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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 43373, this bound is strict. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
gbowge7 | ⊢ (𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gbowgt5 43363 | . 2 ⊢ (𝑍 ∈ GoldbachOddW → 5 < 𝑍) | |
2 | gbowpos 43360 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ) | |
3 | 5nn 11560 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
4 | 3 | nnzi 11844 | . . . . . 6 ⊢ 5 ∈ ℤ |
5 | nnz 11842 | . . . . . 6 ⊢ (𝑍 ∈ ℕ → 𝑍 ∈ ℤ) | |
6 | zltp1le 11870 | . . . . . 6 ⊢ ((5 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) | |
7 | 4, 5, 6 | sylancr 587 | . . . . 5 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) |
8 | 7 | biimpd 230 | . . . 4 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 → (5 + 1) ≤ 𝑍)) |
9 | 2, 8 | syl 17 | . . 3 ⊢ (𝑍 ∈ GoldbachOddW → (5 < 𝑍 → (5 + 1) ≤ 𝑍)) |
10 | 5p1e6 11621 | . . . . . 6 ⊢ (5 + 1) = 6 | |
11 | 10 | breq1i 4963 | . . . . 5 ⊢ ((5 + 1) ≤ 𝑍 ↔ 6 ≤ 𝑍) |
12 | 6re 11564 | . . . . . 6 ⊢ 6 ∈ ℝ | |
13 | 2 | nnred 11490 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℝ) |
14 | leloe 10563 | . . . . . 6 ⊢ ((6 ∈ ℝ ∧ 𝑍 ∈ ℝ) → (6 ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) | |
15 | 12, 13, 14 | sylancr 587 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) |
16 | 11, 15 | syl5bb 284 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → ((5 + 1) ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) |
17 | 6nn 11563 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
18 | 17 | nnzi 11844 | . . . . . . 7 ⊢ 6 ∈ ℤ |
19 | 2 | nnzd 11924 | . . . . . . 7 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℤ) |
20 | zltp1le 11870 | . . . . . . . 8 ⊢ ((6 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (6 < 𝑍 ↔ (6 + 1) ≤ 𝑍)) | |
21 | 20 | biimpd 230 | . . . . . . 7 ⊢ ((6 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (6 < 𝑍 → (6 + 1) ≤ 𝑍)) |
22 | 18, 19, 21 | sylancr 587 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW → (6 < 𝑍 → (6 + 1) ≤ 𝑍)) |
23 | 6p1e7 11622 | . . . . . . 7 ⊢ (6 + 1) = 7 | |
24 | 23 | breq1i 4963 | . . . . . 6 ⊢ ((6 + 1) ≤ 𝑍 ↔ 7 ≤ 𝑍) |
25 | 22, 24 | syl6ib 252 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 < 𝑍 → 7 ≤ 𝑍)) |
26 | isgbow 43353 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) | |
27 | eleq1 2868 | . . . . . . . . 9 ⊢ (6 = 𝑍 → (6 ∈ Odd ↔ 𝑍 ∈ Odd )) | |
28 | 6even 43312 | . . . . . . . . . 10 ⊢ 6 ∈ Even | |
29 | evennodd 43244 | . . . . . . . . . 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 255 | . . . . . . . 8 ⊢ (6 = 𝑍 → (𝑍 ∈ Odd → 7 ≤ 𝑍)) |
33 | 32 | com12 32 | . . . . . . 7 ⊢ (𝑍 ∈ Odd → (6 = 𝑍 → 7 ≤ 𝑍)) |
34 | 33 | adantr 481 | . . . . . 6 ⊢ ((𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → (6 = 𝑍 → 7 ≤ 𝑍)) |
35 | 26, 34 | sylbi 218 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 = 𝑍 → 7 ≤ 𝑍)) |
36 | 25, 35 | jaod 854 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → ((6 < 𝑍 ∨ 6 = 𝑍) → 7 ≤ 𝑍)) |
37 | 16, 36 | sylbid 241 | . . 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 207 ∧ wa 396 ∨ wo 842 = wceq 1520 ∈ wcel 2079 ∃wrex 3104 class class class wbr 4956 (class class class)co 7007 ℝcr 10371 1c1 10373 + caddc 10375 < clt 10510 ≤ cle 10511 ℕcn 11475 5c5 11532 6c6 11533 7c7 11534 ℤcz 11818 ℙcprime 15832 Even ceven 43225 Odd codd 43226 GoldbachOddW cgbow 43347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-n0 11735 df-z 11819 df-uz 12083 df-rp 12229 df-seq 13208 df-exp 13268 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-dvds 15429 df-prm 15833 df-even 43227 df-odd 43228 df-gbow 43350 |
This theorem is referenced by: (None) |
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