![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 8gbe | Structured version Visualization version GIF version |
Description: 8 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
8gbe | ⊢ 8 ∈ GoldbachEven |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8even 46371 | . 2 ⊢ 8 ∈ Even | |
2 | 5prm 17041 | . . 3 ⊢ 5 ∈ ℙ | |
3 | 3prm 16630 | . . 3 ⊢ 3 ∈ ℙ | |
4 | 5odd 46368 | . . . 4 ⊢ 5 ∈ Odd | |
5 | 3odd 46366 | . . . 4 ⊢ 3 ∈ Odd | |
6 | 5p3e8 12368 | . . . . 5 ⊢ (5 + 3) = 8 | |
7 | 6 | eqcomi 2741 | . . . 4 ⊢ 8 = (5 + 3) |
8 | 4, 5, 7 | 3pm3.2i 1339 | . . 3 ⊢ (5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = (5 + 3)) |
9 | eleq1 2821 | . . . . 5 ⊢ (𝑝 = 5 → (𝑝 ∈ Odd ↔ 5 ∈ Odd )) | |
10 | biidd 261 | . . . . 5 ⊢ (𝑝 = 5 → (𝑞 ∈ Odd ↔ 𝑞 ∈ Odd )) | |
11 | oveq1 7415 | . . . . . 6 ⊢ (𝑝 = 5 → (𝑝 + 𝑞) = (5 + 𝑞)) | |
12 | 11 | eqeq2d 2743 | . . . . 5 ⊢ (𝑝 = 5 → (8 = (𝑝 + 𝑞) ↔ 8 = (5 + 𝑞))) |
13 | 9, 10, 12 | 3anbi123d 1436 | . . . 4 ⊢ (𝑝 = 5 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (𝑝 + 𝑞)) ↔ (5 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (5 + 𝑞)))) |
14 | biidd 261 | . . . . 5 ⊢ (𝑞 = 3 → (5 ∈ Odd ↔ 5 ∈ Odd )) | |
15 | eleq1 2821 | . . . . 5 ⊢ (𝑞 = 3 → (𝑞 ∈ Odd ↔ 3 ∈ Odd )) | |
16 | oveq2 7416 | . . . . . 6 ⊢ (𝑞 = 3 → (5 + 𝑞) = (5 + 3)) | |
17 | 16 | eqeq2d 2743 | . . . . 5 ⊢ (𝑞 = 3 → (8 = (5 + 𝑞) ↔ 8 = (5 + 3))) |
18 | 14, 15, 17 | 3anbi123d 1436 | . . . 4 ⊢ (𝑞 = 3 → ((5 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (5 + 𝑞)) ↔ (5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = (5 + 3)))) |
19 | 13, 18 | rspc2ev 3624 | . . 3 ⊢ ((5 ∈ ℙ ∧ 3 ∈ ℙ ∧ (5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = (5 + 3))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (𝑝 + 𝑞))) |
20 | 2, 3, 8, 19 | mp3an 1461 | . 2 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (𝑝 + 𝑞)) |
21 | isgbe 46409 | . 2 ⊢ (8 ∈ GoldbachEven ↔ (8 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (𝑝 + 𝑞)))) | |
22 | 1, 20, 21 | mpbir2an 709 | 1 ⊢ 8 ∈ GoldbachEven |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 (class class class)co 7408 + caddc 11112 3c3 12267 5c5 12269 8c8 12272 ℙcprime 16607 Even ceven 46282 Odd codd 46283 GoldbachEven cgbe 46403 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-rp 12974 df-fz 13484 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-dvds 16197 df-prm 16608 df-even 46284 df-odd 46285 df-gbe 46406 |
This theorem is referenced by: nnsum3primesle9 46452 |
Copyright terms: Public domain | W3C validator |