Theorem 41prothprm 41536

Description: 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)

Hypothesis

Ref	Expression
41prothprm.p	|- P = ; 4 1

Assertion

Ref	Expression
41prothprm	|- ( P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) /\ P e. Prime )

Proof of Theorem 41prothprm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		41prothprm.p	. . 3 |- P = ; 4 1
2	1	41prothprmlem2 41535	. 2 |- ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P )
3		dfdec10 11497	. . 3 |- ; 4 1 = ( (; 1 0 x. 4 ) + 1 )
4		4t2e8 11181	. . . . . . . 8 |- ( 4 x. 2 ) = 8
5		4cn 11098	. . . . . . . . 9 |- 4 e. CC
6		2cn 11091	. . . . . . . . 9 |- 2 e. CC
7	5, 6	mulcomi 10046	. . . . . . . 8 |- ( 4 x. 2 ) = ( 2 x. 4 )
8	4, 7	eqtr3i 2646	. . . . . . 7 |- 8 = ( 2 x. 4 )
9	8	oveq2i 6661	. . . . . 6 |- ( 5 x. 8 ) = ( 5 x. ( 2 x. 4 ) )
10		5cn 11100	. . . . . . 7 |- 5 e. CC
11	10, 6, 5	mulassi 10049	. . . . . 6 |- ( ( 5 x. 2 ) x. 4 ) = ( 5 x. ( 2 x. 4 ) )
12		5t2e10 11634	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
13	12	oveq1i 6660	. . . . . 6 |- ( ( 5 x. 2 ) x. 4 ) = (; 1 0 x. 4 )
14	9, 11, 13	3eqtr2i 2650	. . . . 5 |- ( 5 x. 8 ) = (; 1 0 x. 4 )
15		cu2 12963	. . . . . . 7 |- ( 2 ^ 3 ) = 8
16	15	eqcomi 2631	. . . . . 6 |- 8 = ( 2 ^ 3 )
17	16	oveq2i 6661	. . . . 5 |- ( 5 x. 8 ) = ( 5 x. ( 2 ^ 3 ) )
18	14, 17	eqtr3i 2646	. . . 4 |- (; 1 0 x. 4 ) = ( 5 x. ( 2 ^ 3 ) )
19	18	oveq1i 6660	. . 3 |- ( (; 1 0 x. 4 ) + 1 ) = ( ( 5 x. ( 2 ^ 3 ) ) + 1 )
20	1, 3, 19	3eqtri 2648	. 2 |- P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 )
21		simpr 477	. . 3 |- ( ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) /\ P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) ) -> P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) )
22		3nn 11186	. . . . 5 |- 3 e. NN
23	22	a1i 11	. . . 4 |- ( ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) /\ P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) ) -> 3 e. NN )
24		5nn 11188	. . . . 5 |- 5 e. NN
25	24	a1i 11	. . . 4 |- ( ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) /\ P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) ) -> 5 e. NN )
26		5lt8 11217	. . . . . 6 |- 5 < 8
27	26, 15	breqtrri 4680	. . . . 5 |- 5 < ( 2 ^ 3 )
28	27	a1i 11	. . . 4 |- ( ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) /\ P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) ) -> 5 < ( 2 ^ 3 ) )
29		3z 11410	. . . . . . 7 |- 3 e. ZZ
30	29	a1i 11	. . . . . 6 |- ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) -> 3 e. ZZ )
31		oveq1 6657	. . . . . . . . 9 |- ( x = 3 -> ( x ^ ( ( P - 1 ) / 2 ) ) = ( 3 ^ ( ( P - 1 ) / 2 ) ) )
32	31	oveq1d 6665	. . . . . . . 8 |- ( x = 3 -> ( ( x ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) )
33	32	eqeq1d 2624	. . . . . . 7 |- ( x = 3 -> ( ( ( x ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) <-> ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) ) )
34	33	adantl 482	. . . . . 6 |- ( ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) /\ x = 3 ) -> ( ( ( x ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) <-> ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) ) )
35		id 22	. . . . . 6 |- ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) -> ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) )
36	30, 34, 35	rspcedvd 3317	. . . . 5 |- ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) -> E. x e. ZZ ( ( x ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) )
37	36	adantr 481	. . . 4 |- ( ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) /\ P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) ) -> E. x e. ZZ ( ( x ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) )
38	23, 25, 21, 28, 37	proththd 41531	. . 3 |- ( ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) /\ P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) ) -> P e. Prime )
39	21, 38	jca 554	. 2 |- ( ( ( ( 3 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( -u 1 mod P ) /\ P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) ) -> ( P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) /\ P e. Prime ) )
40	2, 20, 39	mp2an 708	1 |- ( P = ( ( 5 x. ( 2 ^ 3 ) ) + 1 ) /\ P e. Prime )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   5c5 11073   8c8 11076   ZZcz 11377  ;cdc 11493   mod cmo 12668   ^cexp 12860   Primecprime 15385

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-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-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-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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  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-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  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-dvds 14984  df-gcd 15217  df-prm 15386  df-odz 15470  df-phi 15471  df-pc 15542

This theorem is referenced by: (None)