fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 45009

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 12039	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6777	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12252	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 44989	. . . 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 2766	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12250	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12452	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12257	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12452	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12452	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12452	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12254	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12452	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12255	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12452	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12452	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12452	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12121	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12253	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12452	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12452	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12251	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12452	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12249	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 45004	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12118	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2738	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12468	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12064	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 10929	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12041	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11238	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12500	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7285	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 45008	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2766	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12468	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2766	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 1c1 10872 + caddc 10874 − cmin 11205 2c2 12028 3c3 12029 4c4 12030 5c5 12031 6c6 12032 7c7 12033 9c9 12035 ℕ₀cn0 12233 cdc 12437 ↑cexp 13782 FermatNocfmtno 44979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-seq 13722 df-exp 13783 df-fmtno 44980

This theorem is referenced by: fmtno5fac 45034

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