Theorem 2lgslem3b1 16434

Description: Lemma 2 for 2lgslem3 16437. (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 9582	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9484	. . . . 5 |- 8 e. NN
3		nnq 10050	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5		8pos 9417	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10827	. . . 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 1278	. . 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 9585	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9400	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8354	. . . . . . . . . 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 6103	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 3 ) = ( ( 8 x. k ) + 3 ) )
16	15	eqeq2d 2250	. . . . . . 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 16430	. . . . . 6 |- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 3 ) ) -> N = ( ( 2 x. k ) + 1 ) )
20	9, 17, 19	syl2an2r 603	. . . . 5 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 3 ) ) -> N = ( ( 2 x. k ) + 1 ) )
21		oveq1 6095	. . . . . 6 |- ( N = ( ( 2 x. k ) + 1 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 1 ) mod 2 ) )
22		nn0z 9676	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
23		eqidd 2239	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. k ) + 1 ) )
24		2tp1odd 12677	. . . . . . . 8 |- ( ( k e. ZZ /\ ( ( 2 x. k ) + 1 ) = ( ( 2 x. k ) + 1 ) ) -> -. 2 || ( ( 2 x. k ) + 1 ) )
25	22, 23, 24	syl2anc 415	. . . . . . 7 |- ( k e. NN0 -> -. 2 || ( ( 2 x. k ) + 1 ) )
26		2z 9684	. . . . . . . . . . 11 |- 2 e. ZZ
27	26	a1i 9	. . . . . . . . . 10 |- ( k e. NN0 -> 2 e. ZZ )
28	27, 22	zmulcld 9786	. . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. k ) e. ZZ )
29	28	peano2zd 9783	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 1 ) e. ZZ )
30		mod2eq1n2dvds 12672	. . . . . . . 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 2293	. . . . 5 |- ( ( k e. NN0 /\ N = ( ( 2 x. k ) + 1 ) ) -> ( N mod 2 ) = 1 )
34	9, 20, 33	syl2an2r 603	. . . 4 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 3 ) ) -> ( N mod 2 ) = 1 )
35	34	rexlimdva2 2671	. . 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 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8184   0cc0 8186   1c1 8187   + caddc 8189   x. cmul 8191   < clt 8367   - cmin 8505   / cdiv 9011   NNcn 9314   2c2 9365   3c3 9366   4c4 9367   8c8 9371   NN0cn0 9575   ZZcz 9656   QQcq 10036   |_cfl 10721   mod cmo 10781   || cdvds 12580

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-5 9376  df-6 9377  df-7 9378  df-8 9379  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-ico 10314  df-fz 10430  df-fl 10723  df-mod 10782  df-dvds 12581

This theorem is used by:  2lgslem3  16437