Theorem 2lgslem3b1 15619

Description: Lemma 2 for 2lgslem3 15622. (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 9309	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9211	. . . . 5 |- 8 e. NN
3		nnq 9761	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5		8pos 9146	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10520	. . . 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 1250	. . 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 9312	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9129	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8101	. . . . . . . . . 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 5966	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 3 ) = ( ( 8 x. k ) + 3 ) )
16	15	eqeq2d 2218	. . . . . . 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 15615	. . . . . 6 |- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 3 ) ) -> N = ( ( 2 x. k ) + 1 ) )
20	9, 17, 19	syl2an2r 595	. . . . 5 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 3 ) ) -> N = ( ( 2 x. k ) + 1 ) )
21		oveq1 5958	. . . . . 6 |- ( N = ( ( 2 x. k ) + 1 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 1 ) mod 2 ) )
22		nn0z 9399	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
23		eqidd 2207	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. k ) + 1 ) )
24		2tp1odd 12239	. . . . . . . 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 9407	. . . . . . . . . . 11 |- 2 e. ZZ
27	26	a1i 9	. . . . . . . . . 10 |- ( k e. NN0 -> 2 e. ZZ )
28	27, 22	zmulcld 9508	. . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. k ) e. ZZ )
29	28	peano2zd 9505	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) e. ZZ )
30		mod2eq1n2dvds 12234	. . . . . . . 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 2261	. . . . 5 |- ( ( k e. NN0 /\ N = ( ( 2 x. k ) + 1 ) ) -> ( N mod 2 ) = 1 )
34	9, 20, 33	syl2an2r 595	. . . 4 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 3 ) ) -> ( N mod 2 ) = 1 )
35	34	rexlimdva2 2627	. . 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 1373   e. wcel 2177   E.wrex 2486   class class class wbr 4047   ` cfv 5276  (class class class)co 5951   CCcc 7930   0cc0 7932   1c1 7933   + caddc 7935   x. cmul 7937   < clt 8114   - cmin 8250   / cdiv 8752   NNcn 9043   2c2 9094   3c3 9095   4c4 9096   8c8 9100   NN0cn0 9302   ZZcz 9379   QQcq 9747   |_cfl 10418   mod cmo 10474   || cdvds 12142

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-po 4347  df-iso 4348  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-ico 10023  df-fz 10138  df-fl 10420  df-mod 10475  df-dvds 12143

This theorem is referenced by:  2lgslem3  15622