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Mirrors > Home > MPE Home > Th. List > prm23ge5 | Structured version Visualization version GIF version |
Description: A prime is either 2 or 3 or greater than or equal to 5. (Contributed by AV, 5-Jul-2021.) |
Ref | Expression |
---|---|
prm23ge5 | ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . 2 ⊢ ((𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)) → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) | |
2 | 3ioran 1049 | . . 3 ⊢ (¬ (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)) ↔ (¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3 ∧ ¬ 𝑃 ∈ (ℤ≥‘5))) | |
3 | 3ianor 1048 | . . . . . . 7 ⊢ (¬ (5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃) ↔ (¬ 5 ∈ ℤ ∨ ¬ 𝑃 ∈ ℤ ∨ ¬ 5 ≤ 𝑃)) | |
4 | eluz2 11569 | . . . . . . 7 ⊢ (𝑃 ∈ (ℤ≥‘5) ↔ (5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃)) | |
5 | 3, 4 | xchnxbir 322 | . . . . . 6 ⊢ (¬ 𝑃 ∈ (ℤ≥‘5) ↔ (¬ 5 ∈ ℤ ∨ ¬ 𝑃 ∈ ℤ ∨ ¬ 5 ≤ 𝑃)) |
6 | 5nn 11065 | . . . . . . . . 9 ⊢ 5 ∈ ℕ | |
7 | 6 | nnzi 11278 | . . . . . . . 8 ⊢ 5 ∈ ℤ |
8 | 7 | pm2.24i 145 | . . . . . . 7 ⊢ (¬ 5 ∈ ℤ → ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))))) |
9 | prmz 15227 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
10 | pm2.24 120 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℤ → (¬ 𝑃 ∈ ℤ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 ∈ ℤ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
12 | 11 | com12 32 | . . . . . . . 8 ⊢ (¬ 𝑃 ∈ ℤ → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
13 | 12 | a1d 25 | . . . . . . 7 ⊢ (¬ 𝑃 ∈ ℤ → ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))))) |
14 | 9 | zred 11358 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ) |
15 | 5re 10976 | . . . . . . . . . . 11 ⊢ 5 ∈ ℝ | |
16 | 15 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 5 ∈ ℝ) |
17 | 14, 16 | ltnled 10063 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃 < 5 ↔ ¬ 5 ≤ 𝑃)) |
18 | prm23lt5 15357 | . . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 < 5) → (𝑃 = 2 ∨ 𝑃 = 3)) | |
19 | ioran 510 | . . . . . . . . . . . 12 ⊢ (¬ (𝑃 = 2 ∨ 𝑃 = 3) ↔ (¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3)) | |
20 | pm2.24 120 | . . . . . . . . . . . 12 ⊢ ((𝑃 = 2 ∨ 𝑃 = 3) → (¬ (𝑃 = 2 ∨ 𝑃 = 3) → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) | |
21 | 19, 20 | syl5bir 232 | . . . . . . . . . . 11 ⊢ ((𝑃 = 2 ∨ 𝑃 = 3) → ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
22 | 18, 21 | syl 17 | . . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 < 5) → ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
23 | 22 | ex 449 | . . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃 < 5 → ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))))) |
24 | 17, 23 | sylbird 249 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (¬ 5 ≤ 𝑃 → ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))))) |
25 | 24 | com3l 87 | . . . . . . 7 ⊢ (¬ 5 ≤ 𝑃 → ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))))) |
26 | 8, 13, 25 | 3jaoi 1383 | . . . . . 6 ⊢ ((¬ 5 ∈ ℤ ∨ ¬ 𝑃 ∈ ℤ ∨ ¬ 5 ≤ 𝑃) → ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))))) |
27 | 5, 26 | sylbi 206 | . . . . 5 ⊢ (¬ 𝑃 ∈ (ℤ≥‘5) → ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))))) |
28 | 27 | com12 32 | . . . 4 ⊢ ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3) → (¬ 𝑃 ∈ (ℤ≥‘5) → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))))) |
29 | 28 | 3impia 1253 | . . 3 ⊢ ((¬ 𝑃 = 2 ∧ ¬ 𝑃 = 3 ∧ ¬ 𝑃 ∈ (ℤ≥‘5)) → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
30 | 2, 29 | sylbi 206 | . 2 ⊢ (¬ (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)) → (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5)))) |
31 | 1, 30 | pm2.61i 175 | 1 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 ℝcr 9814 < clt 9953 ≤ cle 9954 2c2 10947 3c3 10948 5c5 10950 ℤcz 11254 ℤ≥cuz 11563 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-n0 11170 df-z 11255 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: gausslemma2dlem0f 24886 gausslemma2dlem4 24894 |
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