fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 47803

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 12211	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6837	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12420	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 47783	. . . 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 2759	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12418	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12622	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12425	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12622	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12622	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12622	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12422	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12622	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12423	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12622	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12622	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12622	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12288	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12421	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12622	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12622	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12419	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12622	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12417	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 47798	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12285	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2736	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12638	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12236	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11084	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12213	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11397	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12670	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7368	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 47802	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2759	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12638	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2759	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2113 ‘cfv 6492 (class class class)co 7358 1c1 11027 + caddc 11029 − cmin 11364 2c2 12200 3c3 12201 4c4 12202 5c5 12203 6c6 12204 7c7 12205 9c9 12207 ℕ₀cn0 12401 cdc 12607 ↑cexp 13984 FermatNocfmtno 47773

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-seq 13925 df-exp 13985 df-fmtno 47774

This theorem is referenced by: fmtno5fac 47828

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