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Mirrors > Home > MPE Home > Th. List > prmlem1 | Structured version Visualization version GIF version |
Description: A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
prmlem1.n | ⊢ 𝑁 ∈ ℕ |
prmlem1.gt | ⊢ 1 < 𝑁 |
prmlem1.2 | ⊢ ¬ 2 ∥ 𝑁 |
prmlem1.3 | ⊢ ¬ 3 ∥ 𝑁 |
prmlem1.lt | ⊢ 𝑁 < ;25 |
Ref | Expression |
---|---|
prmlem1 | ⊢ 𝑁 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmlem1.n | . 2 ⊢ 𝑁 ∈ ℕ | |
2 | prmlem1.gt | . 2 ⊢ 1 < 𝑁 | |
3 | prmlem1.2 | . 2 ⊢ ¬ 2 ∥ 𝑁 | |
4 | prmlem1.3 | . 2 ⊢ ¬ 3 ∥ 𝑁 | |
5 | eluzelre 11574 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘5) → 𝑥 ∈ ℝ) | |
6 | 5 | resqcld 12897 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘5) → (𝑥↑2) ∈ ℝ) |
7 | eluzle 11576 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘5) → 5 ≤ 𝑥) | |
8 | 5re 10976 | . . . . . . . . 9 ⊢ 5 ∈ ℝ | |
9 | 5nn0 11189 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
10 | 9 | nn0ge0i 11197 | . . . . . . . . 9 ⊢ 0 ≤ 5 |
11 | le2sq2 12801 | . . . . . . . . 9 ⊢ (((5 ∈ ℝ ∧ 0 ≤ 5) ∧ (𝑥 ∈ ℝ ∧ 5 ≤ 𝑥)) → (5↑2) ≤ (𝑥↑2)) | |
12 | 8, 10, 11 | mpanl12 714 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 5 ≤ 𝑥) → (5↑2) ≤ (𝑥↑2)) |
13 | 5, 7, 12 | syl2anc 691 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘5) → (5↑2) ≤ (𝑥↑2)) |
14 | 1 | nnrei 10906 | . . . . . . . 8 ⊢ 𝑁 ∈ ℝ |
15 | 8 | resqcli 12811 | . . . . . . . 8 ⊢ (5↑2) ∈ ℝ |
16 | prmlem1.lt | . . . . . . . . . 10 ⊢ 𝑁 < ;25 | |
17 | 5cn 10977 | . . . . . . . . . . . 12 ⊢ 5 ∈ ℂ | |
18 | 17 | sqvali 12805 | . . . . . . . . . . 11 ⊢ (5↑2) = (5 · 5) |
19 | 5t5e25 11515 | . . . . . . . . . . 11 ⊢ (5 · 5) = ;25 | |
20 | 18, 19 | eqtri 2632 | . . . . . . . . . 10 ⊢ (5↑2) = ;25 |
21 | 16, 20 | breqtrri 4610 | . . . . . . . . 9 ⊢ 𝑁 < (5↑2) |
22 | ltletr 10008 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ (5↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((𝑁 < (5↑2) ∧ (5↑2) ≤ (𝑥↑2)) → 𝑁 < (𝑥↑2))) | |
23 | 21, 22 | mpani 708 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ (5↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((5↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2))) |
24 | 14, 15, 23 | mp3an12 1406 | . . . . . . 7 ⊢ ((𝑥↑2) ∈ ℝ → ((5↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2))) |
25 | 6, 13, 24 | sylc 63 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘5) → 𝑁 < (𝑥↑2)) |
26 | ltnle 9996 | . . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁)) | |
27 | 14, 6, 26 | sylancr 694 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘5) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁)) |
28 | 25, 27 | mpbid 221 | . . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘5) → ¬ (𝑥↑2) ≤ 𝑁) |
29 | 28 | pm2.21d 117 | . . . 4 ⊢ (𝑥 ∈ (ℤ≥‘5) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)) |
30 | 29 | adantld 482 | . . 3 ⊢ (𝑥 ∈ (ℤ≥‘5) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
31 | 30 | adantl 481 | . 2 ⊢ ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ≥‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
32 | 1, 2, 3, 4, 31 | prmlem1a 15651 | 1 ⊢ 𝑁 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ∖ cdif 3537 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ≤ cle 9954 ℕcn 10897 2c2 10947 3c3 10948 5c5 10950 ;cdc 11369 ℤ≥cuz 11563 ↑cexp 12722 ∥ cdvds 14821 ℙcprime 15223 |
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-int 4411 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-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 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-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-prm 15224 |
This theorem is referenced by: 5prm 15653 7prm 15655 11prm 15660 13prm 15661 17prm 15662 19prm 15663 23prm 15664 |
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