . 7 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵)) |
7 | 1, 6 | mpbid 231 | . . . . . 6 ⊢ (𝐵 ∈ (0..^𝐴) → ¬ 𝐴 ≤ 𝐵) |
8 | 7 | adantr 482 | . . . . 5 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → ¬ 𝐴 ≤ 𝐵) |
9 | elfzonn0 13546 | . . . . . . . . 9 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℕ0) | |
10 | 9 | adantr 482 | . . . . . . . 8 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℕ0) |
11 | simpr 486 | . . . . . . . 8 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ≠ 0) | |
12 | eldifsn 4746 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℕ0 ∖ {0}) ↔ (𝐵 ∈ ℕ0 ∧ 𝐵 ≠ 0)) | |
13 | 10, 11, 12 | sylanbrc 584 | . . . . . . 7 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ (ℕ0 ∖ {0})) |
14 | dfn2 12360 | . . . . . . 7 ⊢ ℕ = (ℕ0 ∖ {0}) | |
15 | 13, 14 | eleqtrrdi 2850 | . . . . . 6 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℕ) |
16 | dvdsle 16127 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 ∥ 𝐵 → 𝐴 ≤ 𝐵)) | |
17 | 4, 15, 16 | syl2an2r 684 | . . . . 5 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → (𝐴 ∥ 𝐵 → 𝐴 ≤ 𝐵)) |
18 | 8, 17 | mtod 197 | . . . 4 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → ¬ 𝐴 ∥ 𝐵) |
19 | 18 | ex 414 | . . 3 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 ≠ 0 → ¬ 𝐴 ∥ 𝐵)) |
20 | 19 | necon4ad 2961 | . 2 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 → 𝐵 = 0)) |
21 | dvds0 16089 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∥ 0) | |
22 | 4, 21 | syl 17 | . . 3 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∥ 0) |
23 | breq2 5108 | . . 3 ⊢ (𝐵 = 0 → (𝐴 ∥ 𝐵 ↔ 𝐴 ∥ 0)) | |
24 | 22, 23 | syl5ibrcom 247 | . 2 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 = 0 → 𝐴 ∥ 𝐵)) |
25 | 20, 24 | impbid 211 | 1 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 ↔ 𝐵 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∖ cdif 3906 {csn 4585 class class class wbr 5104 (class class class)co 7350 0cc0 10985 < clt 11123 ≤ cle 11124 ℕcn 12087 ℕ0cn0 12347 ℤcz 12433 ..^cfzo 13496 ∥ cdvds 16071 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-n0 12348 df-z 12434 df-uz 12697 df-fz 13354 df-fzo 13497 df-dvds 16072 |
This theorem is referenced by: fzocongeq 16141 |
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