Theorem 2lgslem3d 15956

Description: Lemma for 2lgslem3d1 15960. (Contributed by AV, 16-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
2lgslem3d	|- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 7 ) ) -> N = ( ( 2 x. K ) + 2 ) )

Proof of Theorem 2lgslem3d

Step	Hyp	Ref	Expression
1		2lgslem2.n	. . 3 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
2		oveq1 6056	. . . . 5 |- ( P = ( ( 8 x. K ) + 7 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 7 ) - 1 ) )
3	2	oveq1d 6064	. . . 4 |- ( P = ( ( 8 x. K ) + 7 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) )
4		fvoveq1 6072	. . . 4 |- ( P = ( ( 8 x. K ) + 7 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) )
5	3, 4	oveq12d 6067	. . 3 |- ( P = ( ( 8 x. K ) + 7 ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) )
6	1, 5	eqtrid 2277	. 2 |- ( P = ( ( 8 x. K ) + 7 ) -> N = ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) )
7		8nn0 9515	. . . . . . . . . . 11 |- 8 e. NN0
8	7	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. NN0 )
9		id 19	. . . . . . . . . 10 |- ( K e. NN0 -> K e. NN0 )
10	8, 9	nn0mulcld 9554	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9551	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		7cn 9317	. . . . . . . . 9 |- 7 e. CC
13	12	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 7 e. CC )
14		1cnd 8286	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
15	11, 13, 14	addsubassd 8600	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( 8 x. K ) + ( 7 - 1 ) ) )
16		4t2e8 9392	. . . . . . . . . . . 12 |- ( 4 x. 2 ) = 8
17	16	eqcomi 2236	. . . . . . . . . . 11 |- 8 = ( 4 x. 2 )
18	17	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 = ( 4 x. 2 ) )
19	18	oveq1d 6064	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. 2 ) x. K ) )
20		4cn 9311	. . . . . . . . . . 11 |- 4 e. CC
21	20	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 4 e. CC )
22		2cn 9304	. . . . . . . . . . 11 |- 2 e. CC
23	22	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 2 e. CC )
24		nn0cn 9502	. . . . . . . . . 10 |- ( K e. NN0 -> K e. CC )
25	21, 23, 24	mul32d 8422	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 4 x. 2 ) x. K ) = ( ( 4 x. K ) x. 2 ) )
26	19, 25	eqtrd 2265	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. K ) x. 2 ) )
27		7m1e6 9357	. . . . . . . . 9 |- ( 7 - 1 ) = 6
28	27	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> ( 7 - 1 ) = 6 )
29	26, 28	oveq12d 6067	. . . . . . 7 |- ( K e. NN0 -> ( ( 8 x. K ) + ( 7 - 1 ) ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) )
30	15, 29	eqtrd 2265	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) )
31	30	oveq1d 6064	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) )
32		4nn0 9511	. . . . . . . . . 10 |- 4 e. NN0
33	32	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 4 e. NN0 )
34	33, 9	nn0mulcld 9554	. . . . . . . 8 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
35	34	nn0cnd 9551	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
36	35, 23	mulcld 8290	. . . . . 6 |- ( K e. NN0 -> ( ( 4 x. K ) x. 2 ) e. CC )
37		6cn 9315	. . . . . . 7 |- 6 e. CC
38	37	a1i 9	. . . . . 6 |- ( K e. NN0 -> 6 e. CC )
39		2rp 9987	. . . . . . . 8 |- 2 e. RR+
40	39	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 2 e. RR+ )
41	40	rpap0d 10031	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
42	36, 38, 23, 41	divdirapd 9099	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) )
43	35, 23, 41	divcanap4d 9066	. . . . . 6 |- ( K e. NN0 -> ( ( ( 4 x. K ) x. 2 ) / 2 ) = ( 4 x. K ) )
44		3t2e6 9390	. . . . . . . . . 10 |- ( 3 x. 2 ) = 6
45	44	eqcomi 2236	. . . . . . . . 9 |- 6 = ( 3 x. 2 )
46	45	oveq1i 6059	. . . . . . . 8 |- ( 6 / 2 ) = ( ( 3 x. 2 ) / 2 )
47		3cn 9308	. . . . . . . . 9 |- 3 e. CC
48		2ap0 9326	. . . . . . . . 9 |- 2 # 0
49	47, 22, 48	divcanap4i 9029	. . . . . . . 8 |- ( ( 3 x. 2 ) / 2 ) = 3
50	46, 49	eqtri 2253	. . . . . . 7 |- ( 6 / 2 ) = 3
51	50	a1i 9	. . . . . 6 |- ( K e. NN0 -> ( 6 / 2 ) = 3 )
52	43, 51	oveq12d 6067	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) = ( ( 4 x. K ) + 3 ) )
53	31, 42, 52	3eqtrd 2269	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( 4 x. K ) + 3 ) )
54		4ap0 9332	. . . . . . . . 9 |- 4 # 0
55	54	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
56	11, 13, 21, 55	divdirapd 9099	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) )
57		8cn 9319	. . . . . . . . . . 11 |- 8 e. CC
58	57	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
59	58, 24, 21, 55	div23apd 9098	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
60	17	oveq1i 6059	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
61	22, 20, 54	divcanap3i 9028	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
62	60, 61	eqtri 2253	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
63	62	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
64	63	oveq1d 6064	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
65	59, 64	eqtrd 2265	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
66	65	oveq1d 6064	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
67	56, 66	eqtrd 2265	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
68	67	fveq2d 5673	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) )
69		3lt4 9406	. . . . . 6 |- 3 < 4
70		2nn0 9509	. . . . . . . . . . . 12 |- 2 e. NN0
71	70	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> 2 e. NN0 )
72	71, 9	nn0mulcld 9554	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
73	72	nn0zd 9694	. . . . . . . . 9 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
74	73	peano2zd 9699	. . . . . . . 8 |- ( K e. NN0 -> ( ( 2 x. K ) + 1 ) e. ZZ )
75		3nn0 9510	. . . . . . . . 9 |- 3 e. NN0
76	75	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 3 e. NN0 )
77		4nn 9397	. . . . . . . . 9 |- 4 e. NN
78	77	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN )
79		adddivflid 10648	. . . . . . . 8 |- ( ( ( ( 2 x. K ) + 1 ) e. ZZ /\ 3 e. NN0 /\ 4 e. NN ) -> ( 3 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
80	74, 76, 78, 79	syl3anc 1274	. . . . . . 7 |- ( K e. NN0 -> ( 3 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
81	23, 24	mulcld 8290	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
82	47	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> 3 e. CC )
83	82, 21, 55	divclapd 9060	. . . . . . . . . 10 |- ( K e. NN0 -> ( 3 / 4 ) e. CC )
84	81, 14, 83	addassd 8292	. . . . . . . . 9 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) )
85		4p3e7 9378	. . . . . . . . . . . . . . 15 |- ( 4 + 3 ) = 7
86	85	eqcomi 2236	. . . . . . . . . . . . . 14 |- 7 = ( 4 + 3 )
87	86	oveq1i 6059	. . . . . . . . . . . . 13 |- ( 7 / 4 ) = ( ( 4 + 3 ) / 4 )
88	20, 47, 20, 54	divdirapi 9039	. . . . . . . . . . . . 13 |- ( ( 4 + 3 ) / 4 ) = ( ( 4 / 4 ) + ( 3 / 4 ) )
89	20, 54	dividapi 9015	. . . . . . . . . . . . . 14 |- ( 4 / 4 ) = 1
90	89	oveq1i 6059	. . . . . . . . . . . . 13 |- ( ( 4 / 4 ) + ( 3 / 4 ) ) = ( 1 + ( 3 / 4 ) )
91	87, 88, 90	3eqtri 2257	. . . . . . . . . . . 12 |- ( 7 / 4 ) = ( 1 + ( 3 / 4 ) )
92	91	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> ( 7 / 4 ) = ( 1 + ( 3 / 4 ) ) )
93	92	eqcomd 2238	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 + ( 3 / 4 ) ) = ( 7 / 4 ) )
94	93	oveq2d 6065	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
95	84, 94	eqtrd 2265	. . . . . . . 8 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
96	95	fveqeq2d 5677	. . . . . . 7 |- ( K e. NN0 -> ( ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) <-> ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
97	80, 96	bitrd 188	. . . . . 6 |- ( K e. NN0 -> ( 3 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
98	69, 97	mpbii 148	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) )
99	68, 98	eqtrd 2265	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( ( 2 x. K ) + 1 ) )
100	53, 99	oveq12d 6067	. . 3 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) = ( ( ( 4 x. K ) + 3 ) - ( ( 2 x. K ) + 1 ) ) )
101	72	nn0cnd 9551	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
102	35, 82, 101, 14	addsub4d 8627	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) + 3 ) - ( ( 2 x. K ) + 1 ) ) = ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) )
103		2t2e4 9388	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
104	103	eqcomi 2236	. . . . . . . . 9 |- 4 = ( 2 x. 2 )
105	104	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
106	105	oveq1d 6064	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
107	23, 23, 24	mulassd 8293	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
108	106, 107	eqtrd 2265	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
109	108	oveq1d 6064	. . . . 5 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
110		2txmxeqx 9365	. . . . . 6 |- ( ( 2 x. K ) e. CC -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
111	101, 110	syl 14	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
112	109, 111	eqtrd 2265	. . . 4 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( 2 x. K ) )
113		3m1e2 9353	. . . . 5 |- ( 3 - 1 ) = 2
114	113	a1i 9	. . . 4 |- ( K e. NN0 -> ( 3 - 1 ) = 2 )
115	112, 114	oveq12d 6067	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) = ( ( 2 x. K ) + 2 ) )
116	100, 102, 115	3eqtrd 2269	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) = ( ( 2 x. K ) + 2 ) )
117	6, 116	sylan9eqr 2287	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 7 ) ) -> N = ( ( 2 x. K ) + 2 ) )

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   3c3 9285   4c4 9286   6c6 9288   7c7 9289   8c8 9290   NN0cn0 9492   ZZcz 9573   RR+crp 9982   |_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:  2lgslem3d1  15960