Theorem 2lgslem3c1 15743

Description: Lemma 3 for 2lgslem3 15745. (Contributed by AV, 16-Jul-2021.)

Hypothesis

Ref	Expression
2lgslem2.n	|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )

Assertion

Ref	Expression
2lgslem3c1	|- ( ( P e. NN /\ ( P mod 8 ) = 5 ) -> ( N mod 2 ) = 1 )

Proof of Theorem 2lgslem3c1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 9344	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9246	. . . . 5 |- 8 e. NN
3		nnq 9796	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5		8pos 9181	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10557	. . . 4 |- ( ( P e. NN0 /\ 8 e. QQ /\ 0 < 8 ) -> ( ( P mod 8 ) = 5 -> E. k e. NN0 P = ( ( k x. 8 ) + 5 ) ) )
8	1, 4, 6, 7	syl3anc 1252	. . 3 |- ( P e. NN -> ( ( P mod 8 ) = 5 -> E. k e. NN0 P = ( ( k x. 8 ) + 5 ) ) )
9		simpr 110	. . . . 5 |- ( ( P e. NN /\ k e. NN0 ) -> k e. NN0 )
10		nn0cn 9347	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9164	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8136	. . . . . . . . . 10 |- ( k e. NN0 -> ( k x. 8 ) = ( 8 x. k ) )
14	13	adantl 277	. . . . . . . . 9 |- ( ( P e. NN /\ k e. NN0 ) -> ( k x. 8 ) = ( 8 x. k ) )
15	14	oveq1d 5989	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 5 ) = ( ( 8 x. k ) + 5 ) )
16	15	eqeq2d 2221	. . . . . . 7 |- ( ( P e. NN /\ k e. NN0 ) -> ( P = ( ( k x. 8 ) + 5 ) <-> P = ( ( 8 x. k ) + 5 ) ) )
17	16	biimpa 296	. . . . . 6 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 5 ) ) -> P = ( ( 8 x. k ) + 5 ) )
18		2lgslem2.n	. . . . . . 7 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
19	18	2lgslem3c 15739	. . . . . 6 |- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 5 ) ) -> N = ( ( 2 x. k ) + 1 ) )
20	9, 17, 19	syl2an2r 597	. . . . 5 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 5 ) ) -> N = ( ( 2 x. k ) + 1 ) )
21		oveq1 5981	. . . . . 6 |- ( N = ( ( 2 x. k ) + 1 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 1 ) mod 2 ) )
22		nn0z 9434	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
23		eqidd 2210	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. k ) + 1 ) )
24		2tp1odd 12361	. . . . . . . 8 |- ( ( k e. ZZ /\ ( ( 2 x. k ) + 1 ) = ( ( 2 x. k ) + 1 ) ) -> -. 2 || ( ( 2 x. k ) + 1 ) )
25	22, 23, 24	syl2anc 411	. . . . . . 7 |- ( k e. NN0 -> -. 2 || ( ( 2 x. k ) + 1 ) )
26		2z 9442	. . . . . . . . . . 11 |- 2 e. ZZ
27	26	a1i 9	. . . . . . . . . 10 |- ( k e. NN0 -> 2 e. ZZ )
28	27, 22	zmulcld 9543	. . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. k ) e. ZZ )
29	28	peano2zd 9540	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) e. ZZ )
30		mod2eq1n2dvds 12356	. . . . . . . 8 |- ( ( ( 2 x. k ) + 1 ) e. ZZ -> ( ( ( ( 2 x. k ) + 1 ) mod 2 ) = 1 <-> -. 2 || ( ( 2 x. k ) + 1 ) ) )
31	29, 30	syl 14	. . . . . . 7 |- ( k e. NN0 -> ( ( ( ( 2 x. k ) + 1 ) mod 2 ) = 1 <-> -. 2 || ( ( 2 x. k ) + 1 ) ) )
32	25, 31	mpbird 167	. . . . . 6 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 1 ) mod 2 ) = 1 )
33	21, 32	sylan9eqr 2264	. . . . 5 |- ( ( k e. NN0 /\ N = ( ( 2 x. k ) + 1 ) ) -> ( N mod 2 ) = 1 )
34	9, 20, 33	syl2an2r 597	. . . 4 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 5 ) ) -> ( N mod 2 ) = 1 )
35	34	rexlimdva2 2631	. . 3 |- ( P e. NN -> ( E. k e. NN0 P = ( ( k x. 8 ) + 5 ) -> ( N mod 2 ) = 1 ) )
36	8, 35	syld 45	. 2 |- ( P e. NN -> ( ( P mod 8 ) = 5 -> ( N mod 2 ) = 1 ) )
37	36	imp 124	1 |- ( ( P e. NN /\ ( P mod 8 ) = 5 ) -> ( N mod 2 ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   e. wcel 2180   E.wrex 2489   class class class wbr 4062   ` cfv 5294  (class class class)co 5974   CCcc 7965   0cc0 7967   1c1 7968   + caddc 7970   x. cmul 7972   < clt 8149   - cmin 8285   / cdiv 8787   NNcn 9078   2c2 9129   4c4 9131   5c5 9132   8c8 9135   NN0cn0 9337   ZZcz 9414   QQcq 9782   |_cfl 10455   mod cmo 10511   || cdvds 12264

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086

This theorem depends on definitions:  df-bi 117  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-xor 1398  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-po 4364  df-iso 4365  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-n0 9338  df-z 9415  df-uz 9691  df-q 9783  df-rp 9818  df-ico 10058  df-fz 10173  df-fl 10457  df-mod 10512  df-dvds 12265

This theorem is referenced by:  2lgslem3  15745