Theorem 2lgslem3a 15825

Description: Lemma for 2lgslem3a1 15829. (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 6025	. . . . 5 |- ( P = ( ( 8 x. K ) + 1 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 1 ) - 1 ) )
3	2	oveq1d 6033	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) )
4		fvoveq1 6041	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) )
5	3, 4	oveq12d 6036	. . 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 2276	. 2 |- ( P = ( ( 8 x. K ) + 1 ) -> N = ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) )
7		8nn0 9425	. . . . . . . . . 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 9460	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9457	. . . . . . 7 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		pncan1 8556	. . . . . . 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 6033	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( ( 8 x. K ) / 2 ) )
15		4cn 9221	. . . . . . . . . . 11 |- 4 e. CC
16		2cn 9214	. . . . . . . . . . 11 |- 2 e. CC
17		4t2e8 9302	. . . . . . . . . . 11 |- ( 4 x. 2 ) = 8
18	15, 16, 17	mulcomli 8186	. . . . . . . . . 10 |- ( 2 x. 4 ) = 8
19	18	eqcomi 2235	. . . . . . . . 9 |- 8 = ( 2 x. 4 )
20	19	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 8 = ( 2 x. 4 ) )
21	20	oveq1d 6033	. . . . . . 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 9412	. . . . . . . 8 |- ( K e. NN0 -> K e. CC )
25	22, 23, 24	mulassd 8203	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 4 ) x. K ) = ( 2 x. ( 4 x. K ) ) )
26	21, 25	eqtrd 2264	. . . . . 6 |- ( K e. NN0 -> ( 8 x. K ) = ( 2 x. ( 4 x. K ) ) )
27	26	oveq1d 6033	. . . . 5 |- ( K e. NN0 -> ( ( 8 x. K ) / 2 ) = ( ( 2 x. ( 4 x. K ) ) / 2 ) )
28		4nn0 9421	. . . . . . . . 9 |- 4 e. NN0
29	28	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN0 )
30	29, 9	nn0mulcld 9460	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
31	30	nn0cnd 9457	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
32		2ap0 9236	. . . . . . 7 |- 2 # 0
33	32	a1i 9	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
34	31, 22, 33	divcanap3d 8975	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. ( 4 x. K ) ) / 2 ) = ( 4 x. K ) )
35	14, 27, 34	3eqtrd 2268	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( 4 x. K ) )
36		1cnd 8195	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
37		4ap0 9242	. . . . . . . . 9 |- 4 # 0
38	37	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
39	11, 36, 23, 38	divdirapd 9009	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) )
40		8cn 9229	. . . . . . . . . . 11 |- 8 e. CC
41	40	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
42	41, 24, 23, 38	div23apd 9008	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
43	17	eqcomi 2235	. . . . . . . . . . . . 13 |- 8 = ( 4 x. 2 )
44	43	oveq1i 6028	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
45	16, 15, 37	divcanap3i 8938	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
46	44, 45	eqtri 2252	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
47	46	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
48	47	oveq1d 6033	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
49	42, 48	eqtrd 2264	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
50	49	oveq1d 6033	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
51	39, 50	eqtrd 2264	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
52	51	fveq2d 5643	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) )
53		1lt4 9318	. . . . . 6 |- 1 < 4
54		2nn0 9419	. . . . . . . . . 10 |- 2 e. NN0
55	54	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 2 e. NN0 )
56	55, 9	nn0mulcld 9460	. . . . . . . 8 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
57	56	nn0zd 9600	. . . . . . 7 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
58		1nn0 9418	. . . . . . . 8 |- 1 e. NN0
59	58	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 1 e. NN0 )
60		4nn 9307	. . . . . . . 8 |- 4 e. NN
61	60	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 4 e. NN )
62		adddivflid 10553	. . . . . . 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 1273	. . . . . 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 2264	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( 2 x. K ) )
66	35, 65	oveq12d 6036	. . 3 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( ( 4 x. K ) - ( 2 x. K ) ) )
67		2t2e4 9298	. . . . . . . 8 |- ( 2 x. 2 ) = 4
68	67	eqcomi 2235	. . . . . . 7 |- 4 = ( 2 x. 2 )
69	68	a1i 9	. . . . . 6 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
70	69	oveq1d 6033	. . . . 5 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
71	22, 22, 24	mulassd 8203	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
72	70, 71	eqtrd 2264	. . . 4 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
73	72	oveq1d 6033	. . 3 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
74	56	nn0cnd 9457	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
75		2txmxeqx 9275	. . . 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 2268	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( 2 x. K ) )
78	6, 77	sylan9eqr 2286	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 1 ) ) -> N = ( 2 x. K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   < clt 8214   - cmin 8350   # cap 8761   / cdiv 8852   NNcn 9143   2c2 9194   4c4 9196   8c8 9200   NN0cn0 9402   ZZcz 9479   |_cfl 10529

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-n0 9403  df-z 9480  df-q 9854  df-rp 9889  df-fl 10531

This theorem is referenced by:  2lgslem3a1  15829