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Mathbox for Alexander van der Vekens |
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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 12309 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | fveq2i 6900 | . . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1)) |
3 | 4nn0 12522 | . . . 4 ⊢ 4 ∈ ℕ0 | |
4 | fmtnorec1 46877 | . . . 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 2756 | . 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1) |
7 | 2nn0 12520 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ0 | |
8 | 3, 7 | deccl 12723 | . . . . . . . . . 10 ⊢ ;42 ∈ ℕ0 |
9 | 9nn0 12527 | . . . . . . . . . 10 ⊢ 9 ∈ ℕ0 | |
10 | 8, 9 | deccl 12723 | . . . . . . . . 9 ⊢ ;;429 ∈ ℕ0 |
11 | 10, 3 | deccl 12723 | . . . . . . . 8 ⊢ ;;;4294 ∈ ℕ0 |
12 | 11, 9 | deccl 12723 | . . . . . . 7 ⊢ ;;;;42949 ∈ ℕ0 |
13 | 6nn0 12524 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
14 | 12, 13 | deccl 12723 | . . . . . 6 ⊢ ;;;;;429496 ∈ ℕ0 |
15 | 7nn0 12525 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
16 | 14, 15 | deccl 12723 | . . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ0 |
17 | 16, 7 | deccl 12723 | . . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ0 |
18 | 17, 9 | deccl 12723 | . . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ0 |
19 | 6p1e7 12391 | . . 3 ⊢ (6 + 1) = 7 | |
20 | 5nn0 12523 | . . . . . . . . 9 ⊢ 5 ∈ ℕ0 | |
21 | 13, 20 | deccl 12723 | . . . . . . . 8 ⊢ ;65 ∈ ℕ0 |
22 | 21, 20 | deccl 12723 | . . . . . . 7 ⊢ ;;655 ∈ ℕ0 |
23 | 3nn0 12521 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
24 | 22, 23 | deccl 12723 | . . . . . 6 ⊢ ;;;6553 ∈ ℕ0 |
25 | 1nn0 12519 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
26 | fmtno4 46892 | . . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537 | |
27 | 3p1e4 12388 | . . . . . . 7 ⊢ (3 + 1) = 4 | |
28 | eqid 2728 | . . . . . . 7 ⊢ ;;;6553 = ;;;6553 | |
29 | 22, 23, 27, 28 | decsuc 12739 | . . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554 |
30 | 6cn 12334 | . . . . . . 7 ⊢ 6 ∈ ℂ | |
31 | ax-1cn 11197 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
32 | df-7 12311 | . . . . . . 7 ⊢ 7 = (6 + 1) | |
33 | 30, 31, 32 | mvrraddi 11508 | . . . . . 6 ⊢ (7 − 1) = 6 |
34 | 24, 15, 25, 26, 29, 33 | decsubi 12771 | . . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536 |
35 | 34 | oveq1i 7430 | . . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2) |
36 | fmtno5lem4 46896 | . . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296 | |
37 | 35, 36 | eqtri 2756 | . . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296 |
38 | 18, 13, 19, 37 | decsuc 12739 | . 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297 |
39 | 6, 38 | eqtri 2756 | 1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 1c1 11140 + caddc 11142 − cmin 11475 2c2 12298 3c3 12299 4c4 12300 5c5 12301 6c6 12302 7c7 12303 9c9 12305 ℕ0cn0 12503 ;cdc 12708 ↑cexp 14059 FermatNocfmtno 46867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-seq 14000 df-exp 14060 df-fmtno 46868 |
This theorem is referenced by: fmtno5fac 46922 |
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