fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 44074

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 11691	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6648	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 11904	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 44054	. . . 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 2821	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 11902	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12101	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 11909	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12101	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12101	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12101	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 11906	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12101	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 11907	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12101	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12101	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12101	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 11773	. . 3 ⊢ (6 + 1) = 7
20		5nn0 11905	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12101	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12101	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 11903	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12101	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 11901	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 44069	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 11770	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2798	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12117	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 11716	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 10584	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 11693	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 10892	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12149	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7145	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 44073	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2821	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12117	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2821	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 1c1 10527 + caddc 10529 − cmin 10859 2c2 11680 3c3 11681 4c4 11682 5c5 11683 6c6 11684 7c7 11685 9c9 11687 ℕ₀cn0 11885 cdc 12086 ↑cexp 13425 FermatNocfmtno 44044

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-seq 13365 df-exp 13426 df-fmtno 44045

This theorem is referenced by: fmtno5fac 44099

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