fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 47431

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 12359	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6923	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12572	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 47411	. . . 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 2768	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12570	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12773	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12577	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12773	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12773	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12773	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12574	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12773	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12575	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12773	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12773	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12773	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12441	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12573	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12773	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12773	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12571	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12773	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12569	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 47426	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12438	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2740	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12789	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12384	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11242	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12361	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11553	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12821	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7458	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 47430	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2768	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12789	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2768	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2108 ‘cfv 6573 (class class class)co 7448 1c1 11185 + caddc 11187 − cmin 11520 2c2 12348 3c3 12349 4c4 12350 5c5 12351 6c6 12352 7c7 12353 9c9 12355 ℕ₀cn0 12553 cdc 12758 ↑cexp 14112 FermatNocfmtno 47401

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-seq 14053 df-exp 14113 df-fmtno 47402

This theorem is referenced by: fmtno5fac 47456

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