Theorem fmtnodvds 41456

Description: Any Fermat number divides a greater Fermat number minus 2. Corrolary of fmtnorec2 41455, see ProofWiki "Product of Sequence of Fermat Numbers plus 2/Corollary", 31-Jul-2021. (Contributed by AV, 1-Aug-2021.)

Assertion

Ref	Expression
fmtnodvds	|- ( ( N e. NN0 /\ M e. NN ) -> (FermatNo ` N ) || ( (FermatNo ` ( N + M ) ) - 2 ) )

Proof of Theorem fmtnodvds

Dummy variables k n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( N e. NN0 /\ M e. NN ) -> N e. NN0 )
2		nn0nnaddcl 11324	. . . . 5 |- ( ( N e. NN0 /\ M e. NN ) -> ( N + M ) e. NN )
3		nnm1nn0 11334	. . . . 5 |- ( ( N + M ) e. NN -> ( ( N + M ) - 1 ) e. NN0 )
4	2, 3	syl 17	. . . 4 |- ( ( N e. NN0 /\ M e. NN ) -> ( ( N + M ) - 1 ) e. NN0 )
5		1red 10055	. . . . . 6 |- ( ( N e. NN0 /\ M e. NN ) -> 1 e. RR )
6		nnre 11027	. . . . . . 7 |- ( M e. NN -> M e. RR )
7	6	adantl 482	. . . . . 6 |- ( ( N e. NN0 /\ M e. NN ) -> M e. RR )
8		nn0re 11301	. . . . . . 7 |- ( N e. NN0 -> N e. RR )
9	8	adantr 481	. . . . . 6 |- ( ( N e. NN0 /\ M e. NN ) -> N e. RR )
10		nnge1 11046	. . . . . . 7 |- ( M e. NN -> 1 <_ M )
11	10	adantl 482	. . . . . 6 |- ( ( N e. NN0 /\ M e. NN ) -> 1 <_ M )
12	5, 7, 9, 11	leadd2dd 10642	. . . . 5 |- ( ( N e. NN0 /\ M e. NN ) -> ( N + 1 ) <_ ( N + M ) )
13		readdcl 10019	. . . . . . 7 |- ( ( N e. RR /\ M e. RR ) -> ( N + M ) e. RR )
14	8, 6, 13	syl2an 494	. . . . . 6 |- ( ( N e. NN0 /\ M e. NN ) -> ( N + M ) e. RR )
15		leaddsub 10504	. . . . . 6 |- ( ( N e. RR /\ 1 e. RR /\ ( N + M ) e. RR ) -> ( ( N + 1 ) <_ ( N + M ) <-> N <_ ( ( N + M ) - 1 ) ) )
16	9, 5, 14, 15	syl3anc 1326	. . . . 5 |- ( ( N e. NN0 /\ M e. NN ) -> ( ( N + 1 ) <_ ( N + M ) <-> N <_ ( ( N + M ) - 1 ) ) )
17	12, 16	mpbid 222	. . . 4 |- ( ( N e. NN0 /\ M e. NN ) -> N <_ ( ( N + M ) - 1 ) )
18		elfz2nn0 12431	. . . 4 |- ( N e. ( 0 ... ( ( N + M ) - 1 ) ) <-> ( N e. NN0 /\ ( ( N + M ) - 1 ) e. NN0 /\ N <_ ( ( N + M ) - 1 ) ) )
19	1, 4, 17, 18	syl3anbrc 1246	. . 3 |- ( ( N e. NN0 /\ M e. NN ) -> N e. ( 0 ... ( ( N + M ) - 1 ) ) )
20		fzfid 12772	. . . 4 |- ( ( N e. NN0 /\ M e. NN ) -> ( 0 ... ( ( N + M ) - 1 ) ) e. Fin )
21		fz0ssnn0 12435	. . . . 5 |- ( 0 ... ( ( N + M ) - 1 ) ) C_ NN0
22	21	a1i 11	. . . 4 |- ( ( N e. NN0 /\ M e. NN ) -> ( 0 ... ( ( N + M ) - 1 ) ) C_ NN0 )
23		2nn0 11309	. . . . . . . . . 10 |- 2 e. NN0
24	23	a1i 11	. . . . . . . . 9 |- ( n e. NN0 -> 2 e. NN0 )
25		id 22	. . . . . . . . . 10 |- ( n e. NN0 -> n e. NN0 )
26	24, 25	nn0expcld 13031	. . . . . . . . 9 |- ( n e. NN0 -> ( 2 ^ n ) e. NN0 )
27	24, 26	nn0expcld 13031	. . . . . . . 8 |- ( n e. NN0 -> ( 2 ^ ( 2 ^ n ) ) e. NN0 )
28	27	nn0zd 11480	. . . . . . 7 |- ( n e. NN0 -> ( 2 ^ ( 2 ^ n ) ) e. ZZ )
29	28	peano2zd 11485	. . . . . 6 |- ( n e. NN0 -> ( ( 2 ^ ( 2 ^ n ) ) + 1 ) e. ZZ )
30	29	adantl 482	. . . . 5 |- ( ( ( N e. NN0 /\ M e. NN ) /\ n e. NN0 ) -> ( ( 2 ^ ( 2 ^ n ) ) + 1 ) e. ZZ )
31		df-fmtno 41440	. . . . 5 |- FermatNo = ( n e. NN0 |-> ( ( 2 ^ ( 2 ^ n ) ) + 1 ) )
32	30, 31	fmptd 6385	. . . 4 |- ( ( N e. NN0 /\ M e. NN ) -> FermatNo : NN0 --> ZZ )
33	20, 22, 32	fprodfvdvdsd 15058	. . 3 |- ( ( N e. NN0 /\ M e. NN ) -> A. n e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` n ) || prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) )
34		fveq2 6191	. . . . 5 |- ( n = N -> (FermatNo ` n ) = (FermatNo ` N ) )
35	34	breq1d 4663	. . . 4 |- ( n = N -> ( (FermatNo ` n ) || prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) <-> (FermatNo ` N ) || prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) ) )
36	35	rspcv 3305	. . 3 |- ( N e. ( 0 ... ( ( N + M ) - 1 ) ) -> ( A. n e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` n ) || prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) -> (FermatNo ` N ) || prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) ) )
37	19, 33, 36	sylc 65	. 2 |- ( ( N e. NN0 /\ M e. NN ) -> (FermatNo ` N ) || prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) )
38		elfznn0 12433	. . . . . . 7 |- ( k e. ( 0 ... ( ( N + M ) - 1 ) ) -> k e. NN0 )
39	38	adantl 482	. . . . . 6 |- ( ( ( N e. NN0 /\ M e. NN ) /\ k e. ( 0 ... ( ( N + M ) - 1 ) ) ) -> k e. NN0 )
40		fmtnonn 41443	. . . . . 6 |- ( k e. NN0 -> (FermatNo ` k ) e. NN )
41	39, 40	syl 17	. . . . 5 |- ( ( ( N e. NN0 /\ M e. NN ) /\ k e. ( 0 ... ( ( N + M ) - 1 ) ) ) -> (FermatNo ` k ) e. NN )
42	41	nncnd 11036	. . . 4 |- ( ( ( N e. NN0 /\ M e. NN ) /\ k e. ( 0 ... ( ( N + M ) - 1 ) ) ) -> (FermatNo ` k ) e. CC )
43	20, 42	fprodcl 14682	. . 3 |- ( ( N e. NN0 /\ M e. NN ) -> prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) e. CC )
44		2cnd 11093	. . 3 |- ( ( N e. NN0 /\ M e. NN ) -> 2 e. CC )
45		nn0cn 11302	. . . . . . . 8 |- ( N e. NN0 -> N e. CC )
46		nncn 11028	. . . . . . . 8 |- ( M e. NN -> M e. CC )
47		addcl 10018	. . . . . . . 8 |- ( ( N e. CC /\ M e. CC ) -> ( N + M ) e. CC )
48	45, 46, 47	syl2an 494	. . . . . . 7 |- ( ( N e. NN0 /\ M e. NN ) -> ( N + M ) e. CC )
49		npcan1 10455	. . . . . . 7 |- ( ( N + M ) e. CC -> ( ( ( N + M ) - 1 ) + 1 ) = ( N + M ) )
50	48, 49	syl 17	. . . . . 6 |- ( ( N e. NN0 /\ M e. NN ) -> ( ( ( N + M ) - 1 ) + 1 ) = ( N + M ) )
51	50	eqcomd 2628	. . . . 5 |- ( ( N e. NN0 /\ M e. NN ) -> ( N + M ) = ( ( ( N + M ) - 1 ) + 1 ) )
52	51	fveq2d 6195	. . . 4 |- ( ( N e. NN0 /\ M e. NN ) -> (FermatNo ` ( N + M ) ) = (FermatNo ` ( ( ( N + M ) - 1 ) + 1 ) ) )
53		fmtnorec2 41455	. . . . 5 |- ( ( ( N + M ) - 1 ) e. NN0 -> (FermatNo ` ( ( ( N + M ) - 1 ) + 1 ) ) = ( prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) + 2 ) )
54	4, 53	syl 17	. . . 4 |- ( ( N e. NN0 /\ M e. NN ) -> (FermatNo ` ( ( ( N + M ) - 1 ) + 1 ) ) = ( prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) + 2 ) )
55	52, 54	eqtrd 2656	. . 3 |- ( ( N e. NN0 /\ M e. NN ) -> (FermatNo ` ( N + M ) ) = ( prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) + 2 ) )
56	43, 44, 55	mvrraddd 10445	. 2 |- ( ( N e. NN0 /\ M e. NN ) -> ( (FermatNo ` ( N + M ) ) - 2 ) = prod_ k e. ( 0 ... ( ( N + M ) - 1 ) ) (FermatNo ` k ) )
57	37, 56	breqtrrd 4681	1 |- ( ( N e. NN0 /\ M e. NN ) -> (FermatNo ` N ) || ( (FermatNo ` ( N + M ) ) - 2 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ...cfz 12326   ^cexp 12860   prod_cprod 14635   || cdvds 14983  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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-int 4476  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  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-5 11082  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-dvds 14984  df-fmtno 41440

This theorem is referenced by:  goldbachthlem1  41457