Theorem 2lgslem3d1 16202

Description: Lemma 4 for 2lgslem3 16203. (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 9553	. . . 4 |- ( P e. NN -> P e. NN0 )
2		8nn 9455	. . . . 5 |- 8 e. NN
3		nnq 10016	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( P e. NN -> 8 e. QQ )
5	2	nngt0i 9317	. . . . 5 |- 0 < 8
6	5	a1i 9	. . . 4 |- ( P e. NN -> 0 < 8 )
7		modqmuladdnn0 10788	. . . 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 1278	. . 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 9556	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11		8cn 9373	. . . . . . . . . . . 12 |- 8 e. CC
12	11	a1i 9	. . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
13	10, 12	mulcomd 8341	. . . . . . . . . 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 6094	. . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 7 ) = ( ( 8 x. k ) + 7 ) )
16	15	eqeq2d 2250	. . . . . . 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 16198	. . . . . 6 |- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) )
20	9, 17, 19	syl2an2r 603	. . . . 5 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) )
21		oveq1 6086	. . . . . 6 |- ( N = ( ( 2 x. k ) + 2 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 2 ) mod 2 ) )
22		2t1e2 9441	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = 2
23	22	eqcomi 2242	. . . . . . . . . . 11 |- 2 = ( 2 x. 1 )
24	23	a1i 9	. . . . . . . . . 10 |- ( k e. NN0 -> 2 = ( 2 x. 1 ) )
25	24	oveq2d 6095	. . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
26		2cnd 9360	. . . . . . . . . 10 |- ( k e. NN0 -> 2 e. CC )
27		1cnd 8336	. . . . . . . . . 10 |- ( k e. NN0 -> 1 e. CC )
28		adddi 8305	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ k e. CC /\ 1 e. CC ) -> ( 2 x. ( k + 1 ) ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
29	28	eqcomd 2244	. . . . . . . . . 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 1278	. . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + ( 2 x. 1 ) ) = ( 2 x. ( k + 1 ) ) )
31	10, 27	addcld 8339	. . . . . . . . . 10 |- ( k e. NN0 -> ( k + 1 ) e. CC )
32	26, 31	mulcomd 8341	. . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. ( k + 1 ) ) = ( ( k + 1 ) x. 2 ) )
33	25, 30, 32	3eqtrd 2275	. . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( k + 1 ) x. 2 ) )
34	33	oveq1d 6094	. . . . . . 7 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = ( ( ( k + 1 ) x. 2 ) mod 2 ) )
35		peano2nn0 9586	. . . . . . . . 9 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
36	35	nn0zd 9749	. . . . . . . 8 |- ( k e. NN0 -> ( k + 1 ) e. ZZ )
37		2nn 9449	. . . . . . . . 9 |- 2 e. NN
38		nnq 10016	. . . . . . . . 9 |- ( 2 e. NN -> 2 e. QQ )
39	37, 38	mp1i 10	. . . . . . . 8 |- ( k e. NN0 -> 2 e. QQ )
40	37	nngt0i 9317	. . . . . . . . 9 |- 0 < 2
41	40	a1i 9	. . . . . . . 8 |- ( k e. NN0 -> 0 < 2 )
42		mulqmod0 10750	. . . . . . . 8 |- ( ( ( k + 1 ) e. ZZ /\ 2 e. QQ /\ 0 < 2 ) -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
43	36, 39, 41, 42	syl3anc 1278	. . . . . . 7 |- ( k e. NN0 -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
44	34, 43	eqtrd 2271	. . . . . 6 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = 0 )
45	21, 44	sylan9eqr 2293	. . . . 5 |- ( ( k e. NN0 /\ N = ( ( 2 x. k ) + 2 ) ) -> ( N mod 2 ) = 0 )
46	9, 20, 45	syl2an2r 603	. . . 4 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> ( N mod 2 ) = 0 )
47	46	rexlimdva2 2671	. . 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 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   CCcc 8171   0cc0 8173   1c1 8174   + caddc 8176   x. cmul 8178   < clt 8354   - cmin 8491   / cdiv 8996   NNcn 9287   2c2 9338   4c4 9340   7c7 9343   8c8 9344   NN0cn0 9546   ZZcz 9627   QQcq 10002   |_cfl 10686   mod cmo 10742

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 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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-n0 9547  df-z 9628  df-q 10003  df-rp 10038  df-ico 10279  df-fl 10688  df-mod 10743

This theorem is referenced by:  2lgslem3  16203