![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > oddprmALTV | Structured version Visualization version GIF version |
Description: A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by AV, 21-Jun-2020.) |
Ref | Expression |
---|---|
oddprmALTV | ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ Odd ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4767 | . 2 ⊢ (𝑁 ∈ (ℙ ∖ {2}) ↔ (𝑁 ∈ ℙ ∧ 𝑁 ≠ 2)) | |
2 | prmz 16577 | . . . 4 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
3 | 2 | adantr 481 | . . 3 ⊢ ((𝑁 ∈ ℙ ∧ 𝑁 ≠ 2) → 𝑁 ∈ ℤ) |
4 | necom 2993 | . . . . . . 7 ⊢ (𝑁 ≠ 2 ↔ 2 ≠ 𝑁) | |
5 | df-ne 2940 | . . . . . . 7 ⊢ (2 ≠ 𝑁 ↔ ¬ 2 = 𝑁) | |
6 | 4, 5 | sylbb 218 | . . . . . 6 ⊢ (𝑁 ≠ 2 → ¬ 2 = 𝑁) |
7 | 6 | adantl 482 | . . . . 5 ⊢ ((𝑁 ∈ ℙ ∧ 𝑁 ≠ 2) → ¬ 2 = 𝑁) |
8 | 1ne2 12385 | . . . . . . 7 ⊢ 1 ≠ 2 | |
9 | 8 | nesymi 2997 | . . . . . 6 ⊢ ¬ 2 = 1 |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((𝑁 ∈ ℙ ∧ 𝑁 ≠ 2) → ¬ 2 = 1) |
11 | ioran 982 | . . . . 5 ⊢ (¬ (2 = 𝑁 ∨ 2 = 1) ↔ (¬ 2 = 𝑁 ∧ ¬ 2 = 1)) | |
12 | 7, 10, 11 | sylanbrc 583 | . . . 4 ⊢ ((𝑁 ∈ ℙ ∧ 𝑁 ≠ 2) → ¬ (2 = 𝑁 ∨ 2 = 1)) |
13 | 2nn 12250 | . . . . . 6 ⊢ 2 ∈ ℕ | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (𝑁 ≠ 2 → 2 ∈ ℕ) |
15 | dvdsprime 16589 | . . . . 5 ⊢ ((𝑁 ∈ ℙ ∧ 2 ∈ ℕ) → (2 ∥ 𝑁 ↔ (2 = 𝑁 ∨ 2 = 1))) | |
16 | 14, 15 | sylan2 593 | . . . 4 ⊢ ((𝑁 ∈ ℙ ∧ 𝑁 ≠ 2) → (2 ∥ 𝑁 ↔ (2 = 𝑁 ∨ 2 = 1))) |
17 | 12, 16 | mtbird 324 | . . 3 ⊢ ((𝑁 ∈ ℙ ∧ 𝑁 ≠ 2) → ¬ 2 ∥ 𝑁) |
18 | isodd3 45997 | . . 3 ⊢ (𝑁 ∈ Odd ↔ (𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁)) | |
19 | 3, 17, 18 | sylanbrc 583 | . 2 ⊢ ((𝑁 ∈ ℙ ∧ 𝑁 ≠ 2) → 𝑁 ∈ Odd ) |
20 | 1, 19 | sylbi 216 | 1 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ Odd ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∖ cdif 3925 {csn 4606 class class class wbr 5125 1c1 11076 ℕcn 12177 2c2 12232 ℤcz 12523 ∥ cdvds 16162 ℙcprime 16573 Odd codd 45970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-sup 9402 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-3 12241 df-n0 12438 df-z 12524 df-uz 12788 df-rp 12940 df-seq 13932 df-exp 13993 df-cj 15011 df-re 15012 df-im 15013 df-sqrt 15147 df-abs 15148 df-dvds 16163 df-prm 16574 df-odd 45972 |
This theorem is referenced by: evenprm2 46059 odd2prm2 46063 even3prm2 46064 bgoldbtbndlem2 46151 bgoldbtbndlem3 46152 bgoldbtbndlem4 46153 bgoldbtbnd 46154 |
Copyright terms: Public domain | W3C validator |