Theorem 2lgslem3a 16015

Description: Lemma for 2lgslem3a1 16019. (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 6059	. . . . 5 |- ( P = ( ( 8 x. K ) + 1 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 1 ) - 1 ) )
3	2	oveq1d 6067	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) )
4		fvoveq1 6075	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) )
5	3, 4	oveq12d 6070	. . 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 2279	. 2 |- ( P = ( ( 8 x. K ) + 1 ) -> N = ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) )
7		8nn0 9524	. . . . . . . . . 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 9563	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9560	. . . . . . 7 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		pncan1 8655	. . . . . . 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 6067	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( ( 8 x. K ) / 2 ) )
15		4cn 9320	. . . . . . . . . . 11 |- 4 e. CC
16		2cn 9313	. . . . . . . . . . 11 |- 2 e. CC
17		4t2e8 9401	. . . . . . . . . . 11 |- ( 4 x. 2 ) = 8
18	15, 16, 17	mulcomli 8286	. . . . . . . . . 10 |- ( 2 x. 4 ) = 8
19	18	eqcomi 2238	. . . . . . . . 9 |- 8 = ( 2 x. 4 )
20	19	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 8 = ( 2 x. 4 ) )
21	20	oveq1d 6067	. . . . . . 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 9511	. . . . . . . 8 |- ( K e. NN0 -> K e. CC )
25	22, 23, 24	mulassd 8302	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 4 ) x. K ) = ( 2 x. ( 4 x. K ) ) )
26	21, 25	eqtrd 2267	. . . . . 6 |- ( K e. NN0 -> ( 8 x. K ) = ( 2 x. ( 4 x. K ) ) )
27	26	oveq1d 6067	. . . . 5 |- ( K e. NN0 -> ( ( 8 x. K ) / 2 ) = ( ( 2 x. ( 4 x. K ) ) / 2 ) )
28		4nn0 9520	. . . . . . . . 9 |- 4 e. NN0
29	28	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN0 )
30	29, 9	nn0mulcld 9563	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
31	30	nn0cnd 9560	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
32		2ap0 9335	. . . . . . 7 |- 2 # 0
33	32	a1i 9	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
34	31, 22, 33	divcanap3d 9074	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. ( 4 x. K ) ) / 2 ) = ( 4 x. K ) )
35	14, 27, 34	3eqtrd 2271	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( 4 x. K ) )
36		1cnd 8295	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
37		4ap0 9341	. . . . . . . . 9 |- 4 # 0
38	37	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
39	11, 36, 23, 38	divdirapd 9108	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) )
40		8cn 9328	. . . . . . . . . . 11 |- 8 e. CC
41	40	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
42	41, 24, 23, 38	div23apd 9107	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
43	17	eqcomi 2238	. . . . . . . . . . . . 13 |- 8 = ( 4 x. 2 )
44	43	oveq1i 6062	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
45	16, 15, 37	divcanap3i 9037	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
46	44, 45	eqtri 2255	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
47	46	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
48	47	oveq1d 6067	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
49	42, 48	eqtrd 2267	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
50	49	oveq1d 6067	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
51	39, 50	eqtrd 2267	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
52	51	fveq2d 5676	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) )
53		1lt4 9417	. . . . . 6 |- 1 < 4
54		2nn0 9518	. . . . . . . . . 10 |- 2 e. NN0
55	54	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 2 e. NN0 )
56	55, 9	nn0mulcld 9563	. . . . . . . 8 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
57	56	nn0zd 9704	. . . . . . 7 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
58		1nn0 9517	. . . . . . . 8 |- 1 e. NN0
59	58	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 1 e. NN0 )
60		4nn 9406	. . . . . . . 8 |- 4 e. NN
61	60	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 4 e. NN )
62		adddivflid 10659	. . . . . . 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 2267	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( 2 x. K ) )
66	35, 65	oveq12d 6070	. . 3 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( ( 4 x. K ) - ( 2 x. K ) ) )
67		2t2e4 9397	. . . . . . . 8 |- ( 2 x. 2 ) = 4
68	67	eqcomi 2238	. . . . . . 7 |- 4 = ( 2 x. 2 )
69	68	a1i 9	. . . . . 6 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
70	69	oveq1d 6067	. . . . 5 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
71	22, 22, 24	mulassd 8302	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
72	70, 71	eqtrd 2267	. . . 4 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
73	72	oveq1d 6067	. . 3 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
74	56	nn0cnd 9560	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
75		2txmxeqx 9374	. . . 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 2271	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( 2 x. K ) )
78	6, 77	sylan9eqr 2289	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 2205   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   CCcc 8130   0cc0 8132   1c1 8133   + caddc 8135   x. cmul 8137   < clt 8313   - cmin 8449   # cap 8860   / cdiv 8951   NNcn 9242   2c2 9293   4c4 9295   8c8 9299   NN0cn0 9501   ZZcz 9582   |_cfl 10635

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-n0 9502  df-z 9583  df-q 9958  df-rp 9993  df-fl 10637

This theorem is referenced by:  2lgslem3a1  16019