fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 46778

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 12279	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6887	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12492	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 46758	. . . 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 2754	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12490	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12693	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12497	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12693	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12693	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12693	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12494	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12693	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12495	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12693	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12693	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12693	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12361	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12493	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12693	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12693	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12491	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12693	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12489	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 46773	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12358	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2726	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12709	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12304	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11167	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12281	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11478	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12741	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7414	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 46777	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2754	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12709	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2754	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 ‘cfv 6536 (class class class)co 7404 1c1 11110 + caddc 11112 − cmin 11445 2c2 12268 3c3 12269 4c4 12270 5c5 12271 6c6 12272 7c7 12273 9c9 12275 ℕ₀cn0 12473 cdc 12678 ↑cexp 14030 FermatNocfmtno 46748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-seq 13970 df-exp 14031 df-fmtno 46749

This theorem is referenced by: fmtno5fac 46803

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