Theorem lgslem4 15513

Description: Lemma for lgsfcl2 15516. (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 3295	. . . . . . . 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 12615	. . . . . . . 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 12503	. . . . . . 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 1250	. . . . . 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 12487	. . . . . . 7 |- ( P e. Prime -> 1 < P )
10	1, 9	syl 14	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 1 < P )
11	10	ad2antlr 489	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 < P )
12		p1modz1 12138	. . . . 5 |- ( ( P || ( A ^ ( ( P - 1 ) / 2 ) ) /\ 1 < P ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 1 )
13	8, 11, 12	syl2anc 411	. . . 4 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 1 )
14	13	oveq1d 5961	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 1 - 1 ) )
15		1m1e0 9107	. . . 4 |- ( 1 - 1 ) = 0
16		lgslem2.z	. . . . . 6 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
17	16	lgslem2 15511	. . . . 5 |- ( -u 1 e. Z /\ 0 e. Z /\ 1 e. Z )
18	17	simp2i 1010	. . . 4 |- 0 e. Z
19	15, 18	eqeltri 2278	. . 3 |- ( 1 - 1 ) e. Z
20	14, 19	eqeltrdi 2296	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
21		lgslem1 15510	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )
22		elpri 3656	. . . 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 5953	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 0 - 1 ) )
24		df-neg 8248	. . . . . . 7 |- -u 1 = ( 0 - 1 )
25	17	simp1i 1009	. . . . . . 7 |- -u 1 e. Z
26	24, 25	eqeltrri 2279	. . . . . 6 |- ( 0 - 1 ) e. Z
27	23, 26	eqeltrdi 2296	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
28		oveq1 5953	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 2 - 1 ) )
29		2m1e1 9156	. . . . . . 7 |- ( 2 - 1 ) = 1
30	17	simp3i 1011	. . . . . . 7 |- 1 e. Z
31	29, 30	eqeltri 2278	. . . . . 6 |- ( 2 - 1 ) e. Z
32	28, 31	eqeltrdi 2296	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
33	27, 32	jaoi 718	. . . 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 1206	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
36		prmnn 12465	. . . . . 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 12142	. . . 4 |- ( ( P e. NN /\ A e. ZZ ) -> DECID P || A )
40	38, 3, 39	syl2anc 411	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> DECID P || A )
41		exmiddc 838	. . 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 800	1 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2176   {crab 2488   \ cdif 3163   {csn 3633   {cpr 3634   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   0cc0 7927   1c1 7928   + caddc 7930   < clt 8109   <_ cle 8110   - cmin 8245   -ucneg 8246   / cdiv 8747   NNcn 9038   2c2 9089   ZZcz 9374   mod cmo 10469   ^cexp 10685   abscabs 11341   || cdvds 12131   Primecprime 12462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-2o 6505  df-oadd 6508  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-sup 7088  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-fzo 10267  df-fl 10415  df-mod 10470  df-seqfrec 10595  df-exp 10686  df-ihash 10923  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-proddc 11895  df-dvds 12132  df-gcd 12308  df-prm 12463  df-phi 12566

This theorem is referenced by:  lgsfcl2  15516