fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 47594

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 12191	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6825	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12400	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 47574	. . . 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 2754	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12398	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12603	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12405	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12603	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12603	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12603	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12402	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12603	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12403	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12603	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12603	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12603	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12268	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12401	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12603	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12603	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12399	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12603	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12397	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 47589	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12265	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2731	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12619	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12216	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11064	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12193	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11377	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12651	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7356	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 47593	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2754	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12619	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2754	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 ‘cfv 6481 (class class class)co 7346 1c1 11007 + caddc 11009 − cmin 11344 2c2 12180 3c3 12181 4c4 12182 5c5 12183 6c6 12184 7c7 12185 9c9 12187 ℕ₀cn0 12381 cdc 12588 ↑cexp 13968 FermatNocfmtno 47564

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-seq 13909 df-exp 13969 df-fmtno 47565

This theorem is referenced by: fmtno5fac 47619

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