Theorem 2lgslem3b1 15897

Description: Lemma 2 for 2lgslem3 15900. (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 9452	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9354	. . . . 5 |- 8 e. NN
3		nnq 9910	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5		8pos 9289	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10674	. . . 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 9455	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9272	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8244	. . . . . . . . . 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 6043	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 3 ) = ( ( 8 x. k ) + 3 ) )
16	15	eqeq2d 2243	. . . . . . 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 15893	. . . . . 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 6035	. . . . . 6 |- ( N = ( ( 2 x. k ) + 1 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 1 ) mod 2 ) )
22		nn0z 9542	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
23		eqidd 2232	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. k ) + 1 ) )
24		2tp1odd 12506	. . . . . . . 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 9550	. . . . . . . . . . 11 |- 2 e. ZZ
27	26	a1i 9	. . . . . . . . . 10 |- ( k e. NN0 -> 2 e. ZZ )
28	27, 22	zmulcld 9651	. . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. k ) e. ZZ )
29	28	peano2zd 9648	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) e. ZZ )
30		mod2eq1n2dvds 12501	. . . . . . . 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 2286	. . . . 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 2654	. . 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 2202   E.wrex 2512   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8257   - cmin 8393   / cdiv 8895   NNcn 9186   2c2 9237   3c3 9238   4c4 9239   8c8 9243   NN0cn0 9445   ZZcz 9522   QQcq 9896   |_cfl 10572   mod cmo 10628   || cdvds 12409

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-n0 9446  df-z 9523  df-uz 9799  df-q 9897  df-rp 9932  df-ico 10172  df-fz 10287  df-fl 10574  df-mod 10629  df-dvds 12410

This theorem is referenced by:  2lgslem3  15900