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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbachgnn | Structured version Visualization version GIF version |
Description: Lemma for tgoldbachgtd 34347. (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 2835 | . . 3 ⊢ (𝜑 → 𝑁 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}) |
4 | elrabi 3670 | . . 3 ⊢ (𝑁 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} → 𝑁 ∈ ℤ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
6 | 1red 11240 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
7 | 10nn0 12720 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
8 | 7 | nn0rei 12508 | . . . . 5 ⊢ ;10 ∈ ℝ |
9 | 2nn0 12514 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 7nn0 12519 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
11 | 9, 10 | deccl 12717 | . . . . 5 ⊢ ;27 ∈ ℕ0 |
12 | reexpcl 14070 | . . . . 5 ⊢ ((;10 ∈ ℝ ∧ ;27 ∈ ℕ0) → (;10↑;27) ∈ ℝ) | |
13 | 8, 11, 12 | mp2an 690 | . . . 4 ⊢ (;10↑;27) ∈ ℝ |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → (;10↑;27) ∈ ℝ) |
15 | 5 | zred 12691 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
16 | 1re 11239 | . . . . . 6 ⊢ 1 ∈ ℝ | |
17 | 1lt10 12841 | . . . . . 6 ⊢ 1 < ;10 | |
18 | 16, 8, 17 | ltleii 11362 | . . . . 5 ⊢ 1 ≤ ;10 |
19 | expge1 14091 | . . . . 5 ⊢ ((;10 ∈ ℝ ∧ ;27 ∈ ℕ0 ∧ 1 ≤ ;10) → 1 ≤ (;10↑;27)) | |
20 | 8, 11, 18, 19 | mp3an 1457 | . . . 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 11396 | . 2 ⊢ (𝜑 → 1 ≤ 𝑁) |
24 | elnnz1 12613 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) | |
25 | 5, 23, 24 | sylanbrc 581 | 1 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3419 class class class wbr 5144 (class class class)co 7413 ℝcr 11132 0cc0 11133 1c1 11134 ≤ cle 11274 ℕcn 12237 2c2 12292 7c7 12297 ℕ0cn0 12497 ℤcz 12583 ;cdc 12702 ↑cexp 14053 ∥ cdvds 16225 |
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 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-seq 13994 df-exp 14054 |
This theorem is referenced by: tgoldbachgtde 34345 tgoldbachgtda 34346 |
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