fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 47482

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 12330	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6910	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12543	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 47462	. . . 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 12541	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12746	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12548	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12746	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12746	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12746	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12545	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12746	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12546	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12746	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12746	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12746	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12412	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12544	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12746	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12746	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12542	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12746	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12540	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 47477	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12409	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2735	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12762	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12355	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11211	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12332	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11523	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12794	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7441	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 47481	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2763	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12762	. 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 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 − cmin 11490 2c2 12319 3c3 12320 4c4 12321 5c5 12322 6c6 12323 7c7 12324 9c9 12326 ℕ₀cn0 12524 cdc 12731 ↑cexp 14099 FermatNocfmtno 47452

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-seq 14040 df-exp 14100 df-fmtno 47453

This theorem is referenced by: fmtno5fac 47507

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