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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 11733 | . 2 ⊢ 8 ∈ ℕ | |
2 | 4z 12017 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 9nn 11736 | . . . . . 6 ⊢ 9 ∈ ℕ | |
4 | 3 | nnzi 12007 | . . . . 5 ⊢ 9 ∈ ℤ |
5 | 4re 11722 | . . . . . 6 ⊢ 4 ∈ ℝ | |
6 | 9re 11737 | . . . . . 6 ⊢ 9 ∈ ℝ | |
7 | 4lt9 11841 | . . . . . 6 ⊢ 4 < 9 | |
8 | 5, 6, 7 | ltleii 10763 | . . . . 5 ⊢ 4 ≤ 9 |
9 | eluz2 12250 | . . . . 5 ⊢ (9 ∈ (ℤ≥‘4) ↔ (4 ∈ ℤ ∧ 9 ∈ ℤ ∧ 4 ≤ 9)) | |
10 | 2, 4, 8, 9 | mpbir3an 1337 | . . . 4 ⊢ 9 ∈ (ℤ≥‘4) |
11 | 2z 12015 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
12 | 3z 12016 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
13 | 2re 11712 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
14 | 3re 11718 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
15 | 2lt3 11810 | . . . . . . . 8 ⊢ 2 < 3 | |
16 | 13, 14, 15 | ltleii 10763 | . . . . . . 7 ⊢ 2 ≤ 3 |
17 | eluz2 12250 | . . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3)) | |
18 | 11, 12, 16, 17 | mpbir3an 1337 | . . . . . 6 ⊢ 3 ∈ (ℤ≥‘2) |
19 | nprm 16032 | . . . . . 6 ⊢ ((3 ∈ (ℤ≥‘2) ∧ 3 ∈ (ℤ≥‘2)) → ¬ (3 · 3) ∈ ℙ) | |
20 | 18, 18, 19 | mp2an 690 | . . . . 5 ⊢ ¬ (3 · 3) ∈ ℙ |
21 | df-nel 3124 | . . . . . 6 ⊢ (9 ∉ ℙ ↔ ¬ 9 ∈ ℙ) | |
22 | 3t3e9 11805 | . . . . . . . 8 ⊢ (3 · 3) = 9 | |
23 | 22 | eqcomi 2830 | . . . . . . 7 ⊢ 9 = (3 · 3) |
24 | 23 | eleq1i 2903 | . . . . . 6 ⊢ (9 ∈ ℙ ↔ (3 · 3) ∈ ℙ) |
25 | 21, 24 | xchbinx 336 | . . . . 5 ⊢ (9 ∉ ℙ ↔ ¬ (3 · 3) ∈ ℙ) |
26 | 20, 25 | mpbir 233 | . . . 4 ⊢ 9 ∉ ℙ |
27 | 9m1e8 11772 | . . . . . . 7 ⊢ (9 − 1) = 8 | |
28 | 27 | oveq2i 7167 | . . . . . 6 ⊢ (8↑(9 − 1)) = (8↑8) |
29 | 28 | oveq1i 7166 | . . . . 5 ⊢ ((8↑(9 − 1)) mod 9) = ((8↑8) mod 9) |
30 | 8exp8mod9 43950 | . . . . 5 ⊢ ((8↑8) mod 9) = 1 | |
31 | 29, 30 | eqtri 2844 | . . . 4 ⊢ ((8↑(9 − 1)) mod 9) = 1 |
32 | 10, 26, 31 | 3pm3.2i 1335 | . . 3 ⊢ (9 ∈ (ℤ≥‘4) ∧ 9 ∉ ℙ ∧ ((8↑(9 − 1)) mod 9) = 1) |
33 | fpprel 43942 | . . 3 ⊢ (8 ∈ ℕ → (9 ∈ ( FPPr ‘8) ↔ (9 ∈ (ℤ≥‘4) ∧ 9 ∉ ℙ ∧ ((8↑(9 − 1)) mod 9) = 1))) | |
34 | 32, 33 | mpbiri 260 | . 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 1083 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 1c1 10538 · cmul 10542 ≤ cle 10676 − cmin 10870 ℕcn 11638 2c2 11693 3c3 11694 4c4 11695 8c8 11699 9c9 11700 ℤcz 11982 ℤ≥cuz 12244 mod cmo 13238 ↑cexp 13430 ℙcprime 16015 FPPr cfppr 43938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-prm 16016 df-fppr 43939 |
This theorem is referenced by: (None) |
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