fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 47562

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 12259	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6864	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12468	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 47542	. . . 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 2753	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12466	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12671	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12473	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12671	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12671	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12671	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12470	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12671	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12471	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12671	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12671	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12671	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12336	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12469	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12671	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12671	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12467	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12671	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12465	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 47557	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12333	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2730	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12687	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12284	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11133	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12261	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11445	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12719	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7400	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 47561	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2753	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12687	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2753	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 ‘cfv 6514 (class class class)co 7390 1c1 11076 + caddc 11078 − cmin 11412 2c2 12248 3c3 12249 4c4 12250 5c5 12251 6c6 12252 7c7 12253 9c9 12255 ℕ₀cn0 12449 cdc 12656 ↑cexp 14033 FermatNocfmtno 47532

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-seq 13974 df-exp 14034 df-fmtno 47533

This theorem is referenced by: fmtno5fac 47587

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