fmtno5 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5		Structured version Visualization version GIF version

Theorem fmtno5 48049

Description: The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.)

**Assertion**
Ref	Expression
fmtno5	⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

**Proof of Theorem fmtno5**
Step	Hyp	Ref	Expression
1		df-5 12242	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6834	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12451	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 48029	. . . 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 2764	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12449	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12654	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12456	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12654	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12654	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12654	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12453	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12654	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12454	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12654	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12654	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12654	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12319	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12452	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12654	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12654	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12450	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12654	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12448	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 48044	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12316	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2741	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12670	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12267	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11091	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12244	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11405	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12702	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7370	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 48048	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2764	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12670	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2764	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1548 ∈ wcel 2121 ‘cfv 6489 (class class class)co 7360 1c1 11034 + caddc 11036 − cmin 11372 2c2 12231 3c3 12232 4c4 12233 5c5 12234 6c6 12235 7c7 12236 9c9 12238 ℕ₀cn0 12432 cdc 12639 ↑cexp 14018 FermatNocfmtno 48019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-seq 13959 df-exp 14019 df-fmtno 48020

This theorem is referenced by: fmtno5fac 48074

W3C validator