Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gbowge7 | Structured version Visualization version GIF version |
Description: Any weak odd Goldbach number is greater than or equal to 7. Because of 7gbow 43936, this bound is strict. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
gbowge7 | ⊢ (𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gbowgt5 43926 | . 2 ⊢ (𝑍 ∈ GoldbachOddW → 5 < 𝑍) | |
2 | gbowpos 43923 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ) | |
3 | 5nn 11722 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
4 | 3 | nnzi 12005 | . . . . . 6 ⊢ 5 ∈ ℤ |
5 | nnz 12003 | . . . . . 6 ⊢ (𝑍 ∈ ℕ → 𝑍 ∈ ℤ) | |
6 | zltp1le 12031 | . . . . . 6 ⊢ ((5 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) | |
7 | 4, 5, 6 | sylancr 589 | . . . . 5 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) |
8 | 7 | biimpd 231 | . . . 4 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 → (5 + 1) ≤ 𝑍)) |
9 | 2, 8 | syl 17 | . . 3 ⊢ (𝑍 ∈ GoldbachOddW → (5 < 𝑍 → (5 + 1) ≤ 𝑍)) |
10 | 5p1e6 11783 | . . . . . 6 ⊢ (5 + 1) = 6 | |
11 | 10 | breq1i 5072 | . . . . 5 ⊢ ((5 + 1) ≤ 𝑍 ↔ 6 ≤ 𝑍) |
12 | 6re 11726 | . . . . . 6 ⊢ 6 ∈ ℝ | |
13 | 2 | nnred 11652 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℝ) |
14 | leloe 10726 | . . . . . 6 ⊢ ((6 ∈ ℝ ∧ 𝑍 ∈ ℝ) → (6 ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) | |
15 | 12, 13, 14 | sylancr 589 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) |
16 | 11, 15 | syl5bb 285 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → ((5 + 1) ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) |
17 | 6nn 11725 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
18 | 17 | nnzi 12005 | . . . . . . 7 ⊢ 6 ∈ ℤ |
19 | 2 | nnzd 12085 | . . . . . . 7 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℤ) |
20 | zltp1le 12031 | . . . . . . . 8 ⊢ ((6 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (6 < 𝑍 ↔ (6 + 1) ≤ 𝑍)) | |
21 | 20 | biimpd 231 | . . . . . . 7 ⊢ ((6 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (6 < 𝑍 → (6 + 1) ≤ 𝑍)) |
22 | 18, 19, 21 | sylancr 589 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW → (6 < 𝑍 → (6 + 1) ≤ 𝑍)) |
23 | 6p1e7 11784 | . . . . . . 7 ⊢ (6 + 1) = 7 | |
24 | 23 | breq1i 5072 | . . . . . 6 ⊢ ((6 + 1) ≤ 𝑍 ↔ 7 ≤ 𝑍) |
25 | 22, 24 | syl6ib 253 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 < 𝑍 → 7 ≤ 𝑍)) |
26 | isgbow 43916 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) | |
27 | eleq1 2900 | . . . . . . . . 9 ⊢ (6 = 𝑍 → (6 ∈ Odd ↔ 𝑍 ∈ Odd )) | |
28 | 6even 43875 | . . . . . . . . . 10 ⊢ 6 ∈ Even | |
29 | evennodd 43807 | . . . . . . . . . 10 ⊢ (6 ∈ Even → ¬ 6 ∈ Odd ) | |
30 | pm2.21 123 | . . . . . . . . . 10 ⊢ (¬ 6 ∈ Odd → (6 ∈ Odd → 7 ≤ 𝑍)) | |
31 | 28, 29, 30 | mp2b 10 | . . . . . . . . 9 ⊢ (6 ∈ Odd → 7 ≤ 𝑍) |
32 | 27, 31 | syl6bir 256 | . . . . . . . 8 ⊢ (6 = 𝑍 → (𝑍 ∈ Odd → 7 ≤ 𝑍)) |
33 | 32 | com12 32 | . . . . . . 7 ⊢ (𝑍 ∈ Odd → (6 = 𝑍 → 7 ≤ 𝑍)) |
34 | 33 | adantr 483 | . . . . . 6 ⊢ ((𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → (6 = 𝑍 → 7 ≤ 𝑍)) |
35 | 26, 34 | sylbi 219 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 = 𝑍 → 7 ≤ 𝑍)) |
36 | 25, 35 | jaod 855 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → ((6 < 𝑍 ∨ 6 = 𝑍) → 7 ≤ 𝑍)) |
37 | 16, 36 | sylbid 242 | . . 3 ⊢ (𝑍 ∈ GoldbachOddW → ((5 + 1) ≤ 𝑍 → 7 ≤ 𝑍)) |
38 | 9, 37 | syld 47 | . 2 ⊢ (𝑍 ∈ GoldbachOddW → (5 < 𝑍 → 7 ≤ 𝑍)) |
39 | 1, 38 | mpd 15 | 1 ⊢ (𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5065 (class class class)co 7155 ℝcr 10535 1c1 10537 + caddc 10539 < clt 10674 ≤ cle 10675 ℕcn 11637 5c5 11694 6c6 11695 7c7 11696 ℤcz 11980 ℙcprime 16014 Even ceven 43788 Odd codd 43789 GoldbachOddW cgbow 43910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-dvds 15607 df-prm 16015 df-even 43790 df-odd 43791 df-gbow 43913 |
This theorem is referenced by: (None) |
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