fmtno5 - Mathbox for Alexander van der Vekens

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Theorem fmtno5 40007

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 10959	. . . 4 ⊢ 5 = (4 + 1)
2	1	fveq2i 6106	. . 3 ⊢ (FermatNo‘5) = (FermatNo‘(4 + 1))
3		4nn0 11188	. . . 4 ⊢ 4 ∈ ℕ₀
4		fmtnorec1 39987	. . . 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 2632	. 2 ⊢ (FermatNo‘5) = ((((FermatNo‘4) − 1)↑2) + 1)
7		2nn0 11186	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
8	3, 7	deccl 11388	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
9		9nn0 11193	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
10	8, 9	deccl 11388	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
11	10, 3	deccl 11388	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
12	11, 9	deccl 11388	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ₀
13		6nn0 11190	. . . . . . 7 ⊢ 6 ∈ ℕ₀
14	12, 13	deccl 11388	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ₀
15		7nn0 11191	. . . . . 6 ⊢ 7 ∈ ℕ₀
16	14, 15	deccl 11388	. . . . 5 ⊢ ;;;;;;4294967 ∈ ℕ₀
17	16, 7	deccl 11388	. . . 4 ⊢ ;;;;;;;42949672 ∈ ℕ₀
18	17, 9	deccl 11388	. . 3 ⊢ ;;;;;;;;429496729 ∈ ℕ₀
19		6p1e7 11033	. . 3 ⊢ (6 + 1) = 7
20		5nn0 11189	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
21	13, 20	deccl 11388	. . . . . . . 8 ⊢ ;65 ∈ ℕ₀
22	21, 20	deccl 11388	. . . . . . 7 ⊢ ;;655 ∈ ℕ₀
23		3nn0 11187	. . . . . . 7 ⊢ 3 ∈ ℕ₀
24	22, 23	deccl 11388	. . . . . 6 ⊢ ;;;6553 ∈ ℕ₀
25		1nn0 11185	. . . . . 6 ⊢ 1 ∈ ℕ₀
26		fmtno4 40002	. . . . . 6 ⊢ (FermatNo‘4) = ;;;;65537
27		3p1e4 11030	. . . . . . 7 ⊢ (3 + 1) = 4
28		eqid 2610	. . . . . . 7 ⊢ ;;;6553 = ;;;6553
29	22, 23, 27, 28	decsuc 11411	. . . . . 6 ⊢ (;;;6553 + 1) = ;;;6554
30		6cn 10979	. . . . . . 7 ⊢ 6 ∈ ℂ
31		ax-1cn 9873	. . . . . . 7 ⊢ 1 ∈ ℂ
32		df-7 10961	. . . . . . 7 ⊢ 7 = (6 + 1)
33	30, 31, 32	mvrraddi 10177	. . . . . 6 ⊢ (7 − 1) = 6
34	24, 15, 25, 26, 29, 33	decsubi 11459	. . . . 5 ⊢ ((FermatNo‘4) − 1) = ;;;;65536
35	34	oveq1i 6559	. . . 4 ⊢ (((FermatNo‘4) − 1)↑2) = (;;;;65536↑2)
36		fmtno5lem4 40006	. . . 4 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
37	35, 36	eqtri 2632	. . 3 ⊢ (((FermatNo‘4) − 1)↑2) = ;;;;;;;;;4294967296
38	18, 13, 19, 37	decsuc 11411	. 2 ⊢ ((((FermatNo‘4) − 1)↑2) + 1) = ;;;;;;;;;4294967297
39	6, 38	eqtri 2632	1 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297

Colors of variables: wff setvar class

Syntax hints: = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 1c1 9816 + caddc 9818 − cmin 10145 2c2 10947 3c3 10948 4c4 10949 5c5 10950 6c6 10951 7c7 10952 9c9 10954 ℕ₀cn0 11169 cdc 11369 ↑cexp 12722 FermatNocfmtno 39977

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-seq 12664 df-exp 12723 df-fmtno 39978

This theorem is referenced by: fmtno5fac 40032

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