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Mirrors > Home > MPE Home > Th. List > alzdvds | Structured version Visualization version GIF version |
Description: Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
alzdvds | ⊢ (𝑁 ∈ ℤ → (∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 ↔ 𝑁 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssz 11274 | . . . . . . . 8 ⊢ ℕ ⊆ ℤ | |
2 | zcn 11259 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
3 | 2 | abscld 14023 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ) |
4 | arch 11166 | . . . . . . . . 9 ⊢ ((abs‘𝑁) ∈ ℝ → ∃𝑥 ∈ ℕ (abs‘𝑁) < 𝑥) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → ∃𝑥 ∈ ℕ (abs‘𝑁) < 𝑥) |
6 | ssrexv 3630 | . . . . . . . 8 ⊢ (ℕ ⊆ ℤ → (∃𝑥 ∈ ℕ (abs‘𝑁) < 𝑥 → ∃𝑥 ∈ ℤ (abs‘𝑁) < 𝑥)) | |
7 | 1, 5, 6 | mpsyl 66 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → ∃𝑥 ∈ ℤ (abs‘𝑁) < 𝑥) |
8 | zre 11258 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
9 | ltnle 9996 | . . . . . . . . . 10 ⊢ (((abs‘𝑁) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘𝑁) < 𝑥 ↔ ¬ 𝑥 ≤ (abs‘𝑁))) | |
10 | 3, 8, 9 | syl2an 493 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑥 ∈ ℤ) → ((abs‘𝑁) < 𝑥 ↔ ¬ 𝑥 ≤ (abs‘𝑁))) |
11 | 10 | rexbidva 3031 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (∃𝑥 ∈ ℤ (abs‘𝑁) < 𝑥 ↔ ∃𝑥 ∈ ℤ ¬ 𝑥 ≤ (abs‘𝑁))) |
12 | rexnal 2978 | . . . . . . . 8 ⊢ (∃𝑥 ∈ ℤ ¬ 𝑥 ≤ (abs‘𝑁) ↔ ¬ ∀𝑥 ∈ ℤ 𝑥 ≤ (abs‘𝑁)) | |
13 | 11, 12 | syl6bb 275 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (∃𝑥 ∈ ℤ (abs‘𝑁) < 𝑥 ↔ ¬ ∀𝑥 ∈ ℤ 𝑥 ≤ (abs‘𝑁))) |
14 | 7, 13 | mpbid 221 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ¬ ∀𝑥 ∈ ℤ 𝑥 ≤ (abs‘𝑁)) |
15 | 14 | adantl 481 | . . . . 5 ⊢ ((∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 ∧ 𝑁 ∈ ℤ) → ¬ ∀𝑥 ∈ ℤ 𝑥 ≤ (abs‘𝑁)) |
16 | dvdsleabs 14871 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑥 ∥ 𝑁 → 𝑥 ≤ (abs‘𝑁))) | |
17 | 16 | 3expb 1258 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑥 ∥ 𝑁 → 𝑥 ≤ (abs‘𝑁))) |
18 | 17 | expcom 450 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑥 ∈ ℤ → (𝑥 ∥ 𝑁 → 𝑥 ≤ (abs‘𝑁)))) |
19 | 18 | ralrimiv 2948 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ∀𝑥 ∈ ℤ (𝑥 ∥ 𝑁 → 𝑥 ≤ (abs‘𝑁))) |
20 | ralim 2932 | . . . . . . . 8 ⊢ (∀𝑥 ∈ ℤ (𝑥 ∥ 𝑁 → 𝑥 ≤ (abs‘𝑁)) → (∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 → ∀𝑥 ∈ ℤ 𝑥 ≤ (abs‘𝑁))) | |
21 | 19, 20 | syl 17 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 → ∀𝑥 ∈ ℤ 𝑥 ≤ (abs‘𝑁))) |
22 | 21 | com12 32 | . . . . . 6 ⊢ (∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 → ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ∀𝑥 ∈ ℤ 𝑥 ≤ (abs‘𝑁))) |
23 | 22 | expdimp 452 | . . . . 5 ⊢ ((∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 ∧ 𝑁 ∈ ℤ) → (𝑁 ≠ 0 → ∀𝑥 ∈ ℤ 𝑥 ≤ (abs‘𝑁))) |
24 | 15, 23 | mtod 188 | . . . 4 ⊢ ((∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 ∧ 𝑁 ∈ ℤ) → ¬ 𝑁 ≠ 0) |
25 | nne 2786 | . . . 4 ⊢ (¬ 𝑁 ≠ 0 ↔ 𝑁 = 0) | |
26 | 24, 25 | sylib 207 | . . 3 ⊢ ((∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 ∧ 𝑁 ∈ ℤ) → 𝑁 = 0) |
27 | 26 | expcom 450 | . 2 ⊢ (𝑁 ∈ ℤ → (∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 → 𝑁 = 0)) |
28 | dvds0 14835 | . . . 4 ⊢ (𝑥 ∈ ℤ → 𝑥 ∥ 0) | |
29 | breq2 4587 | . . . 4 ⊢ (𝑁 = 0 → (𝑥 ∥ 𝑁 ↔ 𝑥 ∥ 0)) | |
30 | 28, 29 | syl5ibr 235 | . . 3 ⊢ (𝑁 = 0 → (𝑥 ∈ ℤ → 𝑥 ∥ 𝑁)) |
31 | 30 | ralrimiv 2948 | . 2 ⊢ (𝑁 = 0 → ∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁) |
32 | 27, 31 | impbid1 214 | 1 ⊢ (𝑁 ∈ ℤ → (∀𝑥 ∈ ℤ 𝑥 ∥ 𝑁 ↔ 𝑁 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 ℝcr 9814 0cc0 9815 < clt 9953 ≤ cle 9954 ℕcn 10897 ℤcz 11254 abscabs 13822 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 |
This theorem is referenced by: (None) |
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