Theorem 2lgslem3a1 16082

Description: Lemma 1 for 2lgslem3 16086. (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 9520	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9422	. . . . 5 |- 8 e. NN
3		nnq 9983	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5		8pos 9357	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10754	. . . 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 1274	. . 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 9523	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9340	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8311	. . . . . . . . . 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 6073	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 1 ) = ( ( 8 x. k ) + 1 ) )
16	15	eqeq2d 2246	. . . . . . 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 16078	. . . . . 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 6065	. . . . . 6 |- ( N = ( 2 x. k ) -> ( N mod 2 ) = ( ( 2 x. k ) mod 2 ) )
22		2cnd 9327	. . . . . . . . 9 |- ( k e. NN0 -> 2 e. CC )
23	22, 10	mulcomd 8311	. . . . . . . 8 |- ( k e. NN0 -> ( 2 x. k ) = ( k x. 2 ) )
24	23	oveq1d 6073	. . . . . . 7 |- ( k e. NN0 -> ( ( 2 x. k ) mod 2 ) = ( ( k x. 2 ) mod 2 ) )
25		nn0z 9614	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
26		2nn 9416	. . . . . . . . 9 |- 2 e. NN
27		nnq 9983	. . . . . . . . 9 |- ( 2 e. NN -> 2 e. QQ )
28	26, 27	mp1i 10	. . . . . . . 8 |- ( k e. NN0 -> 2 e. QQ )
29		2pos 9345	. . . . . . . . 9 |- 0 < 2
30	29	a1i 9	. . . . . . . 8 |- ( k e. NN0 -> 0 < 2 )
31		mulqmod0 10716	. . . . . . . 8 |- ( ( k e. ZZ /\ 2 e. QQ /\ 0 < 2 ) -> ( ( k x. 2 ) mod 2 ) = 0 )
32	25, 28, 30, 31	syl3anc 1274	. . . . . . 7 |- ( k e. NN0 -> ( ( k x. 2 ) mod 2 ) = 0 )
33	24, 32	eqtrd 2267	. . . . . 6 |- ( k e. NN0 -> ( ( 2 x. k ) mod 2 ) = 0 )
34	21, 33	sylan9eqr 2289	. . . . 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 2665	. . 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 1398   e. wcel 2205   E.wrex 2523   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   - cmin 8460   / cdiv 8963   NNcn 9254   2c2 9305   4c4 9307   8c8 9311   NN0cn0 9513   ZZcz 9594   QQcq 9969   |_cfl 10652   mod cmo 10708

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  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 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-n0 9514  df-z 9595  df-q 9970  df-rp 10005  df-ico 10246  df-fl 10654  df-mod 10709

This theorem is referenced by:  2lgslem3  16086