fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 46897

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 12309	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6900	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12522	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 46877	. . . 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 2756	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12520	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12723	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12527	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12723	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12723	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12723	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12524	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12723	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12525	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12723	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12723	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12723	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12391	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12523	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12723	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12723	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12521	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12723	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12519	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 46892	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12388	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2728	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12739	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12334	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11197	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12311	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11508	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12771	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7430	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 46896	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2756	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12739	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2756	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 1c1 11140 + caddc 11142 − cmin 11475 2c2 12298 3c3 12299 4c4 12300 5c5 12301 6c6 12302 7c7 12303 9c9 12305 ℕ₀cn0 12503 cdc 12708 ↑cexp 14059 FermatNocfmtno 46867

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-seq 14000 df-exp 14060 df-fmtno 46868

This theorem is referenced by: fmtno5fac 46922

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