fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 41469

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 11082	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6194	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 11311	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 41449	. . . 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 2644	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 11309	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 11512	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 11316	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 11512	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 11512	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 11512	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 11313	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 11512	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 11314	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 11512	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 11512	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 11512	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 11156	. . 3 ⊢ (6 + 1) = 7
20		5nn0 11312	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 11512	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 11512	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 11310	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 11512	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 11308	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 41464	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 11153	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2622	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 11535	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 11102	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 9994	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 11084	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 10298	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 11583	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 6660	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 41468	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2644	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 11535	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2644	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 1c1 9937 + caddc 9939 − cmin 10266 2c2 11070 3c3 11071 4c4 11072 5c5 11073 6c6 11074 7c7 11075 9c9 11077 ℕ₀cn0 11292 cdc 11493 ↑cexp 12860 FermatNocfmtno 41439

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-seq 12802 df-exp 12861 df-fmtno 41440

This theorem is referenced by: fmtno5fac 41494

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