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Mirrors > Home > MPE Home > Th. List > Mathboxes > gboge7 | Structured version Visualization version GIF version |
Description: Any odd Goldbach number is greater than or equal to 7. Because of 7gbo 40194, this bound is strict. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
gboge7 | ⊢ (𝑍 ∈ GoldbachOdd → 7 ≤ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gbogt5 40184 | . 2 ⊢ (𝑍 ∈ GoldbachOdd → 5 < 𝑍) | |
2 | gbopos 40181 | . . . 4 ⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ ℕ) | |
3 | 5nn 11065 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
4 | 3 | nnzi 11278 | . . . . . 6 ⊢ 5 ∈ ℤ |
5 | nnz 11276 | . . . . . 6 ⊢ (𝑍 ∈ ℕ → 𝑍 ∈ ℤ) | |
6 | zltp1le 11304 | . . . . . 6 ⊢ ((5 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) | |
7 | 4, 5, 6 | sylancr 694 | . . . . 5 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 ↔ (5 + 1) ≤ 𝑍)) |
8 | 7 | biimpd 218 | . . . 4 ⊢ (𝑍 ∈ ℕ → (5 < 𝑍 → (5 + 1) ≤ 𝑍)) |
9 | 2, 8 | syl 17 | . . 3 ⊢ (𝑍 ∈ GoldbachOdd → (5 < 𝑍 → (5 + 1) ≤ 𝑍)) |
10 | 5p1e6 11032 | . . . . . 6 ⊢ (5 + 1) = 6 | |
11 | 10 | breq1i 4590 | . . . . 5 ⊢ ((5 + 1) ≤ 𝑍 ↔ 6 ≤ 𝑍) |
12 | 6re 10978 | . . . . . 6 ⊢ 6 ∈ ℝ | |
13 | 2 | nnred 10912 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ ℝ) |
14 | leloe 10003 | . . . . . 6 ⊢ ((6 ∈ ℝ ∧ 𝑍 ∈ ℝ) → (6 ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) | |
15 | 12, 13, 14 | sylancr 694 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOdd → (6 ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) |
16 | 11, 15 | syl5bb 271 | . . . 4 ⊢ (𝑍 ∈ GoldbachOdd → ((5 + 1) ≤ 𝑍 ↔ (6 < 𝑍 ∨ 6 = 𝑍))) |
17 | 6nn 11066 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
18 | 17 | nnzi 11278 | . . . . . . 7 ⊢ 6 ∈ ℤ |
19 | 2 | nnzd 11357 | . . . . . . 7 ⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ ℤ) |
20 | zltp1le 11304 | . . . . . . . 8 ⊢ ((6 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (6 < 𝑍 ↔ (6 + 1) ≤ 𝑍)) | |
21 | 20 | biimpd 218 | . . . . . . 7 ⊢ ((6 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (6 < 𝑍 → (6 + 1) ≤ 𝑍)) |
22 | 18, 19, 21 | sylancr 694 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOdd → (6 < 𝑍 → (6 + 1) ≤ 𝑍)) |
23 | 6p1e7 11033 | . . . . . . 7 ⊢ (6 + 1) = 7 | |
24 | 23 | breq1i 4590 | . . . . . 6 ⊢ ((6 + 1) ≤ 𝑍 ↔ 7 ≤ 𝑍) |
25 | 22, 24 | syl6ib 240 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOdd → (6 < 𝑍 → 7 ≤ 𝑍)) |
26 | isgbo 40174 | . . . . . 6 ⊢ (𝑍 ∈ GoldbachOdd ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) | |
27 | eleq1 2676 | . . . . . . . . 9 ⊢ (6 = 𝑍 → (6 ∈ Odd ↔ 𝑍 ∈ Odd )) | |
28 | 6even 40158 | . . . . . . . . . 10 ⊢ 6 ∈ Even | |
29 | evennodd 40094 | . . . . . . . . . 10 ⊢ (6 ∈ Even → ¬ 6 ∈ Odd ) | |
30 | pm2.21 119 | . . . . . . . . . 10 ⊢ (¬ 6 ∈ Odd → (6 ∈ Odd → 7 ≤ 𝑍)) | |
31 | 28, 29, 30 | mp2b 10 | . . . . . . . . 9 ⊢ (6 ∈ Odd → 7 ≤ 𝑍) |
32 | 27, 31 | syl6bir 243 | . . . . . . . 8 ⊢ (6 = 𝑍 → (𝑍 ∈ Odd → 7 ≤ 𝑍)) |
33 | 32 | com12 32 | . . . . . . 7 ⊢ (𝑍 ∈ Odd → (6 = 𝑍 → 7 ≤ 𝑍)) |
34 | 33 | adantr 480 | . . . . . 6 ⊢ ((𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → (6 = 𝑍 → 7 ≤ 𝑍)) |
35 | 26, 34 | sylbi 206 | . . . . 5 ⊢ (𝑍 ∈ GoldbachOdd → (6 = 𝑍 → 7 ≤ 𝑍)) |
36 | 25, 35 | jaod 394 | . . . 4 ⊢ (𝑍 ∈ GoldbachOdd → ((6 < 𝑍 ∨ 6 = 𝑍) → 7 ≤ 𝑍)) |
37 | 16, 36 | sylbid 229 | . . 3 ⊢ (𝑍 ∈ GoldbachOdd → ((5 + 1) ≤ 𝑍 → 7 ≤ 𝑍)) |
38 | 9, 37 | syld 46 | . 2 ⊢ (𝑍 ∈ GoldbachOdd → (5 < 𝑍 → 7 ≤ 𝑍)) |
39 | 1, 38 | mpd 15 | 1 ⊢ (𝑍 ∈ GoldbachOdd → 7 ≤ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 ℕcn 10897 5c5 10950 6c6 10951 7c7 10952 ℤcz 11254 ℙcprime 15223 Even ceven 40075 Odd codd 40076 GoldbachOdd cgbo 40168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-prm 15224 df-even 40077 df-odd 40078 df-gbo 40171 |
This theorem is referenced by: (None) |
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