Theorem lgslem4 16327

Description: Lemma for lgsfcl2 16330. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 19-Mar-2022.)

Hypothesis

Ref	Expression
lgslem2.z	|- Z = { x e. ZZ | ( abs ` x ) <_ 1 }

Assertion

Ref	Expression
lgslem4	|- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )

Distinct variable group:   x, A

Allowed substitution hints:   P( x)   Z( x)

Proof of Theorem lgslem4

Step	Hyp	Ref	Expression
1		eldifi 3351	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2	1	adantl 277	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. Prime )
3		simpl 109	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> A e. ZZ )
4		oddprm 13067	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
5	4	adantl 277	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P - 1 ) / 2 ) e. NN )
6		prmdvdsexp 12952	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN ) -> ( P || ( A ^ ( ( P - 1 ) / 2 ) ) <-> P || A ) )
7	2, 3, 5, 6	syl3anc 1278	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || ( A ^ ( ( P - 1 ) / 2 ) ) <-> P || A ) )
8	7	biimpar 297	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P || ( A ^ ( ( P - 1 ) / 2 ) ) )
9		prmgt1 12936	. . . . . . 7 |- ( P e. Prime -> 1 < P )
10	1, 9	syl 14	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 1 < P )
11	10	ad2antlr 493	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 < P )
12		p1modz1 12586	. . . . 5 |- ( ( P || ( A ^ ( ( P - 1 ) / 2 ) ) /\ 1 < P ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 1 )
13	8, 11, 12	syl2anc 415	. . . 4 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 1 )
14	13	oveq1d 6103	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 1 - 1 ) )
15		1m1e0 9382	. . . 4 |- ( 1 - 1 ) = 0
16		lgslem2.z	. . . . . 6 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
17	16	lgslem2 16325	. . . . 5 |- ( -u 1 e. Z /\ 0 e. Z /\ 1 e. Z )
18	17	simp2i 1038	. . . 4 |- 0 e. Z
19	15, 18	eqeltri 2311	. . 3 |- ( 1 - 1 ) e. Z
20	14, 19	eqeltrdi 2329	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
21		lgslem1 16324	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )
22		elpri 3732	. . . 4 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 \/ ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) )
23		oveq1 6095	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 0 - 1 ) )
24		df-neg 8507	. . . . . . 7 |- -u 1 = ( 0 - 1 )
25	17	simp1i 1037	. . . . . . 7 |- -u 1 e. Z
26	24, 25	eqeltrri 2312	. . . . . 6 |- ( 0 - 1 ) e. Z
27	23, 26	eqeltrdi 2329	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
28		oveq1 6095	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 2 - 1 ) )
29		2m1e1 9431	. . . . . . 7 |- ( 2 - 1 ) = 1
30	17	simp3i 1039	. . . . . . 7 |- 1 e. Z
31	29, 30	eqeltri 2311	. . . . . 6 |- ( 2 - 1 ) e. Z
32	28, 31	eqeltrdi 2329	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
33	27, 32	jaoi 728	. . . 4 |- ( ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 \/ ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
34	21, 22, 33	3syl 17	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
35	34	3expa 1234	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
36		prmnn 12913	. . . . . 6 |- ( P e. Prime -> P e. NN )
37	1, 36	syl 14	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> P e. NN )
38	37	adantl 277	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. NN )
39		dvdsdc 12590	. . . 4 |- ( ( P e. NN /\ A e. ZZ ) -> DECID P || A )
40	38, 3, 39	syl2anc 415	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> DECID P || A )
41		exmiddc 848	. . 3 |- (DECID P || A -> ( P || A \/ -. P || A ) )
42	40, 41	syl 14	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( P || A \/ -. P || A ) )
43	20, 35, 42	mpjaodan 810	1 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   \ cdif 3217   {csn 3709   {cpr 3710   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   0cc0 8185   1c1 8186   + caddc 8188   < clt 8366   <_ cle 8367   - cmin 8504   -ucneg 8505   / cdiv 9010   NNcn 9313   2c2 9364   ZZcz 9655   mod cmo 10780   ^cexp 10996   abscabs 11785   || cdvds 12579   Primecprime 12910

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-proddc 12343  df-dvds 12580  df-gcd 12756  df-prm 12911  df-phi 13018

This theorem is used by:  lgsfcl2  16330