fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 48042

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 12245	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6837	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12454	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 48022	. . . 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 2763	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12452	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12657	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12459	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12657	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12657	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12657	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12456	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12657	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12457	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12657	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12657	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12657	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12322	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12455	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12657	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12657	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12453	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12657	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12451	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 48037	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12319	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2740	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12673	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12270	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11094	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12247	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11408	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12705	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7373	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 48041	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2763	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12673	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2763	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1547 ∈ wcel 2119 ‘cfv 6492 (class class class)co 7363 1c1 11037 + caddc 11039 − cmin 11375 2c2 12234 3c3 12235 4c4 12236 5c5 12237 6c6 12238 7c7 12239 9c9 12241 ℕ₀cn0 12435 cdc 12642 ↑cexp 14021 FermatNocfmtno 48012

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-seq 13962 df-exp 14022 df-fmtno 48013

This theorem is referenced by: fmtno5fac 48067

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