fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 47545

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 12212	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6829	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12421	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 47525	. . . 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 2752	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12419	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12624	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12426	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12624	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12624	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12624	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12423	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12624	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12424	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12624	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12624	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12624	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12289	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12422	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12624	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12624	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12420	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12624	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12418	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 47540	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12286	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2729	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12640	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12237	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11086	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12214	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11398	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12672	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7363	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 47544	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2752	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12640	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2752	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 ‘cfv 6486 (class class class)co 7353 1c1 11029 + caddc 11031 − cmin 11365 2c2 12201 3c3 12202 4c4 12203 5c5 12204 6c6 12205 7c7 12206 9c9 12208 ℕ₀cn0 12402 cdc 12609 ↑cexp 13986 FermatNocfmtno 47515

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-seq 13927 df-exp 13987 df-fmtno 47516

This theorem is referenced by: fmtno5fac 47570

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