![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5 | Structured version Visualization version GIF version |
Description: The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5 | ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 12226 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | fveq2i 6850 | . . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1)) |
3 | 4nn0 12439 | . . . 4 ⊢ 4 ∈ ℕ0 | |
4 | fmtnorec1 45803 | . . . 4 ⊢ (4 ∈ ℕ0 → (FermatNo‘(4 + 1)) = ((((FermatNo‘4) − 1)↑2) + 1)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (FermatNo‘(4 + 1)) = ((((FermatNo‘4) − 1)↑2) + 1) |
6 | 2, 5 | eqtri 2765 | . 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1) |
7 | 2nn0 12437 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ0 | |
8 | 3, 7 | deccl 12640 | . . . . . . . . . 10 ⊢ ;42 ∈ ℕ0 |
9 | 9nn0 12444 | . . . . . . . . . 10 ⊢ 9 ∈ ℕ0 | |
10 | 8, 9 | deccl 12640 | . . . . . . . . 9 ⊢ ;;429 ∈ ℕ0 |
11 | 10, 3 | deccl 12640 | . . . . . . . 8 ⊢ ;;;4294 ∈ ℕ0 |
12 | 11, 9 | deccl 12640 | . . . . . . 7 ⊢ ;;;;42949 ∈ ℕ0 |
13 | 6nn0 12441 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
14 | 12, 13 | deccl 12640 | . . . . . 6 ⊢ ;;;;;429496 ∈ ℕ0 |
15 | 7nn0 12442 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
16 | 14, 15 | deccl 12640 | . . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ0 |
17 | 16, 7 | deccl 12640 | . . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ0 |
18 | 17, 9 | deccl 12640 | . . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ0 |
19 | 6p1e7 12308 | . . 3 ⊢ (6 + 1) = 7 | |
20 | 5nn0 12440 | . . . . . . . . 9 ⊢ 5 ∈ ℕ0 | |
21 | 13, 20 | deccl 12640 | . . . . . . . 8 ⊢ ;65 ∈ ℕ0 |
22 | 21, 20 | deccl 12640 | . . . . . . 7 ⊢ ;;655 ∈ ℕ0 |
23 | 3nn0 12438 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
24 | 22, 23 | deccl 12640 | . . . . . 6 ⊢ ;;;6553 ∈ ℕ0 |
25 | 1nn0 12436 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
26 | fmtno4 45818 | . . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537 | |
27 | 3p1e4 12305 | . . . . . . 7 ⊢ (3 + 1) = 4 | |
28 | eqid 2737 | . . . . . . 7 ⊢ ;;;6553 = ;;;6553 | |
29 | 22, 23, 27, 28 | decsuc 12656 | . . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554 |
30 | 6cn 12251 | . . . . . . 7 ⊢ 6 ∈ ℂ | |
31 | ax-1cn 11116 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
32 | df-7 12228 | . . . . . . 7 ⊢ 7 = (6 + 1) | |
33 | 30, 31, 32 | mvrraddi 11425 | . . . . . 6 ⊢ (7 − 1) = 6 |
34 | 24, 15, 25, 26, 29, 33 | decsubi 12688 | . . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536 |
35 | 34 | oveq1i 7372 | . . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2) |
36 | fmtno5lem4 45822 | . . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296 | |
37 | 35, 36 | eqtri 2765 | . . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296 |
38 | 18, 13, 19, 37 | decsuc 12656 | . 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297 |
39 | 6, 38 | eqtri 2765 | 1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ‘cfv 6501 (class class class)co 7362 1c1 11059 + caddc 11061 − cmin 11392 2c2 12215 3c3 12216 4c4 12217 5c5 12218 6c6 12219 7c7 12220 9c9 12222 ℕ0cn0 12420 ;cdc 12625 ↑cexp 13974 FermatNocfmtno 45793 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-seq 13914 df-exp 13975 df-fmtno 45794 |
This theorem is referenced by: fmtno5fac 45848 |
Copyright terms: Public domain | W3C validator |