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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbachgnn | Structured version Visualization version GIF version |
Description: Lemma for tgoldbachgtd 34655. (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 2848 | . . 3 ⊢ (𝜑 → 𝑁 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}) |
4 | elrabi 3689 | . . 3 ⊢ (𝑁 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} → 𝑁 ∈ ℤ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
6 | 1red 11259 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
7 | 10nn0 12748 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
8 | 7 | nn0rei 12534 | . . . . 5 ⊢ ;10 ∈ ℝ |
9 | 2nn0 12540 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 7nn0 12545 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
11 | 9, 10 | deccl 12745 | . . . . 5 ⊢ ;27 ∈ ℕ0 |
12 | reexpcl 14115 | . . . . 5 ⊢ ((;10 ∈ ℝ ∧ ;27 ∈ ℕ0) → (;10↑;27) ∈ ℝ) | |
13 | 8, 11, 12 | mp2an 692 | . . . 4 ⊢ (;10↑;27) ∈ ℝ |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → (;10↑;27) ∈ ℝ) |
15 | 5 | zred 12719 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
16 | 1re 11258 | . . . . . 6 ⊢ 1 ∈ ℝ | |
17 | 1lt10 12869 | . . . . . 6 ⊢ 1 < ;10 | |
18 | 16, 8, 17 | ltleii 11381 | . . . . 5 ⊢ 1 ≤ ;10 |
19 | expge1 14136 | . . . . 5 ⊢ ((;10 ∈ ℝ ∧ ;27 ∈ ℕ0 ∧ 1 ≤ ;10) → 1 ≤ (;10↑;27)) | |
20 | 8, 11, 18, 19 | mp3an 1460 | . . . 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 11415 | . 2 ⊢ (𝜑 → 1 ≤ 𝑁) |
24 | elnnz1 12640 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) | |
25 | 5, 23, 24 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2105 {crab 3432 class class class wbr 5147 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 ≤ cle 11293 ℕcn 12263 2c2 12318 7c7 12323 ℕ0cn0 12523 ℤcz 12610 ;cdc 12730 ↑cexp 14098 ∥ cdvds 16286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-seq 14039 df-exp 14099 |
This theorem is referenced by: tgoldbachgtde 34653 tgoldbachgtda 34654 |
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