Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 4fppr1 | Structured version Visualization version GIF version |
Description: 4 is the (smallest) Fermat pseudoprime to the base 1. (Contributed by AV, 3-Jun-2023.) |
Ref | Expression |
---|---|
4fppr1 | ⊢ 4 ∈ ( FPPr ‘1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4z 12455 | . . 3 ⊢ 4 ∈ ℤ | |
2 | uzid 12698 | . . 3 ⊢ (4 ∈ ℤ → 4 ∈ (ℤ≥‘4)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 4 ∈ (ℤ≥‘4) |
4 | 4nprm 16497 | . . 3 ⊢ ¬ 4 ∈ ℙ | |
5 | 4 | nelir 3049 | . 2 ⊢ 4 ∉ ℙ |
6 | 4m1e3 12203 | . . . . . 6 ⊢ (4 − 1) = 3 | |
7 | 6 | oveq2i 7348 | . . . . 5 ⊢ (1↑(4 − 1)) = (1↑3) |
8 | 3z 12454 | . . . . . 6 ⊢ 3 ∈ ℤ | |
9 | 1exp 13913 | . . . . . 6 ⊢ (3 ∈ ℤ → (1↑3) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (1↑3) = 1 |
11 | 7, 10 | eqtri 2764 | . . . 4 ⊢ (1↑(4 − 1)) = 1 |
12 | 11 | oveq1i 7347 | . . 3 ⊢ ((1↑(4 − 1)) mod 4) = (1 mod 4) |
13 | 4re 12158 | . . . 4 ⊢ 4 ∈ ℝ | |
14 | 1lt4 12250 | . . . 4 ⊢ 1 < 4 | |
15 | 1mod 13724 | . . . 4 ⊢ ((4 ∈ ℝ ∧ 1 < 4) → (1 mod 4) = 1) | |
16 | 13, 14, 15 | mp2an 689 | . . 3 ⊢ (1 mod 4) = 1 |
17 | 12, 16 | eqtri 2764 | . 2 ⊢ ((1↑(4 − 1)) mod 4) = 1 |
18 | 1nn 12085 | . . 3 ⊢ 1 ∈ ℕ | |
19 | fpprel 45539 | . . 3 ⊢ (1 ∈ ℕ → (4 ∈ ( FPPr ‘1) ↔ (4 ∈ (ℤ≥‘4) ∧ 4 ∉ ℙ ∧ ((1↑(4 − 1)) mod 4) = 1))) | |
20 | 18, 19 | ax-mp 5 | . 2 ⊢ (4 ∈ ( FPPr ‘1) ↔ (4 ∈ (ℤ≥‘4) ∧ 4 ∉ ℙ ∧ ((1↑(4 − 1)) mod 4) = 1)) |
21 | 3, 5, 17, 20 | mpbir3an 1340 | 1 ⊢ 4 ∈ ( FPPr ‘1) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∉ wnel 3046 class class class wbr 5092 ‘cfv 6479 (class class class)co 7337 ℝcr 10971 1c1 10973 < clt 11110 − cmin 11306 ℕcn 12074 3c3 12130 4c4 12131 ℤcz 12420 ℤ≥cuz 12683 mod cmo 13690 ↑cexp 13883 ℙcprime 16473 FPPr cfppr 45535 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-2o 8368 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-sup 9299 df-inf 9300 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-n0 12335 df-z 12421 df-uz 12684 df-rp 12832 df-fl 13613 df-mod 13691 df-seq 13823 df-exp 13884 df-cj 14909 df-re 14910 df-im 14911 df-sqrt 15045 df-abs 15046 df-dvds 16063 df-prm 16474 df-fppr 45536 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |