Theorem 2lgslem3a 15953

Description: Lemma for 2lgslem3a1 15957. (Contributed by AV, 14-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
2lgslem3a	|- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 1 ) ) -> N = ( 2 x. K ) )

Proof of Theorem 2lgslem3a

Step	Hyp	Ref	Expression
1		2lgslem2.n	. . 3 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
2		oveq1 6056	. . . . 5 |- ( P = ( ( 8 x. K ) + 1 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 1 ) - 1 ) )
3	2	oveq1d 6064	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) )
4		fvoveq1 6072	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) )
5	3, 4	oveq12d 6067	. . 3 |- ( P = ( ( 8 x. K ) + 1 ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) )
6	1, 5	eqtrid 2277	. 2 |- ( P = ( ( 8 x. K ) + 1 ) -> N = ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) )
7		8nn0 9515	. . . . . . . . . 10 |- 8 e. NN0
8	7	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 8 e. NN0 )
9		id 19	. . . . . . . . 9 |- ( K e. NN0 -> K e. NN0 )
10	8, 9	nn0mulcld 9554	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9551	. . . . . . 7 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		pncan1 8646	. . . . . . 7 |- ( ( 8 x. K ) e. CC -> ( ( ( 8 x. K ) + 1 ) - 1 ) = ( 8 x. K ) )
13	11, 12	syl 14	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) - 1 ) = ( 8 x. K ) )
14	13	oveq1d 6064	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( ( 8 x. K ) / 2 ) )
15		4cn 9311	. . . . . . . . . . 11 |- 4 e. CC
16		2cn 9304	. . . . . . . . . . 11 |- 2 e. CC
17		4t2e8 9392	. . . . . . . . . . 11 |- ( 4 x. 2 ) = 8
18	15, 16, 17	mulcomli 8277	. . . . . . . . . 10 |- ( 2 x. 4 ) = 8
19	18	eqcomi 2236	. . . . . . . . 9 |- 8 = ( 2 x. 4 )
20	19	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 8 = ( 2 x. 4 ) )
21	20	oveq1d 6064	. . . . . . 7 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 2 x. 4 ) x. K ) )
22	16	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 2 e. CC )
23	15	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. CC )
24		nn0cn 9502	. . . . . . . 8 |- ( K e. NN0 -> K e. CC )
25	22, 23, 24	mulassd 8293	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 4 ) x. K ) = ( 2 x. ( 4 x. K ) ) )
26	21, 25	eqtrd 2265	. . . . . 6 |- ( K e. NN0 -> ( 8 x. K ) = ( 2 x. ( 4 x. K ) ) )
27	26	oveq1d 6064	. . . . 5 |- ( K e. NN0 -> ( ( 8 x. K ) / 2 ) = ( ( 2 x. ( 4 x. K ) ) / 2 ) )
28		4nn0 9511	. . . . . . . . 9 |- 4 e. NN0
29	28	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN0 )
30	29, 9	nn0mulcld 9554	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
31	30	nn0cnd 9551	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
32		2ap0 9326	. . . . . . 7 |- 2 # 0
33	32	a1i 9	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
34	31, 22, 33	divcanap3d 9065	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. ( 4 x. K ) ) / 2 ) = ( 4 x. K ) )
35	14, 27, 34	3eqtrd 2269	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( 4 x. K ) )
36		1cnd 8286	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
37		4ap0 9332	. . . . . . . . 9 |- 4 # 0
38	37	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
39	11, 36, 23, 38	divdirapd 9099	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) )
40		8cn 9319	. . . . . . . . . . 11 |- 8 e. CC
41	40	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
42	41, 24, 23, 38	div23apd 9098	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
43	17	eqcomi 2236	. . . . . . . . . . . . 13 |- 8 = ( 4 x. 2 )
44	43	oveq1i 6059	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
45	16, 15, 37	divcanap3i 9028	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
46	44, 45	eqtri 2253	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
47	46	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
48	47	oveq1d 6064	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
49	42, 48	eqtrd 2265	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
50	49	oveq1d 6064	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
51	39, 50	eqtrd 2265	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
52	51	fveq2d 5673	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) )
53		1lt4 9408	. . . . . 6 |- 1 < 4
54		2nn0 9509	. . . . . . . . . 10 |- 2 e. NN0
55	54	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 2 e. NN0 )
56	55, 9	nn0mulcld 9554	. . . . . . . 8 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
57	56	nn0zd 9694	. . . . . . 7 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
58		1nn0 9508	. . . . . . . 8 |- 1 e. NN0
59	58	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 1 e. NN0 )
60		4nn 9397	. . . . . . . 8 |- 4 e. NN
61	60	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 4 e. NN )
62		adddivflid 10648	. . . . . . 7 |- ( ( ( 2 x. K ) e. ZZ /\ 1 e. NN0 /\ 4 e. NN ) -> ( 1 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) = ( 2 x. K ) ) )
63	57, 59, 61, 62	syl3anc 1274	. . . . . 6 |- ( K e. NN0 -> ( 1 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) = ( 2 x. K ) ) )
64	53, 63	mpbii 148	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) = ( 2 x. K ) )
65	52, 64	eqtrd 2265	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( 2 x. K ) )
66	35, 65	oveq12d 6067	. . 3 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( ( 4 x. K ) - ( 2 x. K ) ) )
67		2t2e4 9388	. . . . . . . 8 |- ( 2 x. 2 ) = 4
68	67	eqcomi 2236	. . . . . . 7 |- 4 = ( 2 x. 2 )
69	68	a1i 9	. . . . . 6 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
70	69	oveq1d 6064	. . . . 5 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
71	22, 22, 24	mulassd 8293	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
72	70, 71	eqtrd 2265	. . . 4 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
73	72	oveq1d 6064	. . 3 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
74	56	nn0cnd 9551	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
75		2txmxeqx 9365	. . . 4 |- ( ( 2 x. K ) e. CC -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
76	74, 75	syl 14	. . 3 |- ( K e. NN0 -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
77	66, 73, 76	3eqtrd 2269	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( 2 x. K ) )
78	6, 77	sylan9eqr 2287	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 1 ) ) -> N = ( 2 x. K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   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   # cap 8851   / cdiv 8942   NNcn 9233   2c2 9284   4c4 9286   8c8 9290   NN0cn0 9492   ZZcz 9573   |_cfl 10624

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-fl 10626

This theorem is referenced by:  2lgslem3a1  15957