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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbachgnn | Structured version Visualization version GIF version |
Description: Lemma for tgoldbachgtd 32043. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
Ref | Expression |
---|---|
tgoldbachgtda.o | ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} |
tgoldbachgtda.n | ⊢ (𝜑 → 𝑁 ∈ 𝑂) |
tgoldbachgtda.0 | ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) |
Ref | Expression |
---|---|
tgoldbachgnn | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgoldbachgtda.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑂) | |
2 | tgoldbachgtda.o | . . . 4 ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} | |
3 | 1, 2 | eleqtrdi 2900 | . . 3 ⊢ (𝜑 → 𝑁 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}) |
4 | elrabi 3623 | . . 3 ⊢ (𝑁 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} → 𝑁 ∈ ℤ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
6 | 1red 10631 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
7 | 10nn0 12104 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
8 | 7 | nn0rei 11896 | . . . . 5 ⊢ ;10 ∈ ℝ |
9 | 2nn0 11902 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 7nn0 11907 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
11 | 9, 10 | deccl 12101 | . . . . 5 ⊢ ;27 ∈ ℕ0 |
12 | reexpcl 13442 | . . . . 5 ⊢ ((;10 ∈ ℝ ∧ ;27 ∈ ℕ0) → (;10↑;27) ∈ ℝ) | |
13 | 8, 11, 12 | mp2an 691 | . . . 4 ⊢ (;10↑;27) ∈ ℝ |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → (;10↑;27) ∈ ℝ) |
15 | 5 | zred 12075 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
16 | 1re 10630 | . . . . . 6 ⊢ 1 ∈ ℝ | |
17 | 1lt10 12225 | . . . . . 6 ⊢ 1 < ;10 | |
18 | 16, 8, 17 | ltleii 10752 | . . . . 5 ⊢ 1 ≤ ;10 |
19 | expge1 13462 | . . . . 5 ⊢ ((;10 ∈ ℝ ∧ ;27 ∈ ℕ0 ∧ 1 ≤ ;10) → 1 ≤ (;10↑;27)) | |
20 | 8, 11, 18, 19 | mp3an 1458 | . . . 4 ⊢ 1 ≤ (;10↑;27) |
21 | 20 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ≤ (;10↑;27)) |
22 | tgoldbachgtda.0 | . . 3 ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) | |
23 | 6, 14, 15, 21, 22 | letrd 10786 | . 2 ⊢ (𝜑 → 1 ≤ 𝑁) |
24 | elnnz1 11996 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) | |
25 | 5, 23, 24 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 ≤ cle 10665 ℕcn 11625 2c2 11680 7c7 11685 ℕ0cn0 11885 ℤcz 11969 ;cdc 12086 ↑cexp 13425 ∥ cdvds 15599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-seq 13365 df-exp 13426 |
This theorem is referenced by: tgoldbachgtde 32041 tgoldbachgtda 32042 |
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