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Mirrors > Home > MPE Home > Th. List > oddprm | Structured version Visualization version GIF version |
Description: A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 10-Jul-2022.) |
Ref | Expression |
---|---|
oddprm | ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 4087 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℙ) | |
2 | prmz 16552 | . . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℤ) |
4 | eldifsni 4751 | . . . . . . 7 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ≠ 2) | |
5 | 4 | necomd 3000 | . . . . . 6 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ≠ 𝑁) |
6 | 5 | neneqd 2949 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 = 𝑁) |
7 | 2z 12536 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
8 | uzid 12779 | . . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
10 | dvdsprm 16580 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℙ) → (2 ∥ 𝑁 ↔ 2 = 𝑁)) | |
11 | 9, 1, 10 | sylancr 588 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (2 ∥ 𝑁 ↔ 2 = 𝑁)) |
12 | 6, 11 | mtbird 325 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑁) |
13 | 1z 12534 | . . . . 5 ⊢ 1 ∈ ℤ | |
14 | n2dvds1 16251 | . . . . 5 ⊢ ¬ 2 ∥ 1 | |
15 | omoe 16247 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) ∧ (1 ∈ ℤ ∧ ¬ 2 ∥ 1)) → 2 ∥ (𝑁 − 1)) | |
16 | 13, 14, 15 | mpanr12 704 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → 2 ∥ (𝑁 − 1)) |
17 | 3, 12, 16 | syl2anc 585 | . . 3 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ∥ (𝑁 − 1)) |
18 | prmnn 16551 | . . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ) | |
19 | nnm1nn0 12455 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
20 | 1, 18, 19 | 3syl 18 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (𝑁 − 1) ∈ ℕ0) |
21 | nn0z 12525 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ0 → (𝑁 − 1) ∈ ℤ) | |
22 | evend2 16240 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℤ → (2 ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | |
23 | 20, 21, 22 | 3syl 18 | . . 3 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (2 ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / 2) ∈ ℤ)) |
24 | 17, 23 | mpbid 231 | . 2 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℤ) |
25 | prmuz2 16573 | . . 3 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ≥‘2)) | |
26 | uz2m1nn 12849 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ) | |
27 | nngt0 12185 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → 0 < (𝑁 − 1)) | |
28 | nnre 12161 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℝ) | |
29 | 2rp 12921 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
30 | 29 | a1i 11 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → 2 ∈ ℝ+) |
31 | 28, 30 | gt0divd 12995 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → (0 < (𝑁 − 1) ↔ 0 < ((𝑁 − 1) / 2))) |
32 | 27, 31 | mpbid 231 | . . 3 ⊢ ((𝑁 − 1) ∈ ℕ → 0 < ((𝑁 − 1) / 2)) |
33 | 1, 25, 26, 32 | 4syl 19 | . 2 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 0 < ((𝑁 − 1) / 2)) |
34 | elnnz 12510 | . 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 − 1) / 2))) | |
35 | 24, 33, 34 | sylanbrc 584 | 1 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3908 {csn 4587 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 0cc0 11052 1c1 11053 < clt 11190 − cmin 11386 / cdiv 11813 ℕcn 12154 2c2 12209 ℕ0cn0 12414 ℤcz 12500 ℤ≥cuz 12764 ℝ+crp 12916 ∥ cdvds 16137 ℙcprime 16548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-dvds 16138 df-prm 16549 |
This theorem is referenced by: nnoddn2prm 16684 4sqlem19 16836 lgslem1 26648 lgslem4 26651 lgsval2lem 26658 lgsvalmod 26667 lgsmod 26674 lgsdirprm 26682 lgsne0 26686 lgsqrlem1 26697 lgsqrlem2 26698 lgsqrlem3 26699 lgsqrlem4 26700 gausslemma2dlem4 26720 lgseisenlem1 26726 lgseisenlem2 26727 lgseisenlem4 26729 lgseisen 26730 m1lgs 26739 2lgslem2 26746 fmtnoprmfac2 45766 |
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