fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 42500

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 11445	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6451	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 11667	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 42480	. . . 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 2802	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 11665	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 11864	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 11672	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 11864	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 11864	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 11864	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 11669	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 11864	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 11670	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 11864	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 11864	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 11864	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 11534	. . 3 ⊢ (6 + 1) = 7
20		5nn0 11668	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 11864	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 11864	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 11666	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 11864	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 11664	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 42495	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 11531	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2778	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 11881	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 11473	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 10332	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 11447	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 10642	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 11913	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 6934	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 42499	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2802	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 11881	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2802	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 1c1 10275 + caddc 10277 − cmin 10608 2c2 11434 3c3 11435 4c4 11436 5c5 11437 6c6 11438 7c7 11439 9c9 11441 ℕ₀cn0 11646 cdc 11849 ↑cexp 13182 FermatNocfmtno 42470

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-seq 13124 df-exp 13183 df-fmtno 42471

This theorem is referenced by: fmtno5fac 42525

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