Theorem 2lgslem3d1 15960

Description: Lemma 4 for 2lgslem3 15961. (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 9499	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9401	. . . . 5 |- 8 e. NN
3		nnq 9961	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5	2	nngt0i 9263	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10726	. . . 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 1274	. . 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 9502	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9319	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8291	. . . . . . . . . 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 6064	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 7 ) = ( ( 8 x. k ) + 7 ) )
16	15	eqeq2d 2244	. . . . . . 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 15956	. . . . . 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 6056	. . . . . 6 |- ( N = ( ( 2 x. k ) + 2 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 2 ) mod 2 ) )
22		2t1e2 9387	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = 2
23	22	eqcomi 2236	. . . . . . . . . . 11 |- 2 = ( 2 x. 1 )
24	23	a1i 9	. . . . . . . . . 10 |- ( k e. NN0 -> 2 = ( 2 x. 1 ) )
25	24	oveq2d 6065	. . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
26		2cnd 9306	. . . . . . . . . 10 |- ( k e. NN0 -> 2 e. CC )
27		1cnd 8286	. . . . . . . . . 10 |- ( k e. NN0 -> 1 e. CC )
28		adddi 8255	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ k e. CC /\ 1 e. CC ) -> ( 2 x. ( k + 1 ) ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
29	28	eqcomd 2238	. . . . . . . . . 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 1274	. . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + ( 2 x. 1 ) ) = ( 2 x. ( k + 1 ) ) )
31	10, 27	addcld 8289	. . . . . . . . . 10 |- ( k e. NN0 -> ( k + 1 ) e. CC )
32	26, 31	mulcomd 8291	. . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. ( k + 1 ) ) = ( ( k + 1 ) x. 2 ) )
33	25, 30, 32	3eqtrd 2269	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( k + 1 ) x. 2 ) )
34	33	oveq1d 6064	. . . . . . 7 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = ( ( ( k + 1 ) x. 2 ) mod 2 ) )
35		peano2nn0 9532	. . . . . . . . 9 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
36	35	nn0zd 9694	. . . . . . . 8 |- ( k e. NN0 -> ( k + 1 ) e. ZZ )
37		2nn 9395	. . . . . . . . 9 |- 2 e. NN
38		nnq 9961	. . . . . . . . 9 |- ( 2 e. NN -> 2 e. QQ )
39	37, 38	mp1i 10	. . . . . . . 8 |- ( k e. NN0 -> 2 e. QQ )
40	37	nngt0i 9263	. . . . . . . . 9 |- 0 < 2
41	40	a1i 9	. . . . . . . 8 |- ( k e. NN0 -> 0 < 2 )
42		mulqmod0 10688	. . . . . . . 8 |- ( ( ( k + 1 ) e. ZZ /\ 2 e. QQ /\ 0 < 2 ) -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
43	36, 39, 41, 42	syl3anc 1274	. . . . . . 7 |- ( k e. NN0 -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
44	34, 43	eqtrd 2265	. . . . . 6 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = 0 )
45	21, 44	sylan9eqr 2287	. . . . 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 2663	. . 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 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   CCcc 8121   0cc0 8123   1c1 8124   + caddc 8126   x. cmul 8128   < clt 8304   - cmin 8440   / cdiv 8942   NNcn 9233   2c2 9284   4c4 9286   7c7 9289   8c8 9290   NN0cn0 9492   ZZcz 9573   QQcq 9947   |_cfl 10624   mod cmo 10680

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-n0 9493  df-z 9574  df-q 9948  df-rp 9983  df-ico 10223  df-fl 10626  df-mod 10681

This theorem is referenced by:  2lgslem3  15961