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Mirrors > Home > ILE Home > Th. List > fzo0dvdseq | 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 9165 | . . . . . . 7 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 < 𝐴) | |
2 | elfzoelz 9157 | . . . . . . . 8 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℤ) | |
3 | elfzoel2 9156 | . . . . . . . 8 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∈ ℤ) | |
4 | zltnle 8397 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵)) | |
5 | 2, 3, 4 | syl2anc 403 | . . . . . . 7 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵)) |
6 | 1, 5 | mpbid 145 | . . . . . 6 ⊢ (𝐵 ∈ (0..^𝐴) → ¬ 𝐴 ≤ 𝐵) |
7 | 6 | adantr 270 | . . . . 5 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → ¬ 𝐴 ≤ 𝐵) |
8 | 3 | adantr 270 | . . . . . . 7 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐴 ∈ ℤ) |
9 | elfzonn0 9195 | . . . . . . . . . 10 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℕ0) | |
10 | 9 | adantr 270 | . . . . . . . . 9 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℕ0) |
11 | simpr 108 | . . . . . . . . 9 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ≠ 0) | |
12 | eldifsn 3517 | . . . . . . . . 9 ⊢ (𝐵 ∈ (ℕ0 ∖ {0}) ↔ (𝐵 ∈ ℕ0 ∧ 𝐵 ≠ 0)) | |
13 | 10, 11, 12 | sylanbrc 408 | . . . . . . . 8 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ (ℕ0 ∖ {0})) |
14 | dfn2 8301 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
15 | 13, 14 | syl6eleqr 2172 | . . . . . . 7 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℕ) |
16 | dvdsle 10244 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 ∥ 𝐵 → 𝐴 ≤ 𝐵)) | |
17 | 8, 15, 16 | syl2anc 403 | . . . . . 6 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → (𝐴 ∥ 𝐵 → 𝐴 ≤ 𝐵)) |
18 | 17 | impancom 256 | . . . . 5 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → (𝐵 ≠ 0 → 𝐴 ≤ 𝐵)) |
19 | 7, 18 | mtod 621 | . . . 4 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → ¬ 𝐵 ≠ 0) |
20 | 0z 8362 | . . . . . . . 8 ⊢ 0 ∈ ℤ | |
21 | zdceq 8423 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝐵 = 0) | |
22 | 20, 21 | mpan2 415 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → DECID 𝐵 = 0) |
23 | nnedc 2250 | . . . . . . 7 ⊢ (DECID 𝐵 = 0 → (¬ 𝐵 ≠ 0 ↔ 𝐵 = 0)) | |
24 | 22, 23 | syl 14 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → (¬ 𝐵 ≠ 0 ↔ 𝐵 = 0)) |
25 | 2, 24 | syl 14 | . . . . 5 ⊢ (𝐵 ∈ (0..^𝐴) → (¬ 𝐵 ≠ 0 ↔ 𝐵 = 0)) |
26 | 25 | adantr 270 | . . . 4 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → (¬ 𝐵 ≠ 0 ↔ 𝐵 = 0)) |
27 | 19, 26 | mpbid 145 | . . 3 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → 𝐵 = 0) |
28 | 27 | ex 113 | . 2 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 → 𝐵 = 0)) |
29 | dvds0 10210 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∥ 0) | |
30 | 3, 29 | syl 14 | . . 3 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∥ 0) |
31 | breq2 3789 | . . 3 ⊢ (𝐵 = 0 → (𝐴 ∥ 𝐵 ↔ 𝐴 ∥ 0)) | |
32 | 30, 31 | syl5ibrcom 155 | . 2 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 = 0 → 𝐴 ∥ 𝐵)) |
33 | 28, 32 | impbid 127 | 1 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 ↔ 𝐵 = 0)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 DECID wdc 775 = wceq 1284 ∈ wcel 1433 ≠ wne 2245 ∖ cdif 2970 {csn 3398 class class class wbr 3785 (class class class)co 5532 0cc0 6981 < clt 7153 ≤ cle 7154 ℕcn 8039 ℕ0cn0 8288 ℤcz 8351 ..^cfzo 9152 ∥ cdvds 10195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-q 8705 df-fz 9030 df-fzo 9153 df-dvds 10196 |
This theorem is referenced by: fzocongeq 10258 |
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