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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbachgnn | Structured version Visualization version GIF version |
Description: Lemma for tgoldbachgtd 30740. (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 | syl6eleq 2711 | . . 3 ⊢ (𝜑 → 𝑁 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}) |
4 | elrabi 3359 | . . 3 ⊢ (𝑁 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} → 𝑁 ∈ ℤ) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
6 | 1red 10055 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
7 | 10nn0 11516 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
8 | 7 | nn0rei 11303 | . . . . 5 ⊢ ;10 ∈ ℝ |
9 | 2nn0 11309 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 7nn0 11314 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
11 | 9, 10 | deccl 11512 | . . . . 5 ⊢ ;27 ∈ ℕ0 |
12 | reexpcl 12877 | . . . . 5 ⊢ ((;10 ∈ ℝ ∧ ;27 ∈ ℕ0) → (;10↑;27) ∈ ℝ) | |
13 | 8, 11, 12 | mp2an 708 | . . . 4 ⊢ (;10↑;27) ∈ ℝ |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → (;10↑;27) ∈ ℝ) |
15 | 5 | zred 11482 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
16 | 1re 10039 | . . . . . 6 ⊢ 1 ∈ ℝ | |
17 | 1lt10 11681 | . . . . . 6 ⊢ 1 < ;10 | |
18 | 16, 8, 17 | ltleii 10160 | . . . . 5 ⊢ 1 ≤ ;10 |
19 | expge1 12897 | . . . . 5 ⊢ ((;10 ∈ ℝ ∧ ;27 ∈ ℕ0 ∧ 1 ≤ ;10) → 1 ≤ (;10↑;27)) | |
20 | 8, 11, 18, 19 | mp3an 1424 | . . . 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 10194 | . 2 ⊢ (𝜑 → 1 ≤ 𝑁) |
24 | elnnz1 11403 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) | |
25 | 5, 23, 24 | sylanbrc 698 | 1 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 {crab 2916 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 ≤ cle 10075 ℕcn 11020 2c2 11070 7c7 11075 ℕ0cn0 11292 ℤcz 11377 ;cdc 11493 ↑cexp 12860 ∥ cdvds 14983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: tgoldbachgtde 30738 tgoldbachgtda 30739 |
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