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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 12253 | . 2 ⊢ 8 ∈ ℕ | |
2 | 4z 12542 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 9nn 12256 | . . . . . 6 ⊢ 9 ∈ ℕ | |
4 | 3 | nnzi 12532 | . . . . 5 ⊢ 9 ∈ ℤ |
5 | 4re 12242 | . . . . . 6 ⊢ 4 ∈ ℝ | |
6 | 9re 12257 | . . . . . 6 ⊢ 9 ∈ ℝ | |
7 | 4lt9 12361 | . . . . . 6 ⊢ 4 < 9 | |
8 | 5, 6, 7 | ltleii 11283 | . . . . 5 ⊢ 4 ≤ 9 |
9 | eluz2 12774 | . . . . 5 ⊢ (9 ∈ (ℤ≥‘4) ↔ (4 ∈ ℤ ∧ 9 ∈ ℤ ∧ 4 ≤ 9)) | |
10 | 2, 4, 8, 9 | mpbir3an 1342 | . . . 4 ⊢ 9 ∈ (ℤ≥‘4) |
11 | 2z 12540 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
12 | 3z 12541 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
13 | 2re 12232 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
14 | 3re 12238 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
15 | 2lt3 12330 | . . . . . . . 8 ⊢ 2 < 3 | |
16 | 13, 14, 15 | ltleii 11283 | . . . . . . 7 ⊢ 2 ≤ 3 |
17 | eluz2 12774 | . . . . . . 7 ⊢ (3 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3)) | |
18 | 11, 12, 16, 17 | mpbir3an 1342 | . . . . . 6 ⊢ 3 ∈ (ℤ≥‘2) |
19 | nprm 16569 | . . . . . 6 ⊢ ((3 ∈ (ℤ≥‘2) ∧ 3 ∈ (ℤ≥‘2)) → ¬ (3 · 3) ∈ ℙ) | |
20 | 18, 18, 19 | mp2an 691 | . . . . 5 ⊢ ¬ (3 · 3) ∈ ℙ |
21 | df-nel 3047 | . . . . . 6 ⊢ (9 ∉ ℙ ↔ ¬ 9 ∈ ℙ) | |
22 | 3t3e9 12325 | . . . . . . . 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 12292 | . . . . . . 7 ⊢ (9 − 1) = 8 | |
28 | 27 | oveq2i 7369 | . . . . . 6 ⊢ (8↑(9 − 1)) = (8↑8) |
29 | 28 | oveq1i 7368 | . . . . 5 ⊢ ((8↑(9 − 1)) mod 9) = ((8↑8) mod 9) |
30 | 8exp8mod9 46014 | . . . . 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 46006 | . . 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 3046 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 1c1 11057 · cmul 11061 ≤ cle 11195 − cmin 11390 ℕcn 12158 2c2 12213 3c3 12214 4c4 12215 8c8 12219 9c9 12220 ℤcz 12504 ℤ≥cuz 12768 mod cmo 13780 ↑cexp 13973 ℙcprime 16552 FPPr cfppr 46002 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-rp 12921 df-fl 13703 df-mod 13781 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-dvds 16142 df-prm 16553 df-fppr 46003 |
This theorem is referenced by: (None) |
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