Theorem 2lgslem3b1 16102

Description: Lemma 2 for 2lgslem3 16105. (Contributed by AV, 16-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
2lgslem3b1	|- ( ( P e. NN /\ ( P mod 8 ) = 3 ) -> ( N mod 2 ) = 1 )

Proof of Theorem 2lgslem3b1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 9524	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9426	. . . . 5 |- 8 e. NN
3		nnq 9987	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5		8pos 9361	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10758	. . . 4 |- ( ( P e. NN0 /\ 8 e. QQ /\ 0 < 8 ) -> ( ( P mod 8 ) = 3 -> E. k e. NN0 P = ( ( k x. 8 ) + 3 ) ) )
8	1, 4, 6, 7	syl3anc 1274	. . 3 |- ( P e. NN -> ( ( P mod 8 ) = 3 -> E. k e. NN0 P = ( ( k x. 8 ) + 3 ) ) )
9		simpr 110	. . . . 5 |- ( ( P e. NN /\ k e. NN0 ) -> k e. NN0 )
10		nn0cn 9527	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9344	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8312	. . . . . . . . . 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 6074	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 3 ) = ( ( 8 x. k ) + 3 ) )
16	15	eqeq2d 2246	. . . . . . 7 |- ( ( P e. NN /\ k e. NN0 ) -> ( P = ( ( k x. 8 ) + 3 ) <-> P = ( ( 8 x. k ) + 3 ) ) )
17	16	biimpa 296	. . . . . 6 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 3 ) ) -> P = ( ( 8 x. k ) + 3 ) )
18		2lgslem2.n	. . . . . . 7 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
19	18	2lgslem3b 16098	. . . . . 6 |- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 3 ) ) -> N = ( ( 2 x. k ) + 1 ) )
20	9, 17, 19	syl2an2r 599	. . . . 5 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 3 ) ) -> N = ( ( 2 x. k ) + 1 ) )
21		oveq1 6066	. . . . . 6 |- ( N = ( ( 2 x. k ) + 1 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 1 ) mod 2 ) )
22		nn0z 9618	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
23		eqidd 2235	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. k ) + 1 ) )
24		2tp1odd 12600	. . . . . . . 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 9626	. . . . . . . . . . 11 |- 2 e. ZZ
27	26	a1i 9	. . . . . . . . . 10 |- ( k e. NN0 -> 2 e. ZZ )
28	27, 22	zmulcld 9728	. . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. k ) e. ZZ )
29	28	peano2zd 9725	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) e. ZZ )
30		mod2eq1n2dvds 12595	. . . . . . . 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 2289	. . . . 5 |- ( ( k e. NN0 /\ N = ( ( 2 x. k ) + 1 ) ) -> ( N mod 2 ) = 1 )
34	9, 20, 33	syl2an2r 599	. . . 4 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 3 ) ) -> ( N mod 2 ) = 1 )
35	34	rexlimdva2 2665	. . 3 |- ( P e. NN -> ( E. k e. NN0 P = ( ( k x. 8 ) + 3 ) -> ( N mod 2 ) = 1 ) )
36	8, 35	syld 45	. 2 |- ( P e. NN -> ( ( P mod 8 ) = 3 -> ( N mod 2 ) = 1 ) )
37	36	imp 124	1 |- ( ( P e. NN /\ ( P mod 8 ) = 3 ) -> ( N mod 2 ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   E.wrex 2523   class class class wbr 4115   ` cfv 5358  (class class class)co 6059   CCcc 8142   0cc0 8144   1c1 8145   + caddc 8147   x. cmul 8149   < clt 8325   - cmin 8462   / cdiv 8967   NNcn 9258   2c2 9309   3c3 9310   4c4 9311   8c8 9315   NN0cn0 9517   ZZcz 9598   QQcq 9973   |_cfl 10656   mod cmo 10712   || cdvds 12503

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-mulrcl 8243  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-precex 8254  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-apti 8259  ax-pre-ltadd 8260  ax-pre-mulgt0 8261  ax-pre-mulext 8262  ax-arch 8263

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-po 4423  df-iso 4424  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-reap 8868  df-ap 8875  df-div 8968  df-inn 9259  df-2 9317  df-3 9318  df-4 9319  df-5 9320  df-6 9321  df-7 9322  df-8 9323  df-n0 9518  df-z 9599  df-uz 9876  df-q 9974  df-rp 10009  df-ico 10250  df-fz 10366  df-fl 10658  df-mod 10713  df-dvds 12504

This theorem is referenced by:  2lgslem3  16105