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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 45995 | . 2 ⊢ 8 ∈ Even | |
2 | 5prm 16989 | . . 3 ⊢ 5 ∈ ℙ | |
3 | 3prm 16578 | . . 3 ⊢ 3 ∈ ℙ | |
4 | 5odd 45992 | . . . 4 ⊢ 5 ∈ Odd | |
5 | 3odd 45990 | . . . 4 ⊢ 3 ∈ Odd | |
6 | 5p3e8 12318 | . . . . 5 ⊢ (5 + 3) = 8 | |
7 | 6 | eqcomi 2742 | . . . 4 ⊢ 8 = (5 + 3) |
8 | 4, 5, 7 | 3pm3.2i 1340 | . . 3 ⊢ (5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = (5 + 3)) |
9 | eleq1 2822 | . . . . 5 ⊢ (𝑝 = 5 → (𝑝 ∈ Odd ↔ 5 ∈ Odd )) | |
10 | biidd 262 | . . . . 5 ⊢ (𝑝 = 5 → (𝑞 ∈ Odd ↔ 𝑞 ∈ Odd )) | |
11 | oveq1 7368 | . . . . . 6 ⊢ (𝑝 = 5 → (𝑝 + 𝑞) = (5 + 𝑞)) | |
12 | 11 | eqeq2d 2744 | . . . . 5 ⊢ (𝑝 = 5 → (8 = (𝑝 + 𝑞) ↔ 8 = (5 + 𝑞))) |
13 | 9, 10, 12 | 3anbi123d 1437 | . . . 4 ⊢ (𝑝 = 5 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (𝑝 + 𝑞)) ↔ (5 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (5 + 𝑞)))) |
14 | biidd 262 | . . . . 5 ⊢ (𝑞 = 3 → (5 ∈ Odd ↔ 5 ∈ Odd )) | |
15 | eleq1 2822 | . . . . 5 ⊢ (𝑞 = 3 → (𝑞 ∈ Odd ↔ 3 ∈ Odd )) | |
16 | oveq2 7369 | . . . . . 6 ⊢ (𝑞 = 3 → (5 + 𝑞) = (5 + 3)) | |
17 | 16 | eqeq2d 2744 | . . . . 5 ⊢ (𝑞 = 3 → (8 = (5 + 𝑞) ↔ 8 = (5 + 3))) |
18 | 14, 15, 17 | 3anbi123d 1437 | . . . 4 ⊢ (𝑞 = 3 → ((5 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (5 + 𝑞)) ↔ (5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = (5 + 3)))) |
19 | 13, 18 | rspc2ev 3594 | . . 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 1462 | . 2 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (𝑝 + 𝑞)) |
21 | isgbe 46033 | . 2 ⊢ (8 ∈ GoldbachEven ↔ (8 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (𝑝 + 𝑞)))) | |
22 | 1, 20, 21 | mpbir2an 710 | 1 ⊢ 8 ∈ GoldbachEven |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 (class class class)co 7361 + caddc 11062 3c3 12217 5c5 12219 8c8 12222 ℙcprime 16555 Even ceven 45906 Odd codd 45907 GoldbachEven cgbe 46027 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-rp 12924 df-fz 13434 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-dvds 16145 df-prm 16556 df-even 45908 df-odd 45909 df-gbe 46030 |
This theorem is referenced by: nnsum3primesle9 46076 |
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