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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 13705 | . . . . . . 7 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 < 𝐴) | |
2 | elfzoelz 13696 | . . . . . . . . 9 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℤ) | |
3 | 2 | zred 12720 | . . . . . . . 8 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℝ) |
4 | elfzoel2 13695 | . . . . . . . . 9 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∈ ℤ) | |
5 | 4 | zred 12720 | . . . . . . . 8 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∈ ℝ) |
6 | 3, 5 | ltnled 11406 | . . . . . . 7 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵)) |
7 | 1, 6 | mpbid 232 | . . . . . 6 ⊢ (𝐵 ∈ (0..^𝐴) → ¬ 𝐴 ≤ 𝐵) |
8 | 7 | adantr 480 | . . . . 5 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → ¬ 𝐴 ≤ 𝐵) |
9 | elfzonn0 13744 | . . . . . . . . 9 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℕ0) | |
10 | 9 | adantr 480 | . . . . . . . 8 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℕ0) |
11 | simpr 484 | . . . . . . . 8 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ≠ 0) | |
12 | eldifsn 4791 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℕ0 ∖ {0}) ↔ (𝐵 ∈ ℕ0 ∧ 𝐵 ≠ 0)) | |
13 | 10, 11, 12 | sylanbrc 583 | . . . . . . 7 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ (ℕ0 ∖ {0})) |
14 | dfn2 12537 | . . . . . . 7 ⊢ ℕ = (ℕ0 ∖ {0}) | |
15 | 13, 14 | eleqtrrdi 2850 | . . . . . 6 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℕ) |
16 | dvdsle 16344 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 ∥ 𝐵 → 𝐴 ≤ 𝐵)) | |
17 | 4, 15, 16 | syl2an2r 685 | . . . . 5 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → (𝐴 ∥ 𝐵 → 𝐴 ≤ 𝐵)) |
18 | 8, 17 | mtod 198 | . . . 4 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → ¬ 𝐴 ∥ 𝐵) |
19 | 18 | ex 412 | . . 3 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 ≠ 0 → ¬ 𝐴 ∥ 𝐵)) |
20 | 19 | necon4ad 2957 | . 2 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 → 𝐵 = 0)) |
21 | dvds0 16306 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∥ 0) | |
22 | 4, 21 | syl 17 | . . 3 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∥ 0) |
23 | breq2 5152 | . . 3 ⊢ (𝐵 = 0 → (𝐴 ∥ 𝐵 ↔ 𝐴 ∥ 0)) | |
24 | 22, 23 | syl5ibrcom 247 | . 2 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 = 0 → 𝐴 ∥ 𝐵)) |
25 | 20, 24 | impbid 212 | 1 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 ↔ 𝐵 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 class class class wbr 5148 (class class class)co 7431 0cc0 11153 < clt 11293 ≤ cle 11294 ℕcn 12264 ℕ0cn0 12524 ℤcz 12611 ..^cfzo 13691 ∥ cdvds 16287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-dvds 16288 |
This theorem is referenced by: fzocongeq 16358 |
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