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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 11445 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | fveq2i 6451 | . . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1)) |
3 | 4nn0 11667 | . . . 4 ⊢ 4 ∈ ℕ0 | |
4 | fmtnorec1 42480 | . . . 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 2802 | . 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1) |
7 | 2nn0 11665 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ0 | |
8 | 3, 7 | deccl 11864 | . . . . . . . . . 10 ⊢ ;42 ∈ ℕ0 |
9 | 9nn0 11672 | . . . . . . . . . 10 ⊢ 9 ∈ ℕ0 | |
10 | 8, 9 | deccl 11864 | . . . . . . . . 9 ⊢ ;;429 ∈ ℕ0 |
11 | 10, 3 | deccl 11864 | . . . . . . . 8 ⊢ ;;;4294 ∈ ℕ0 |
12 | 11, 9 | deccl 11864 | . . . . . . 7 ⊢ ;;;;42949 ∈ ℕ0 |
13 | 6nn0 11669 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
14 | 12, 13 | deccl 11864 | . . . . . 6 ⊢ ;;;;;429496 ∈ ℕ0 |
15 | 7nn0 11670 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
16 | 14, 15 | deccl 11864 | . . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ0 |
17 | 16, 7 | deccl 11864 | . . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ0 |
18 | 17, 9 | deccl 11864 | . . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ0 |
19 | 6p1e7 11534 | . . 3 ⊢ (6 + 1) = 7 | |
20 | 5nn0 11668 | . . . . . . . . 9 ⊢ 5 ∈ ℕ0 | |
21 | 13, 20 | deccl 11864 | . . . . . . . 8 ⊢ ;65 ∈ ℕ0 |
22 | 21, 20 | deccl 11864 | . . . . . . 7 ⊢ ;;655 ∈ ℕ0 |
23 | 3nn0 11666 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
24 | 22, 23 | deccl 11864 | . . . . . 6 ⊢ ;;;6553 ∈ ℕ0 |
25 | 1nn0 11664 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
26 | fmtno4 42495 | . . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537 | |
27 | 3p1e4 11531 | . . . . . . 7 ⊢ (3 + 1) = 4 | |
28 | eqid 2778 | . . . . . . 7 ⊢ ;;;6553 = ;;;6553 | |
29 | 22, 23, 27, 28 | decsuc 11881 | . . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554 |
30 | 6cn 11473 | . . . . . . 7 ⊢ 6 ∈ ℂ | |
31 | ax-1cn 10332 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
32 | df-7 11447 | . . . . . . 7 ⊢ 7 = (6 + 1) | |
33 | 30, 31, 32 | mvrraddi 10642 | . . . . . 6 ⊢ (7 − 1) = 6 |
34 | 24, 15, 25, 26, 29, 33 | decsubi 11913 | . . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536 |
35 | 34 | oveq1i 6934 | . . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2) |
36 | fmtno5lem4 42499 | . . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296 | |
37 | 35, 36 | eqtri 2802 | . . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296 |
38 | 18, 13, 19, 37 | decsuc 11881 | . 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297 |
39 | 6, 38 | eqtri 2802 | 1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 1c1 10275 + caddc 10277 − cmin 10608 2c2 11434 3c3 11435 4c4 11436 5c5 11437 6c6 11438 7c7 11439 9c9 11441 ℕ0cn0 11646 ;cdc 11849 ↑cexp 13182 FermatNocfmtno 42470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-seq 13124 df-exp 13183 df-fmtno 42471 |
This theorem is referenced by: fmtno5fac 42525 |
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