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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 45143 | . 2 ⊢ 8 ∈ Even | |
2 | 5prm 16820 | . . 3 ⊢ 5 ∈ ℙ | |
3 | 3prm 16409 | . . 3 ⊢ 3 ∈ ℙ | |
4 | 5odd 45140 | . . . 4 ⊢ 5 ∈ Odd | |
5 | 3odd 45138 | . . . 4 ⊢ 3 ∈ Odd | |
6 | 5p3e8 12140 | . . . . 5 ⊢ (5 + 3) = 8 | |
7 | 6 | eqcomi 2747 | . . . 4 ⊢ 8 = (5 + 3) |
8 | 4, 5, 7 | 3pm3.2i 1338 | . . 3 ⊢ (5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = (5 + 3)) |
9 | eleq1 2826 | . . . . 5 ⊢ (𝑝 = 5 → (𝑝 ∈ Odd ↔ 5 ∈ Odd )) | |
10 | biidd 261 | . . . . 5 ⊢ (𝑝 = 5 → (𝑞 ∈ Odd ↔ 𝑞 ∈ Odd )) | |
11 | oveq1 7274 | . . . . . 6 ⊢ (𝑝 = 5 → (𝑝 + 𝑞) = (5 + 𝑞)) | |
12 | 11 | eqeq2d 2749 | . . . . 5 ⊢ (𝑝 = 5 → (8 = (𝑝 + 𝑞) ↔ 8 = (5 + 𝑞))) |
13 | 9, 10, 12 | 3anbi123d 1435 | . . . 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 2826 | . . . . 5 ⊢ (𝑞 = 3 → (𝑞 ∈ Odd ↔ 3 ∈ Odd )) | |
16 | oveq2 7275 | . . . . . 6 ⊢ (𝑞 = 3 → (5 + 𝑞) = (5 + 3)) | |
17 | 16 | eqeq2d 2749 | . . . . 5 ⊢ (𝑞 = 3 → (8 = (5 + 𝑞) ↔ 8 = (5 + 3))) |
18 | 14, 15, 17 | 3anbi123d 1435 | . . . 4 ⊢ (𝑞 = 3 → ((5 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (5 + 𝑞)) ↔ (5 ∈ Odd ∧ 3 ∈ Odd ∧ 8 = (5 + 3)))) |
19 | 13, 18 | rspc2ev 3571 | . . 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 1460 | . 2 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (𝑝 + 𝑞)) |
21 | isgbe 45181 | . 2 ⊢ (8 ∈ GoldbachEven ↔ (8 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 8 = (𝑝 + 𝑞)))) | |
22 | 1, 20, 21 | mpbir2an 708 | 1 ⊢ 8 ∈ GoldbachEven |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 (class class class)co 7267 + caddc 10884 3c3 12039 5c5 12041 8c8 12044 ℙcprime 16386 Even ceven 45054 Odd codd 45055 GoldbachEven cgbe 45175 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-sup 9188 df-inf 9189 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-rp 12741 df-fz 13250 df-seq 13732 df-exp 13793 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-dvds 15974 df-prm 16387 df-even 45056 df-odd 45057 df-gbe 45178 |
This theorem is referenced by: nnsum3primesle9 45224 |
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