fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 43718

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 11702	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6672	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 11915	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 43698	. . . 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 2844	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 11913	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12112	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 11920	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12112	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12112	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12112	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 11917	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12112	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 11918	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12112	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12112	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12112	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 11784	. . 3 ⊢ (6 + 1) = 7
20		5nn0 11916	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12112	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12112	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 11914	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12112	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 11912	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 43713	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 11781	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2821	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12128	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 11727	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 10594	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 11704	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 10902	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12160	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7165	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 43717	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2844	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12128	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2844	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 1c1 10537 + caddc 10539 − cmin 10869 2c2 11691 3c3 11692 4c4 11693 5c5 11694 6c6 11695 7c7 11696 9c9 11698 ℕ₀cn0 11896 cdc 12097 ↑cexp 13428 FermatNocfmtno 43688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-seq 13369 df-exp 13429 df-fmtno 43689

This theorem is referenced by: fmtno5fac 43743

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