fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 48020

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 12247	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6843	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12456	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 48000	. . . 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 12454	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12659	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12461	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12659	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12659	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12659	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12458	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12659	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12459	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12659	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12659	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12659	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12324	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12457	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12659	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12659	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12455	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12659	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12453	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 48015	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12321	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2736	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12675	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12272	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11096	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12249	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11410	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12707	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7377	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 48019	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2759	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12675	. 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 1542 ∈ wcel 2114 ‘cfv 6498 (class class class)co 7367 1c1 11039 + caddc 11041 − cmin 11377 2c2 12236 3c3 12237 4c4 12238 5c5 12239 6c6 12240 7c7 12241 9c9 12243 ℕ₀cn0 12437 cdc 12644 ↑cexp 14023 FermatNocfmtno 47990

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 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 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-seq 13964 df-exp 14024 df-fmtno 47991

This theorem is referenced by: fmtno5fac 48045

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