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 48032

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 12238	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6837	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12447	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 48012	. . . 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 2760	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12445	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12650	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12452	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12650	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12650	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12650	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12449	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12650	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12450	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12650	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12650	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12650	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12315	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12448	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12650	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12650	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12446	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12650	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12444	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 48027	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12312	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2737	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12666	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12263	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11087	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12240	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11401	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12698	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7370	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 48031	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2760	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12666	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2760	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 ‘cfv 6492 (class class class)co 7360 1c1 11030 + caddc 11032 − cmin 11368 2c2 12227 3c3 12228 4c4 12229 5c5 12230 6c6 12231 7c7 12232 9c9 12234 ℕ₀cn0 12428 cdc 12635 ↑cexp 14014 FermatNocfmtno 48002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-seq 13955 df-exp 14015 df-fmtno 48003

This theorem is referenced by: fmtno5fac 48057

W3C validator