Theorem 2lgslem3d 15740

Description: Lemma for 2lgslem3d1 15744. (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 5981	. . . . 5 |- ( P = ( ( 8 x. K ) + 7 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 7 ) - 1 ) )
3	2	oveq1d 5989	. . . 4 |- ( P = ( ( 8 x. K ) + 7 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) )
4		fvoveq1 5997	. . . 4 |- ( P = ( ( 8 x. K ) + 7 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) )
5	3, 4	oveq12d 5992	. . 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 2254	. 2 |- ( P = ( ( 8 x. K ) + 7 ) -> N = ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) )
7		8nn0 9360	. . . . . . . . . . 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 9395	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9392	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		7cn 9162	. . . . . . . . 9 |- 7 e. CC
13	12	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 7 e. CC )
14		1cnd 8130	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
15	11, 13, 14	addsubassd 8445	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( 8 x. K ) + ( 7 - 1 ) ) )
16		4t2e8 9237	. . . . . . . . . . . 12 |- ( 4 x. 2 ) = 8
17	16	eqcomi 2213	. . . . . . . . . . 11 |- 8 = ( 4 x. 2 )
18	17	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 = ( 4 x. 2 ) )
19	18	oveq1d 5989	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. 2 ) x. K ) )
20		4cn 9156	. . . . . . . . . . 11 |- 4 e. CC
21	20	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 4 e. CC )
22		2cn 9149	. . . . . . . . . . 11 |- 2 e. CC
23	22	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 2 e. CC )
24		nn0cn 9347	. . . . . . . . . 10 |- ( K e. NN0 -> K e. CC )
25	21, 23, 24	mul32d 8267	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 4 x. 2 ) x. K ) = ( ( 4 x. K ) x. 2 ) )
26	19, 25	eqtrd 2242	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. K ) x. 2 ) )
27		7m1e6 9202	. . . . . . . . 9 |- ( 7 - 1 ) = 6
28	27	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> ( 7 - 1 ) = 6 )
29	26, 28	oveq12d 5992	. . . . . . 7 |- ( K e. NN0 -> ( ( 8 x. K ) + ( 7 - 1 ) ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) )
30	15, 29	eqtrd 2242	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) )
31	30	oveq1d 5989	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) )
32		4nn0 9356	. . . . . . . . . 10 |- 4 e. NN0
33	32	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 4 e. NN0 )
34	33, 9	nn0mulcld 9395	. . . . . . . 8 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
35	34	nn0cnd 9392	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
36	35, 23	mulcld 8135	. . . . . 6 |- ( K e. NN0 -> ( ( 4 x. K ) x. 2 ) e. CC )
37		6cn 9160	. . . . . . 7 |- 6 e. CC
38	37	a1i 9	. . . . . 6 |- ( K e. NN0 -> 6 e. CC )
39		2rp 9822	. . . . . . . 8 |- 2 e. RR+
40	39	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 2 e. RR+ )
41	40	rpap0d 9866	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
42	36, 38, 23, 41	divdirapd 8944	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) )
43	35, 23, 41	divcanap4d 8911	. . . . . 6 |- ( K e. NN0 -> ( ( ( 4 x. K ) x. 2 ) / 2 ) = ( 4 x. K ) )
44		3t2e6 9235	. . . . . . . . . 10 |- ( 3 x. 2 ) = 6
45	44	eqcomi 2213	. . . . . . . . 9 |- 6 = ( 3 x. 2 )
46	45	oveq1i 5984	. . . . . . . 8 |- ( 6 / 2 ) = ( ( 3 x. 2 ) / 2 )
47		3cn 9153	. . . . . . . . 9 |- 3 e. CC
48		2ap0 9171	. . . . . . . . 9 |- 2 # 0
49	47, 22, 48	divcanap4i 8874	. . . . . . . 8 |- ( ( 3 x. 2 ) / 2 ) = 3
50	46, 49	eqtri 2230	. . . . . . 7 |- ( 6 / 2 ) = 3
51	50	a1i 9	. . . . . 6 |- ( K e. NN0 -> ( 6 / 2 ) = 3 )
52	43, 51	oveq12d 5992	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) = ( ( 4 x. K ) + 3 ) )
53	31, 42, 52	3eqtrd 2246	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( 4 x. K ) + 3 ) )
54		4ap0 9177	. . . . . . . . 9 |- 4 # 0
55	54	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
56	11, 13, 21, 55	divdirapd 8944	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) )
57		8cn 9164	. . . . . . . . . . 11 |- 8 e. CC
58	57	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
59	58, 24, 21, 55	div23apd 8943	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
60	17	oveq1i 5984	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
61	22, 20, 54	divcanap3i 8873	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
62	60, 61	eqtri 2230	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
63	62	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
64	63	oveq1d 5989	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
65	59, 64	eqtrd 2242	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
66	65	oveq1d 5989	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
67	56, 66	eqtrd 2242	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
68	67	fveq2d 5607	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) )
69		3lt4 9251	. . . . . 6 |- 3 < 4
70		2nn0 9354	. . . . . . . . . . . 12 |- 2 e. NN0
71	70	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> 2 e. NN0 )
72	71, 9	nn0mulcld 9395	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
73	72	nn0zd 9535	. . . . . . . . 9 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
74	73	peano2zd 9540	. . . . . . . 8 |- ( K e. NN0 -> ( ( 2 x. K ) + 1 ) e. ZZ )
75		3nn0 9355	. . . . . . . . 9 |- 3 e. NN0
76	75	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 3 e. NN0 )
77		4nn 9242	. . . . . . . . 9 |- 4 e. NN
78	77	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN )
79		adddivflid 10479	. . . . . . . 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 1252	. . . . . . 7 |- ( K e. NN0 -> ( 3 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
81	23, 24	mulcld 8135	. . . . . . . . . 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 8905	. . . . . . . . . 10 |- ( K e. NN0 -> ( 3 / 4 ) e. CC )
84	81, 14, 83	addassd 8137	. . . . . . . . 9 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) )
85		4p3e7 9223	. . . . . . . . . . . . . . 15 |- ( 4 + 3 ) = 7
86	85	eqcomi 2213	. . . . . . . . . . . . . 14 |- 7 = ( 4 + 3 )
87	86	oveq1i 5984	. . . . . . . . . . . . 13 |- ( 7 / 4 ) = ( ( 4 + 3 ) / 4 )
88	20, 47, 20, 54	divdirapi 8884	. . . . . . . . . . . . 13 |- ( ( 4 + 3 ) / 4 ) = ( ( 4 / 4 ) + ( 3 / 4 ) )
89	20, 54	dividapi 8860	. . . . . . . . . . . . . 14 |- ( 4 / 4 ) = 1
90	89	oveq1i 5984	. . . . . . . . . . . . 13 |- ( ( 4 / 4 ) + ( 3 / 4 ) ) = ( 1 + ( 3 / 4 ) )
91	87, 88, 90	3eqtri 2234	. . . . . . . . . . . 12 |- ( 7 / 4 ) = ( 1 + ( 3 / 4 ) )
92	91	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> ( 7 / 4 ) = ( 1 + ( 3 / 4 ) ) )
93	92	eqcomd 2215	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 + ( 3 / 4 ) ) = ( 7 / 4 ) )
94	93	oveq2d 5990	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
95	84, 94	eqtrd 2242	. . . . . . . 8 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
96	95	fveqeq2d 5611	. . . . . . 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 2242	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( ( 2 x. K ) + 1 ) )
100	53, 99	oveq12d 5992	. . 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 9392	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
102	35, 82, 101, 14	addsub4d 8472	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) + 3 ) - ( ( 2 x. K ) + 1 ) ) = ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) )
103		2t2e4 9233	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
104	103	eqcomi 2213	. . . . . . . . 9 |- 4 = ( 2 x. 2 )
105	104	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
106	105	oveq1d 5989	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
107	23, 23, 24	mulassd 8138	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
108	106, 107	eqtrd 2242	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
109	108	oveq1d 5989	. . . . 5 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
110		2txmxeqx 9210	. . . . . 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 2242	. . . 4 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( 2 x. K ) )
113		3m1e2 9198	. . . . 5 |- ( 3 - 1 ) = 2
114	113	a1i 9	. . . 4 |- ( K e. NN0 -> ( 3 - 1 ) = 2 )
115	112, 114	oveq12d 5992	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) = ( ( 2 x. K ) + 2 ) )
116	100, 102, 115	3eqtrd 2246	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) = ( ( 2 x. K ) + 2 ) )
117	6, 116	sylan9eqr 2264	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 7 ) ) -> N = ( ( 2 x. K ) + 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   e. wcel 2180   class class class wbr 4062   ` cfv 5294  (class class class)co 5974   CCcc 7965   0cc0 7967   1c1 7968   + caddc 7970   x. cmul 7972   < clt 8149   - cmin 8285   # cap 8696   / cdiv 8787   NNcn 9078   2c2 9129   3c3 9130   4c4 9131   6c6 9133   7c7 9134   8c8 9135   NN0cn0 9337   ZZcz 9414   RR+crp 9817   |_cfl 10455

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086

This theorem depends on definitions:  df-bi 117  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-po 4364  df-iso 4365  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-n0 9338  df-z 9415  df-q 9783  df-rp 9818  df-fl 10457

This theorem is referenced by:  2lgslem3d1  15744