![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
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 47114, this bound is strict. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
gbowge7 | ⊢ (𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gbowgt5 47104 | . 2 ⊢ (𝑍 ∈ GoldbachOddW → 5 < 𝑍) | |
2 | gbowpos 47101 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ) | |
3 | 5nn 12334 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
4 | 3 | nnzi 12622 | . . . . . 6 ⊢ 5 ∈ ℤ |
5 | nnz 12615 | . . . . . 6 ⊢ (𝑍 ∈ ℕ → 𝑍 ∈ ℤ) | |
6 | zltp1le 12648 | . . . . . 6 ⊢ ((5 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) | |
7 | 4, 5, 6 | sylancr 585 | . . . . 5 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) |
8 | 7 | biimpd 228 | . . . 4 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 → (5 + 1) ≤ 𝑍)) |
9 | 2, 8 | syl 17 | . . 3 ⊢ (𝑍 ∈ GoldbachOddW → (5 < 𝑍 → (5 + 1) ≤ 𝑍)) |
10 | 5p1e6 12395 | . . . . . 6 ⊢ (5 + 1) = 6 | |
11 | 10 | breq1i 5157 | . . . . 5 ⊢ ((5 + 1) ≤ 𝑍 ↔ 6 ≤ 𝑍) |
12 | 6re 12338 | . . . . . 6 ⊢ 6 ∈ ℝ | |
13 | 2 | nnred 12263 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℝ) |
14 | leloe 11336 | . . . . . 6 ⊢ ((6 ∈ ℝ ∧ 𝑍 ∈ ℝ) → (6 ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) | |
15 | 12, 13, 14 | sylancr 585 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) |
16 | 11, 15 | bitrid 282 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → ((5 + 1) ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) |
17 | 6nn 12337 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
18 | 17 | nnzi 12622 | . . . . . . 7 ⊢ 6 ∈ ℤ |
19 | 2 | nnzd 12621 | . . . . . . 7 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℤ) |
20 | zltp1le 12648 | . . . . . . . 8 ⊢ ((6 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (6 < 𝑍 ↔ (6 + 1) ≤ 𝑍)) | |
21 | 20 | biimpd 228 | . . . . . . 7 ⊢ ((6 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (6 < 𝑍 → (6 + 1) ≤ 𝑍)) |
22 | 18, 19, 21 | sylancr 585 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW → (6 < 𝑍 → (6 + 1) ≤ 𝑍)) |
23 | 6p1e7 12396 | . . . . . . 7 ⊢ (6 + 1) = 7 | |
24 | 23 | breq1i 5157 | . . . . . 6 ⊢ ((6 + 1) ≤ 𝑍 ↔ 7 ≤ 𝑍) |
25 | 22, 24 | imbitrdi 250 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 < 𝑍 → 7 ≤ 𝑍)) |
26 | isgbow 47094 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) | |
27 | eleq1 2816 | . . . . . . . . 9 ⊢ (6 = 𝑍 → (6 ∈ Odd ↔ 𝑍 ∈ Odd )) | |
28 | 6even 47053 | . . . . . . . . . 10 ⊢ 6 ∈ Even | |
29 | evennodd 46985 | . . . . . . . . . 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 253 | . . . . . . . 8 ⊢ (6 = 𝑍 → (𝑍 ∈ Odd → 7 ≤ 𝑍)) |
33 | 32 | com12 32 | . . . . . . 7 ⊢ (𝑍 ∈ Odd → (6 = 𝑍 → 7 ≤ 𝑍)) |
34 | 33 | adantr 479 | . . . . . 6 ⊢ ((𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → (6 = 𝑍 → 7 ≤ 𝑍)) |
35 | 26, 34 | sylbi 216 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 = 𝑍 → 7 ≤ 𝑍)) |
36 | 25, 35 | jaod 857 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → ((6 < 𝑍 ∨ 6 = 𝑍) → 7 ≤ 𝑍)) |
37 | 16, 36 | sylbid 239 | . . 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 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∃wrex 3066 class class class wbr 5150 (class class class)co 7424 ℝcr 11143 1c1 11145 + caddc 11147 < clt 11284 ≤ cle 11285 ℕcn 12248 5c5 12306 6c6 12307 7c7 12308 ℤcz 12594 ℙcprime 16647 Even ceven 46966 Odd codd 46967 GoldbachOddW cgbow 47088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-dvds 16237 df-prm 16648 df-even 46968 df-odd 46969 df-gbow 47091 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |