fmtnorec4 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtnorec4		Structured version Visualization version Unicode version

Theorem fmtnorec4 41461

Description: The fourth recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 31-Jul-2021.)

**Assertion**
Ref	Expression
fmtnorec4	FermatNo FermatNo FermatNo

**Proof of Theorem fmtnorec4**
Step	Hyp	Ref	Expression
1		eluz2nn 11726	. . . . . . 7
2		nnm1nn0 11334	. . . . . . 7
3	1, 2	syl 17	. . . . . 6
4		fmtno 41441	. . . . . 6 FermatNo
5	3, 4	syl 17	. . . . 5 FermatNo
6	5	oveq1d 6665	. . . 4 FermatNo
7		2nn 11185	. . . . . . . 8
8	7	a1i 11	. . . . . . 7
9		2nn0 11309	. . . . . . . . 9
10	9	a1i 11	. . . . . . . 8
11	10, 3	nn0expcld 13031	. . . . . . 7
12	8, 11	nnexpcld 13030	. . . . . 6
13	12	nncnd 11036	. . . . 5
14		binom21 12980	. . . . 5
15	13, 14	syl 17	. . . 4
16		2cn 11091	. . . . . . . . 9
17	16	a1i 11	. . . . . . . 8
18	17, 10, 11	expmuld 13011	. . . . . . 7
19	17, 3	expp1d 13009	. . . . . . . . 9
20	1	nncnd 11036	. . . . . . . . . . 11
21		npcan1 10455	. . . . . . . . . . 11
22	20, 21	syl 17	. . . . . . . . . 10
23	22	oveq2d 6666	. . . . . . . . 9
24	19, 23	eqtr3d 2658	. . . . . . . 8
25	24	oveq2d 6666	. . . . . . 7
26	18, 25	eqtr3d 2658	. . . . . 6
27	26	oveq1d 6665	. . . . 5
28	27	oveq1d 6665	. . . 4
29	6, 15, 28	3eqtrd 2660	. . 3 FermatNo
30		uznn0sub 11719	. . . . . . . 8
31		fmtno 41441	. . . . . . . 8 FermatNo
32	30, 31	syl 17	. . . . . . 7 FermatNo
33	32	oveq1d 6665	. . . . . 6 FermatNo
34	33	oveq1d 6665	. . . . 5 FermatNo
35	10, 30	nn0expcld 13031	. . . . . . . . 9
36	8, 35	nnexpcld 13030	. . . . . . . 8
37	36	nncnd 11036	. . . . . . 7
38		peano2cn 10208	. . . . . . 7
39	37, 38	syl 17	. . . . . 6
40		binom2sub1 12982	. . . . . 6
41	39, 40	syl 17	. . . . 5
42		binom21 12980	. . . . . . . 8
43	37, 42	syl 17	. . . . . . 7
44	43	oveq1d 6665	. . . . . 6
45	44	oveq1d 6665	. . . . 5
46	34, 41, 45	3eqtrd 2660	. . . 4 FermatNo
47	46	oveq2d 6666	. . 3 FermatNo
48	29, 47	oveq12d 6668	. 2 FermatNo FermatNo
49	36, 10	nnexpcld 13030	. . . . . . . . 9
50	49	nncnd 11036	. . . . . . . 8
51	17, 37	mulcld 10060	. . . . . . . 8
52	50, 51	addcld 10059	. . . . . . 7
53		peano2cn 10208	. . . . . . 7
54	52, 53	syl 17	. . . . . 6
55	17, 39	mulcld 10060	. . . . . 6
56	54, 55	subcld 10392	. . . . 5
57		1cnd 10056	. . . . 5
58	17, 56, 57	adddid 10064	. . . 4
59	52, 57	addcld 10059	. . . . . . 7
60	17, 59, 55	subdid 10486	. . . . . 6
61	17, 52, 57	adddid 10064	. . . . . . . 8
62	17, 50, 51	adddid 10064	. . . . . . . . . 10
63	17, 10, 35	expmuld 13011	. . . . . . . . . . . . . 14
64	17, 30	expp1d 13009	. . . . . . . . . . . . . . . 16
65	20, 17, 57	subsubd 10420	. . . . . . . . . . . . . . . . . 18
66	65	eqcomd 2628	. . . . . . . . . . . . . . . . 17
67	66	oveq2d 6666	. . . . . . . . . . . . . . . 16
68	64, 67	eqtr3d 2658	. . . . . . . . . . . . . . 15
69	68	oveq2d 6666	. . . . . . . . . . . . . 14
70	63, 69	eqtr3d 2658	. . . . . . . . . . . . 13
71	70	oveq2d 6666	. . . . . . . . . . . 12
72		2m1e1 11135	. . . . . . . . . . . . . . . . 17
73	72	a1i 11	. . . . . . . . . . . . . . . 16
74	73	oveq2d 6666	. . . . . . . . . . . . . . 15
75	74	oveq2d 6666	. . . . . . . . . . . . . 14
76	75	oveq2d 6666	. . . . . . . . . . . . 13
77	76	oveq2d 6666	. . . . . . . . . . . 12
78	71, 77	eqtrd 2656	. . . . . . . . . . 11
79	17, 17, 37	mulassd 10063	. . . . . . . . . . . 12
80	79	eqcomd 2628	. . . . . . . . . . 11
81	78, 80	oveq12d 6668	. . . . . . . . . 10
82	62, 81	eqtrd 2656	. . . . . . . . 9
83		2t1e2 11176	. . . . . . . . . 10
84	83	a1i 11	. . . . . . . . 9
85	82, 84	oveq12d 6668	. . . . . . . 8
86	61, 85	eqtrd 2656	. . . . . . 7
87	17, 37, 57	adddid 10064	. . . . . . . . . 10
88	84	oveq2d 6666	. . . . . . . . . 10
89	87, 88	eqtrd 2656	. . . . . . . . 9
90	89	oveq2d 6666	. . . . . . . 8
91	17, 51, 17	adddid 10064	. . . . . . . 8
92		2t2e4 11177	. . . . . . . . . 10
93	92	a1i 11	. . . . . . . . 9
94	80, 93	oveq12d 6668	. . . . . . . 8
95	90, 91, 94	3eqtrd 2660	. . . . . . 7
96	86, 95	oveq12d 6668	. . . . . 6
97	60, 96	eqtrd 2656	. . . . 5
98	97, 84	oveq12d 6668	. . . 4
99	58, 98	eqtrd 2656	. . 3
100	99	oveq2d 6666	. 2
101	17, 13	mulcld 10060	. . . . . . . . . . 11
102	16, 16	mulcli 10045	. . . . . . . . . . . . 13
103	102	a1i 11	. . . . . . . . . . . 12
104	103, 37	mulcld 10060	. . . . . . . . . . 11
105	101, 104	addcld 10059	. . . . . . . . . 10
106	105, 17	addcld 10059	. . . . . . . . 9
107		4cn 11098	. . . . . . . . . . 11
108	107	a1i 11	. . . . . . . . . 10
109	104, 108	addcld 10059	. . . . . . . . 9
110	105, 17, 17	addassd 10062	. . . . . . . . . 10
111		2p2e4 11144	. . . . . . . . . . . 12
112	111	a1i 11	. . . . . . . . . . 11
113	112	oveq2d 6666	. . . . . . . . . 10
114	101, 104, 108	addassd 10062	. . . . . . . . . 10
115	110, 113, 114	3eqtrd 2660	. . . . . . . . 9
116	106, 17, 101, 109, 115	subaddeqd 10446	. . . . . . . 8
117	116	eqcomd 2628	. . . . . . 7
118	106, 109	subcld 10392	. . . . . . . 8
119	101, 17, 118	subadd2d 10411	. . . . . . 7
120	117, 119	mpbid 222	. . . . . 6
121	120	oveq2d 6666	. . . . 5
122		eluzge2nn0 11727	. . . . . . . . . 10
123	10, 122	nn0expcld 13031	. . . . . . . . 9
124	8, 123	nnexpcld 13030	. . . . . . . 8
125	124	nncnd 11036	. . . . . . 7
126	125, 101	addcld 10059	. . . . . 6
127	118, 17	addcld 10059	. . . . . 6
128	126, 127, 125	subadd2d 10411	. . . . 5
129	121, 128	mpbird 247	. . . 4
130	129	oveq1d 6665	. . 3
131	126, 57, 127	addsubd 10413	. . 3
132		fmtno 41441	. . . 4 FermatNo
133	122, 132	syl 17	. . 3 FermatNo
134	130, 131, 133	3eqtr4d 2666	. 2 FermatNo
135	48, 100, 134	3eqtrrd 2661	1 FermatNo FermatNo FermatNo

Colors of variables: wff setvar class

Syntax hints:

wi 4

wceq 1483

wcel 1990

cfv 5888 (class class class)co 6650

cc 9934

c1 9937

caddc 9939

cmul 9941

cmin 10266

cn 11020

c2 11070

c4 11072

cn0 11292

cuz 11687

cexp 12860 FermatNocfmtno 41439

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861 df-fmtno 41440

This theorem is referenced by: (None)

W3C validator