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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 9fppr8 | Structured version Visualization version GIF version |
Description: 9 is the (smallest) Fermat pseudoprime to the base 8. (Contributed by AV, 2-Jun-2023.) |
Ref | Expression |
---|---|
9fppr8 | ⊢ 9 ∈ ( FPPr ‘8) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8nn 12289 | . 2 ⊢ 8 ∈ ℕ | |
2 | 4z 12578 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 9nn 12292 | . . . . . 6 ⊢ 9 ∈ ℕ | |
4 | 3 | nnzi 12568 | . . . . 5 ⊢ 9 ∈ ℤ |
5 | 4re 12278 | . . . . . 6 ⊢ 4 ∈ ℝ | |
6 | 9re 12293 | . . . . . 6 ⊢ 9 ∈ ℝ | |
7 | 4lt9 12397 | . . . . . 6 ⊢ 4 < 9 | |
8 | 5, 6, 7 | ltleii 11319 | . . . . 5 ⊢ 4 ≤ 9 |
9 | eluz2 12810 | . . . . 5 ⊢ (9 ∈ (ℤ≥‘4) ↔ (4 ∈ ℤ ∧ 9 ∈ ℤ ∧ 4 ≤ 9)) | |
10 | 2, 4, 8, 9 | mpbir3an 1341 | . . . 4 ⊢ 9 ∈ (ℤ≥‘4) |
11 | 2z 12576 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
12 | 3z 12577 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
13 | 2re 12268 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
14 | 3re 12274 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
15 | 2lt3 12366 | . . . . . . . 8 ⊢ 2 < 3 | |
16 | 13, 14, 15 | ltleii 11319 | . . . . . . 7 ⊢ 2 ≤ 3 |
17 | eluz2 12810 | . . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3)) | |
18 | 11, 12, 16, 17 | mpbir3an 1341 | . . . . . 6 ⊢ 3 ∈ (ℤ≥‘2) |
19 | nprm 16607 | . . . . . 6 ⊢ ((3 ∈ (ℤ≥‘2) ∧ 3 ∈ (ℤ≥‘2)) → ¬ (3 · 3) ∈ ℙ) | |
20 | 18, 18, 19 | mp2an 690 | . . . . 5 ⊢ ¬ (3 · 3) ∈ ℙ |
21 | df-nel 3046 | . . . . . 6 ⊢ (9 ∉ ℙ ↔ ¬ 9 ∈ ℙ) | |
22 | 3t3e9 12361 | . . . . . . . 8 ⊢ (3 · 3) = 9 | |
23 | 22 | eqcomi 2740 | . . . . . . 7 ⊢ 9 = (3 · 3) |
24 | 23 | eleq1i 2823 | . . . . . 6 ⊢ (9 ∈ ℙ ↔ (3 · 3) ∈ ℙ) |
25 | 21, 24 | xchbinx 333 | . . . . 5 ⊢ (9 ∉ ℙ ↔ ¬ (3 · 3) ∈ ℙ) |
26 | 20, 25 | mpbir 230 | . . . 4 ⊢ 9 ∉ ℙ |
27 | 9m1e8 12328 | . . . . . . 7 ⊢ (9 − 1) = 8 | |
28 | 27 | oveq2i 7404 | . . . . . 6 ⊢ (8↑(9 − 1)) = (8↑8) |
29 | 28 | oveq1i 7403 | . . . . 5 ⊢ ((8↑(9 − 1)) mod 9) = ((8↑8) mod 9) |
30 | 8exp8mod9 46174 | . . . . 5 ⊢ ((8↑8) mod 9) = 1 | |
31 | 29, 30 | eqtri 2759 | . . . 4 ⊢ ((8↑(9 − 1)) mod 9) = 1 |
32 | 10, 26, 31 | 3pm3.2i 1339 | . . 3 ⊢ (9 ∈ (ℤ≥‘4) ∧ 9 ∉ ℙ ∧ ((8↑(9 − 1)) mod 9) = 1) |
33 | fpprel 46166 | . . 3 ⊢ (8 ∈ ℕ → (9 ∈ ( FPPr ‘8) ↔ (9 ∈ (ℤ≥‘4) ∧ 9 ∉ ℙ ∧ ((8↑(9 − 1)) mod 9) = 1))) | |
34 | 32, 33 | mpbiri 257 | . 2 ⊢ (8 ∈ ℕ → 9 ∈ ( FPPr ‘8)) |
35 | 1, 34 | ax-mp 5 | 1 ⊢ 9 ∈ ( FPPr ‘8) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∉ wnel 3045 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 1c1 11093 · cmul 11097 ≤ cle 11231 − cmin 11426 ℕcn 12194 2c2 12249 3c3 12250 4c4 12251 8c8 12255 9c9 12256 ℤcz 12540 ℤ≥cuz 12804 mod cmo 13816 ↑cexp 14009 ℙcprime 16590 FPPr cfppr 46162 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-rp 12957 df-fl 13739 df-mod 13817 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-dvds 16180 df-prm 16591 df-fppr 46163 |
This theorem is referenced by: (None) |
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