Theorem 2lgslem3d1 15832

Description: Lemma 4 for 2lgslem3 15833. (Contributed by AV, 15-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
2lgslem3d1	|- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( N mod 2 ) = 0 )

Proof of Theorem 2lgslem3d1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 9409	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9311	. . . . 5 |- 8 e. NN
3		nnq 9867	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5	2	nngt0i 9173	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10631	. . . 4 |- ( ( P e. NN0 /\ 8 e. QQ /\ 0 < 8 ) -> ( ( P mod 8 ) = 7 -> E. k e. NN0 P = ( ( k x. 8 ) + 7 ) ) )
8	1, 4, 6, 7	syl3anc 1273	. . 3 |- ( P e. NN -> ( ( P mod 8 ) = 7 -> E. k e. NN0 P = ( ( k x. 8 ) + 7 ) ) )
9		simpr 110	. . . . 5 |- ( ( P e. NN /\ k e. NN0 ) -> k e. NN0 )
10		nn0cn 9412	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9229	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8201	. . . . . . . . . 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 6033	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 7 ) = ( ( 8 x. k ) + 7 ) )
16	15	eqeq2d 2243	. . . . . . 7 |- ( ( P e. NN /\ k e. NN0 ) -> ( P = ( ( k x. 8 ) + 7 ) <-> P = ( ( 8 x. k ) + 7 ) ) )
17	16	biimpa 296	. . . . . 6 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> P = ( ( 8 x. k ) + 7 ) )
18		2lgslem2.n	. . . . . . 7 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
19	18	2lgslem3d 15828	. . . . . 6 |- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) )
20	9, 17, 19	syl2an2r 599	. . . . 5 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) )
21		oveq1 6025	. . . . . 6 |- ( N = ( ( 2 x. k ) + 2 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 2 ) mod 2 ) )
22		2t1e2 9297	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = 2
23	22	eqcomi 2235	. . . . . . . . . . 11 |- 2 = ( 2 x. 1 )
24	23	a1i 9	. . . . . . . . . 10 |- ( k e. NN0 -> 2 = ( 2 x. 1 ) )
25	24	oveq2d 6034	. . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
26		2cnd 9216	. . . . . . . . . 10 |- ( k e. NN0 -> 2 e. CC )
27		1cnd 8195	. . . . . . . . . 10 |- ( k e. NN0 -> 1 e. CC )
28		adddi 8164	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ k e. CC /\ 1 e. CC ) -> ( 2 x. ( k + 1 ) ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
29	28	eqcomd 2237	. . . . . . . . . 10 |- ( ( 2 e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( 2 x. k ) + ( 2 x. 1 ) ) = ( 2 x. ( k + 1 ) ) )
30	26, 10, 27, 29	syl3anc 1273	. . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + ( 2 x. 1 ) ) = ( 2 x. ( k + 1 ) ) )
31	10, 27	addcld 8199	. . . . . . . . . 10 |- ( k e. NN0 -> ( k + 1 ) e. CC )
32	26, 31	mulcomd 8201	. . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. ( k + 1 ) ) = ( ( k + 1 ) x. 2 ) )
33	25, 30, 32	3eqtrd 2268	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( k + 1 ) x. 2 ) )
34	33	oveq1d 6033	. . . . . . 7 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = ( ( ( k + 1 ) x. 2 ) mod 2 ) )
35		peano2nn0 9442	. . . . . . . . 9 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
36	35	nn0zd 9600	. . . . . . . 8 |- ( k e. NN0 -> ( k + 1 ) e. ZZ )
37		2nn 9305	. . . . . . . . 9 |- 2 e. NN
38		nnq 9867	. . . . . . . . 9 |- ( 2 e. NN -> 2 e. QQ )
39	37, 38	mp1i 10	. . . . . . . 8 |- ( k e. NN0 -> 2 e. QQ )
40	37	nngt0i 9173	. . . . . . . . 9 |- 0 < 2
41	40	a1i 9	. . . . . . . 8 |- ( k e. NN0 -> 0 < 2 )
42		mulqmod0 10593	. . . . . . . 8 |- ( ( ( k + 1 ) e. ZZ /\ 2 e. QQ /\ 0 < 2 ) -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
43	36, 39, 41, 42	syl3anc 1273	. . . . . . 7 |- ( k e. NN0 -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
44	34, 43	eqtrd 2264	. . . . . 6 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = 0 )
45	21, 44	sylan9eqr 2286	. . . . 5 |- ( ( k e. NN0 /\ N = ( ( 2 x. k ) + 2 ) ) -> ( N mod 2 ) = 0 )
46	9, 20, 45	syl2an2r 599	. . . 4 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> ( N mod 2 ) = 0 )
47	46	rexlimdva2 2653	. . 3 |- ( P e. NN -> ( E. k e. NN0 P = ( ( k x. 8 ) + 7 ) -> ( N mod 2 ) = 0 ) )
48	8, 47	syld 45	. 2 |- ( P e. NN -> ( ( P mod 8 ) = 7 -> ( N mod 2 ) = 0 ) )
49	48	imp 124	1 |- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( N mod 2 ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   < clt 8214   - cmin 8350   / cdiv 8852   NNcn 9143   2c2 9194   4c4 9196   7c7 9199   8c8 9200   NN0cn0 9402   ZZcz 9479   QQcq 9853   |_cfl 10529   mod cmo 10585

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-n0 9403  df-z 9480  df-q 9854  df-rp 9889  df-ico 10129  df-fl 10531  df-mod 10586

This theorem is referenced by:  2lgslem3  15833