fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 46225

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 12278	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6895	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12491	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 46205	. . . 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 2761	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12489	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12692	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12496	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12692	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12692	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12692	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12493	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12692	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12494	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12692	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12692	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12692	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12360	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12492	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12692	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12692	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12490	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12692	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12488	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 46220	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12357	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2733	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12708	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12303	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11168	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12280	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11477	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12740	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7419	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 46224	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2761	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12708	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2761	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2107 ‘cfv 6544 (class class class)co 7409 1c1 11111 + caddc 11113 − cmin 11444 2c2 12267 3c3 12268 4c4 12269 5c5 12270 6c6 12271 7c7 12272 9c9 12274 ℕ₀cn0 12472 cdc 12677 ↑cexp 14027 FermatNocfmtno 46195

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-seq 13967 df-exp 14028 df-fmtno 46196

This theorem is referenced by: fmtno5fac 46250

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