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 43814, this bound is strict. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
gbowge7 | ⊢ (𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gbowgt5 43804 | . 2 ⊢ (𝑍 ∈ GoldbachOddW → 5 < 𝑍) | |
2 | gbowpos 43801 | . . . 4 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ) | |
3 | 5nn 11711 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
4 | 3 | nnzi 11994 | . . . . . 6 ⊢ 5 ∈ ℤ |
5 | nnz 11992 | . . . . . 6 ⊢ (𝑍 ∈ ℕ → 𝑍 ∈ ℤ) | |
6 | zltp1le 12020 | . . . . . 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 11772 | . . . . . 6 ⊢ (5 + 1) = 6 | |
11 | 10 | breq1i 5064 | . . . . 5 ⊢ ((5 + 1) ≤ 𝑍 ↔ 6 ≤ 𝑍) |
12 | 6re 11715 | . . . . . 6 ⊢ 6 ∈ ℝ | |
13 | 2 | nnred 11641 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℝ) |
14 | leloe 10715 | . . . . . 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 11714 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
18 | 17 | nnzi 11994 | . . . . . . 7 ⊢ 6 ∈ ℤ |
19 | 2 | nnzd 12074 | . . . . . . 7 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℤ) |
20 | zltp1le 12020 | . . . . . . . 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 11773 | . . . . . . 7 ⊢ (6 + 1) = 7 | |
24 | 23 | breq1i 5064 | . . . . . 6 ⊢ ((6 + 1) ≤ 𝑍 ↔ 7 ≤ 𝑍) |
25 | 22, 24 | syl6ib 252 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOddW → (6 < 𝑍 → 7 ≤ 𝑍)) |
26 | isgbow 43794 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) | |
27 | eleq1 2897 | . . . . . . . . 9 ⊢ (6 = 𝑍 → (6 ∈ Odd ↔ 𝑍 ∈ Odd )) | |
28 | 6even 43753 | . . . . . . . . . 10 ⊢ 6 ∈ Even | |
29 | evennodd 43685 | . . . . . . . . . 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 853 | . . . 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 841 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 1c1 10526 + caddc 10528 < clt 10663 ≤ cle 10664 ℕcn 11626 5c5 11683 6c6 11684 7c7 11685 ℤcz 11969 ℙcprime 16003 Even ceven 43666 Odd codd 43667 GoldbachOddW cgbow 43788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-prm 16004 df-even 43668 df-odd 43669 df-gbow 43791 |
This theorem is referenced by: (None) |
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