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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 12307 | . 2 ⊢ 8 ∈ ℕ | |
2 | 4z 12596 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 9nn 12310 | . . . . . 6 ⊢ 9 ∈ ℕ | |
4 | 3 | nnzi 12586 | . . . . 5 ⊢ 9 ∈ ℤ |
5 | 4re 12296 | . . . . . 6 ⊢ 4 ∈ ℝ | |
6 | 9re 12311 | . . . . . 6 ⊢ 9 ∈ ℝ | |
7 | 4lt9 12415 | . . . . . 6 ⊢ 4 < 9 | |
8 | 5, 6, 7 | ltleii 11337 | . . . . 5 ⊢ 4 ≤ 9 |
9 | eluz2 12828 | . . . . 5 ⊢ (9 ∈ (ℤ≥‘4) ↔ (4 ∈ ℤ ∧ 9 ∈ ℤ ∧ 4 ≤ 9)) | |
10 | 2, 4, 8, 9 | mpbir3an 1342 | . . . 4 ⊢ 9 ∈ (ℤ≥‘4) |
11 | 2z 12594 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
12 | 3z 12595 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
13 | 2re 12286 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
14 | 3re 12292 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
15 | 2lt3 12384 | . . . . . . . 8 ⊢ 2 < 3 | |
16 | 13, 14, 15 | ltleii 11337 | . . . . . . 7 ⊢ 2 ≤ 3 |
17 | eluz2 12828 | . . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3)) | |
18 | 11, 12, 16, 17 | mpbir3an 1342 | . . . . . 6 ⊢ 3 ∈ (ℤ≥‘2) |
19 | nprm 16625 | . . . . . 6 ⊢ ((3 ∈ (ℤ≥‘2) ∧ 3 ∈ (ℤ≥‘2)) → ¬ (3 · 3) ∈ ℙ) | |
20 | 18, 18, 19 | mp2an 691 | . . . . 5 ⊢ ¬ (3 · 3) ∈ ℙ |
21 | df-nel 3048 | . . . . . 6 ⊢ (9 ∉ ℙ ↔ ¬ 9 ∈ ℙ) | |
22 | 3t3e9 12379 | . . . . . . . 8 ⊢ (3 · 3) = 9 | |
23 | 22 | eqcomi 2742 | . . . . . . 7 ⊢ 9 = (3 · 3) |
24 | 23 | eleq1i 2825 | . . . . . 6 ⊢ (9 ∈ ℙ ↔ (3 · 3) ∈ ℙ) |
25 | 21, 24 | xchbinx 334 | . . . . 5 ⊢ (9 ∉ ℙ ↔ ¬ (3 · 3) ∈ ℙ) |
26 | 20, 25 | mpbir 230 | . . . 4 ⊢ 9 ∉ ℙ |
27 | 9m1e8 12346 | . . . . . . 7 ⊢ (9 − 1) = 8 | |
28 | 27 | oveq2i 7420 | . . . . . 6 ⊢ (8↑(9 − 1)) = (8↑8) |
29 | 28 | oveq1i 7419 | . . . . 5 ⊢ ((8↑(9 − 1)) mod 9) = ((8↑8) mod 9) |
30 | 8exp8mod9 46404 | . . . . 5 ⊢ ((8↑8) mod 9) = 1 | |
31 | 29, 30 | eqtri 2761 | . . . 4 ⊢ ((8↑(9 − 1)) mod 9) = 1 |
32 | 10, 26, 31 | 3pm3.2i 1340 | . . 3 ⊢ (9 ∈ (ℤ≥‘4) ∧ 9 ∉ ℙ ∧ ((8↑(9 − 1)) mod 9) = 1) |
33 | fpprel 46396 | . . 3 ⊢ (8 ∈ ℕ → (9 ∈ ( FPPr ‘8) ↔ (9 ∈ (ℤ≥‘4) ∧ 9 ∉ ℙ ∧ ((8↑(9 − 1)) mod 9) = 1))) | |
34 | 32, 33 | mpbiri 258 | . 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 1088 = wceq 1542 ∈ wcel 2107 ∉ wnel 3047 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 1c1 11111 · cmul 11115 ≤ cle 11249 − cmin 11444 ℕcn 12212 2c2 12267 3c3 12268 4c4 12269 8c8 12273 9c9 12274 ℤcz 12558 ℤ≥cuz 12822 mod cmo 13834 ↑cexp 14027 ℙcprime 16608 FPPr cfppr 46392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-prm 16609 df-fppr 46393 |
This theorem is referenced by: (None) |
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