fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 47544

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 12332	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6909	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12545	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 47524	. . . 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 2765	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12543	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12748	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12550	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12748	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12748	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12748	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12547	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12748	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12548	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12748	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12748	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12748	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12414	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12546	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12748	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12748	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12544	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12748	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12542	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 47539	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12411	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2737	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12764	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12357	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11213	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12334	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11525	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12796	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7441	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 47543	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2765	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12764	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2765	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 1c1 11156 + caddc 11158 − cmin 11492 2c2 12321 3c3 12322 4c4 12323 5c5 12324 6c6 12325 7c7 12326 9c9 12328 ℕ₀cn0 12526 cdc 12733 ↑cexp 14102 FermatNocfmtno 47514

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-seq 14043 df-exp 14103 df-fmtno 47515

This theorem is referenced by: fmtno5fac 47569

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