fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 48126

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 12276	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6864	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 12493	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 48106	. . . 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 2784	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 12491	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 12696	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 12498	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 12696	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 12696	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 12696	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 12495	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 12696	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 12496	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 12696	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 12696	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 12696	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 12358	. . 3 ⊢ (6 + 1) = 7
20		5nn0 12494	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 12696	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 12696	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 12492	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 12696	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 12490	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 48121	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 12355	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2761	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 12717	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 12302	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 11124	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 12278	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 11440	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 12749	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 7400	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 48125	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2784	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 12717	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2784	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1559 ∈ wcel 2141 ‘cfv 6515 (class class class)co 7390 1c1 11067 + caddc 11069 − cmin 11407 2c2 12265 3c3 12266 4c4 12267 5c5 12268 6c6 12269 7c7 12270 9c9 12272 ℕ₀cn0 12474 cdc 12681 ↑cexp 14067 FermatNocfmtno 48096

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-er 8671 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-seq 14008 df-exp 14068 df-fmtno 48097

This theorem is referenced by: fmtno5fac 48151

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