Theorem 2lgslem3a1 15852

Description: Lemma 1 for 2lgslem3 15856. (Contributed by AV, 15-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
2lgslem3a1	|- ( ( P e. NN /\ ( P mod 8 ) = 1 ) -> ( N mod 2 ) = 0 )

Proof of Theorem 2lgslem3a1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 9411	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9313	. . . . 5 |- 8 e. NN
3		nnq 9869	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5		8pos 9248	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10633	. . . 4 |- ( ( P e. NN0 /\ 8 e. QQ /\ 0 < 8 ) -> ( ( P mod 8 ) = 1 -> E. k e. NN0 P = ( ( k x. 8 ) + 1 ) ) )
8	1, 4, 6, 7	syl3anc 1273	. . 3 |- ( P e. NN -> ( ( P mod 8 ) = 1 -> E. k e. NN0 P = ( ( k x. 8 ) + 1 ) ) )
9		simpr 110	. . . . 5 |- ( ( P e. NN /\ k e. NN0 ) -> k e. NN0 )
10		nn0cn 9414	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9231	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8203	. . . . . . . . . 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 6035	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 1 ) = ( ( 8 x. k ) + 1 ) )
16	15	eqeq2d 2242	. . . . . . 7 |- ( ( P e. NN /\ k e. NN0 ) -> ( P = ( ( k x. 8 ) + 1 ) <-> P = ( ( 8 x. k ) + 1 ) ) )
17	16	biimpa 296	. . . . . 6 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 1 ) ) -> P = ( ( 8 x. k ) + 1 ) )
18		2lgslem2.n	. . . . . . 7 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
19	18	2lgslem3a 15848	. . . . . 6 |- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 1 ) ) -> N = ( 2 x. k ) )
20	9, 17, 19	syl2an2r 599	. . . . 5 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 1 ) ) -> N = ( 2 x. k ) )
21		oveq1 6027	. . . . . 6 |- ( N = ( 2 x. k ) -> ( N mod 2 ) = ( ( 2 x. k ) mod 2 ) )
22		2cnd 9218	. . . . . . . . 9 |- ( k e. NN0 -> 2 e. CC )
23	22, 10	mulcomd 8203	. . . . . . . 8 |- ( k e. NN0 -> ( 2 x. k ) = ( k x. 2 ) )
24	23	oveq1d 6035	. . . . . . 7 |- ( k e. NN0 -> ( ( 2 x. k ) mod 2 ) = ( ( k x. 2 ) mod 2 ) )
25		nn0z 9501	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
26		2nn 9307	. . . . . . . . 9 |- 2 e. NN
27		nnq 9869	. . . . . . . . 9 |- ( 2 e. NN -> 2 e. QQ )
28	26, 27	mp1i 10	. . . . . . . 8 |- ( k e. NN0 -> 2 e. QQ )
29		2pos 9236	. . . . . . . . 9 |- 0 < 2
30	29	a1i 9	. . . . . . . 8 |- ( k e. NN0 -> 0 < 2 )
31		mulqmod0 10595	. . . . . . . 8 |- ( ( k e. ZZ /\ 2 e. QQ /\ 0 < 2 ) -> ( ( k x. 2 ) mod 2 ) = 0 )
32	25, 28, 30, 31	syl3anc 1273	. . . . . . 7 |- ( k e. NN0 -> ( ( k x. 2 ) mod 2 ) = 0 )
33	24, 32	eqtrd 2263	. . . . . 6 |- ( k e. NN0 -> ( ( 2 x. k ) mod 2 ) = 0 )
34	21, 33	sylan9eqr 2285	. . . . 5 |- ( ( k e. NN0 /\ N = ( 2 x. k ) ) -> ( N mod 2 ) = 0 )
35	9, 20, 34	syl2an2r 599	. . . 4 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 1 ) ) -> ( N mod 2 ) = 0 )
36	35	rexlimdva2 2652	. . 3 |- ( P e. NN -> ( E. k e. NN0 P = ( ( k x. 8 ) + 1 ) -> ( N mod 2 ) = 0 ) )
37	8, 36	syld 45	. 2 |- ( P e. NN -> ( ( P mod 8 ) = 1 -> ( N mod 2 ) = 0 ) )
38	37	imp 124	1 |- ( ( P e. NN /\ ( P mod 8 ) = 1 ) -> ( N mod 2 ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2201   E.wrex 2510   class class class wbr 4087   ` cfv 5325  (class class class)co 6020   CCcc 8032   0cc0 8034   1c1 8035   + caddc 8037   x. cmul 8039   < clt 8216   - cmin 8352   / cdiv 8854   NNcn 9145   2c2 9196   4c4 9198   8c8 9202   NN0cn0 9404   ZZcz 9481   QQcq 9855   |_cfl 10531   mod cmo 10587

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-mulrcl 8133  ax-addcom 8134  ax-mulcom 8135  ax-addass 8136  ax-mulass 8137  ax-distr 8138  ax-i2m1 8139  ax-0lt1 8140  ax-1rid 8141  ax-0id 8142  ax-rnegex 8143  ax-precex 8144  ax-cnre 8145  ax-pre-ltirr 8146  ax-pre-ltwlin 8147  ax-pre-lttrn 8148  ax-pre-apti 8149  ax-pre-ltadd 8150  ax-pre-mulgt0 8151  ax-pre-mulext 8152  ax-arch 8153

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-id 4389  df-po 4392  df-iso 4393  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-fv 5333  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-pnf 8218  df-mnf 8219  df-xr 8220  df-ltxr 8221  df-le 8222  df-sub 8354  df-neg 8355  df-reap 8757  df-ap 8764  df-div 8855  df-inn 9146  df-2 9204  df-3 9205  df-4 9206  df-5 9207  df-6 9208  df-7 9209  df-8 9210  df-n0 9405  df-z 9482  df-q 9856  df-rp 9891  df-ico 10131  df-fl 10533  df-mod 10588

This theorem is referenced by:  2lgslem3  15856