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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gbege6 | Structured version Visualization version GIF version |
Description: Any even Goldbach number is greater than or equal to 6. Because of 6gbe 46053, this bound is strict. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
gbege6 | ⊢ (𝑍 ∈ GoldbachEven → 6 ≤ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gbepos 46040 | . 2 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ ℕ) | |
2 | gbegt5 46043 | . 2 ⊢ (𝑍 ∈ GoldbachEven → 5 < 𝑍) | |
3 | 5nn 12247 | . . . . . 6 ⊢ 5 ∈ ℕ | |
4 | 3 | nnzi 12535 | . . . . 5 ⊢ 5 ∈ ℤ |
5 | nnz 12528 | . . . . 5 ⊢ (𝑍 ∈ ℕ → 𝑍 ∈ ℤ) | |
6 | zltp1le 12561 | . . . . 5 ⊢ ((5 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) | |
7 | 4, 5, 6 | sylancr 588 | . . . 4 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) |
8 | 7 | biimpd 228 | . . 3 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 → (5 + 1) ≤ 𝑍)) |
9 | 5p1e6 12308 | . . . 4 ⊢ (5 + 1) = 6 | |
10 | 9 | breq1i 5116 | . . 3 ⊢ ((5 + 1) ≤ 𝑍 ↔ 6 ≤ 𝑍) |
11 | 8, 10 | syl6ib 251 | . 2 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 → 6 ≤ 𝑍)) |
12 | 1, 2, 11 | sylc 65 | 1 ⊢ (𝑍 ∈ GoldbachEven → 6 ≤ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 class class class wbr 5109 (class class class)co 7361 1c1 11060 + caddc 11062 < clt 11197 ≤ cle 11198 ℕcn 12161 5c5 12219 6c6 12220 ℤcz 12507 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-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-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-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 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: (None) |
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