Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > odd2prm2 | Structured version Visualization version GIF version |
Description: If an odd number is the sum of two prime numbers, one of the prime numbers must be 2. (Contributed by AV, 26-Dec-2021.) |
Ref | Expression |
---|---|
odd2prm2 | ⊢ ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2900 | . . . . . 6 ⊢ (𝑁 = (𝑃 + 𝑄) → (𝑁 ∈ Odd ↔ (𝑃 + 𝑄) ∈ Odd )) | |
2 | evennodd 43828 | . . . . . . . . 9 ⊢ ((𝑃 + 𝑄) ∈ Even → ¬ (𝑃 + 𝑄) ∈ Odd ) | |
3 | 2 | pm2.21d 121 | . . . . . . . 8 ⊢ ((𝑃 + 𝑄) ∈ Even → ((𝑃 + 𝑄) ∈ Odd → (𝑃 = 2 ∨ 𝑄 = 2))) |
4 | df-ne 3017 | . . . . . . . . . . . 12 ⊢ (𝑃 ≠ 2 ↔ ¬ 𝑃 = 2) | |
5 | eldifsn 4719 | . . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) | |
6 | oddprmALTV 43872 | . . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ Odd ) | |
7 | 5, 6 | sylbir 237 | . . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → 𝑃 ∈ Odd ) |
8 | 7 | ex 415 | . . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → (𝑃 ≠ 2 → 𝑃 ∈ Odd )) |
9 | 4, 8 | syl5bir 245 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → 𝑃 ∈ Odd )) |
10 | df-ne 3017 | . . . . . . . . . . . 12 ⊢ (𝑄 ≠ 2 ↔ ¬ 𝑄 = 2) | |
11 | eldifsn 4719 | . . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℙ ∖ {2}) ↔ (𝑄 ∈ ℙ ∧ 𝑄 ≠ 2)) | |
12 | oddprmALTV 43872 | . . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℙ ∖ {2}) → 𝑄 ∈ Odd ) | |
13 | 11, 12 | sylbir 237 | . . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℙ ∧ 𝑄 ≠ 2) → 𝑄 ∈ Odd ) |
14 | 13 | ex 415 | . . . . . . . . . . . 12 ⊢ (𝑄 ∈ ℙ → (𝑄 ≠ 2 → 𝑄 ∈ Odd )) |
15 | 10, 14 | syl5bir 245 | . . . . . . . . . . 11 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 = 2 → 𝑄 ∈ Odd )) |
16 | 9, 15 | im2anan9 621 | . . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd ))) |
17 | 16 | imp 409 | . . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd )) |
18 | opoeALTV 43868 | . . . . . . . . 9 ⊢ ((𝑃 ∈ Odd ∧ 𝑄 ∈ Odd ) → (𝑃 + 𝑄) ∈ Even ) | |
19 | 17, 18 | syl 17 | . . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 + 𝑄) ∈ Even ) |
20 | 3, 19 | syl11 33 | . . . . . . 7 ⊢ ((𝑃 + 𝑄) ∈ Odd → (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 = 2 ∨ 𝑄 = 2))) |
21 | 20 | expd 418 | . . . . . 6 ⊢ ((𝑃 + 𝑄) ∈ Odd → ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2)))) |
22 | 1, 21 | syl6bi 255 | . . . . 5 ⊢ (𝑁 = (𝑃 + 𝑄) → (𝑁 ∈ Odd → ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2))))) |
23 | 22 | 3imp231 1109 | . . . 4 ⊢ ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2))) |
24 | 23 | com12 32 | . . 3 ⊢ ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))) |
25 | 24 | ex 415 | . 2 ⊢ (¬ 𝑃 = 2 → (¬ 𝑄 = 2 → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)))) |
26 | orc 863 | . . 3 ⊢ (𝑃 = 2 → (𝑃 = 2 ∨ 𝑄 = 2)) | |
27 | 26 | a1d 25 | . 2 ⊢ (𝑃 = 2 → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))) |
28 | olc 864 | . . 3 ⊢ (𝑄 = 2 → (𝑃 = 2 ∨ 𝑄 = 2)) | |
29 | 28 | a1d 25 | . 2 ⊢ (𝑄 = 2 → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))) |
30 | 25, 27, 29 | pm2.61ii 185 | 1 ⊢ ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 {csn 4567 (class class class)co 7156 + caddc 10540 2c2 11693 ℙcprime 16015 Even ceven 43809 Odd codd 43810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-prm 16016 df-even 43811 df-odd 43812 |
This theorem is referenced by: even3prm2 43904 |
Copyright terms: Public domain | W3C validator |