Theorem 2lgslem3d1 15787

Description: Lemma 4 for 2lgslem3 15788. (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 9384	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9286	. . . . 5 |- 8 e. NN
3		nnq 9836	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5	2	nngt0i 9148	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10598	. . . 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 1271	. . 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 9387	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9204	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8176	. . . . . . . . . 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 6022	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 7 ) = ( ( 8 x. k ) + 7 ) )
16	15	eqeq2d 2241	. . . . . . 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 15783	. . . . . 6 |- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) )
20	9, 17, 19	syl2an2r 597	. . . . 5 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) )
21		oveq1 6014	. . . . . 6 |- ( N = ( ( 2 x. k ) + 2 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 2 ) mod 2 ) )
22		2t1e2 9272	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = 2
23	22	eqcomi 2233	. . . . . . . . . . 11 |- 2 = ( 2 x. 1 )
24	23	a1i 9	. . . . . . . . . 10 |- ( k e. NN0 -> 2 = ( 2 x. 1 ) )
25	24	oveq2d 6023	. . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
26		2cnd 9191	. . . . . . . . . 10 |- ( k e. NN0 -> 2 e. CC )
27		1cnd 8170	. . . . . . . . . 10 |- ( k e. NN0 -> 1 e. CC )
28		adddi 8139	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ k e. CC /\ 1 e. CC ) -> ( 2 x. ( k + 1 ) ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
29	28	eqcomd 2235	. . . . . . . . . 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 1271	. . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + ( 2 x. 1 ) ) = ( 2 x. ( k + 1 ) ) )
31	10, 27	addcld 8174	. . . . . . . . . 10 |- ( k e. NN0 -> ( k + 1 ) e. CC )
32	26, 31	mulcomd 8176	. . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. ( k + 1 ) ) = ( ( k + 1 ) x. 2 ) )
33	25, 30, 32	3eqtrd 2266	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( k + 1 ) x. 2 ) )
34	33	oveq1d 6022	. . . . . . 7 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = ( ( ( k + 1 ) x. 2 ) mod 2 ) )
35		peano2nn0 9417	. . . . . . . . 9 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
36	35	nn0zd 9575	. . . . . . . 8 |- ( k e. NN0 -> ( k + 1 ) e. ZZ )
37		2nn 9280	. . . . . . . . 9 |- 2 e. NN
38		nnq 9836	. . . . . . . . 9 |- ( 2 e. NN -> 2 e. QQ )
39	37, 38	mp1i 10	. . . . . . . 8 |- ( k e. NN0 -> 2 e. QQ )
40	37	nngt0i 9148	. . . . . . . . 9 |- 0 < 2
41	40	a1i 9	. . . . . . . 8 |- ( k e. NN0 -> 0 < 2 )
42		mulqmod0 10560	. . . . . . . 8 |- ( ( ( k + 1 ) e. ZZ /\ 2 e. QQ /\ 0 < 2 ) -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
43	36, 39, 41, 42	syl3anc 1271	. . . . . . 7 |- ( k e. NN0 -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
44	34, 43	eqtrd 2262	. . . . . 6 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = 0 )
45	21, 44	sylan9eqr 2284	. . . . 5 |- ( ( k e. NN0 /\ N = ( ( 2 x. k ) + 2 ) ) -> ( N mod 2 ) = 0 )
46	9, 20, 45	syl2an2r 597	. . . 4 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> ( N mod 2 ) = 0 )
47	46	rexlimdva2 2651	. . 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 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8005   0cc0 8007   1c1 8008   + caddc 8010   x. cmul 8012   < clt 8189   - cmin 8325   / cdiv 8827   NNcn 9118   2c2 9169   4c4 9171   7c7 9174   8c8 9175   NN0cn0 9377   ZZcz 9454   QQcq 9822   |_cfl 10496   mod cmo 10552

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-n0 9378  df-z 9455  df-q 9823  df-rp 9858  df-ico 10098  df-fl 10498  df-mod 10553

This theorem is referenced by:  2lgslem3  15788