Theorem 2lgslem3d 15421

Description: Lemma for 2lgslem3d1 15425. (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 5932	. . . . 5 |- ( P = ( ( 8 x. K ) + 7 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 7 ) - 1 ) )
3	2	oveq1d 5940	. . . 4 |- ( P = ( ( 8 x. K ) + 7 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) )
4		fvoveq1 5948	. . . 4 |- ( P = ( ( 8 x. K ) + 7 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) )
5	3, 4	oveq12d 5943	. . 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 2241	. 2 |- ( P = ( ( 8 x. K ) + 7 ) -> N = ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) )
7		8nn0 9289	. . . . . . . . . . 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 9324	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9321	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		7cn 9091	. . . . . . . . 9 |- 7 e. CC
13	12	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 7 e. CC )
14		1cnd 8059	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
15	11, 13, 14	addsubassd 8374	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( 8 x. K ) + ( 7 - 1 ) ) )
16		4t2e8 9166	. . . . . . . . . . . 12 |- ( 4 x. 2 ) = 8
17	16	eqcomi 2200	. . . . . . . . . . 11 |- 8 = ( 4 x. 2 )
18	17	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 = ( 4 x. 2 ) )
19	18	oveq1d 5940	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. 2 ) x. K ) )
20		4cn 9085	. . . . . . . . . . 11 |- 4 e. CC
21	20	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 4 e. CC )
22		2cn 9078	. . . . . . . . . . 11 |- 2 e. CC
23	22	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 2 e. CC )
24		nn0cn 9276	. . . . . . . . . 10 |- ( K e. NN0 -> K e. CC )
25	21, 23, 24	mul32d 8196	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 4 x. 2 ) x. K ) = ( ( 4 x. K ) x. 2 ) )
26	19, 25	eqtrd 2229	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. K ) x. 2 ) )
27		7m1e6 9131	. . . . . . . . 9 |- ( 7 - 1 ) = 6
28	27	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> ( 7 - 1 ) = 6 )
29	26, 28	oveq12d 5943	. . . . . . 7 |- ( K e. NN0 -> ( ( 8 x. K ) + ( 7 - 1 ) ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) )
30	15, 29	eqtrd 2229	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) )
31	30	oveq1d 5940	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) )
32		4nn0 9285	. . . . . . . . . 10 |- 4 e. NN0
33	32	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 4 e. NN0 )
34	33, 9	nn0mulcld 9324	. . . . . . . 8 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
35	34	nn0cnd 9321	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
36	35, 23	mulcld 8064	. . . . . 6 |- ( K e. NN0 -> ( ( 4 x. K ) x. 2 ) e. CC )
37		6cn 9089	. . . . . . 7 |- 6 e. CC
38	37	a1i 9	. . . . . 6 |- ( K e. NN0 -> 6 e. CC )
39		2rp 9750	. . . . . . . 8 |- 2 e. RR+
40	39	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 2 e. RR+ )
41	40	rpap0d 9794	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
42	36, 38, 23, 41	divdirapd 8873	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) )
43	35, 23, 41	divcanap4d 8840	. . . . . 6 |- ( K e. NN0 -> ( ( ( 4 x. K ) x. 2 ) / 2 ) = ( 4 x. K ) )
44		3t2e6 9164	. . . . . . . . . 10 |- ( 3 x. 2 ) = 6
45	44	eqcomi 2200	. . . . . . . . 9 |- 6 = ( 3 x. 2 )
46	45	oveq1i 5935	. . . . . . . 8 |- ( 6 / 2 ) = ( ( 3 x. 2 ) / 2 )
47		3cn 9082	. . . . . . . . 9 |- 3 e. CC
48		2ap0 9100	. . . . . . . . 9 |- 2 # 0
49	47, 22, 48	divcanap4i 8803	. . . . . . . 8 |- ( ( 3 x. 2 ) / 2 ) = 3
50	46, 49	eqtri 2217	. . . . . . 7 |- ( 6 / 2 ) = 3
51	50	a1i 9	. . . . . 6 |- ( K e. NN0 -> ( 6 / 2 ) = 3 )
52	43, 51	oveq12d 5943	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) = ( ( 4 x. K ) + 3 ) )
53	31, 42, 52	3eqtrd 2233	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( 4 x. K ) + 3 ) )
54		4ap0 9106	. . . . . . . . 9 |- 4 # 0
55	54	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
56	11, 13, 21, 55	divdirapd 8873	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) )
57		8cn 9093	. . . . . . . . . . 11 |- 8 e. CC
58	57	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
59	58, 24, 21, 55	div23apd 8872	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
60	17	oveq1i 5935	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
61	22, 20, 54	divcanap3i 8802	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
62	60, 61	eqtri 2217	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
63	62	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
64	63	oveq1d 5940	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
65	59, 64	eqtrd 2229	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
66	65	oveq1d 5940	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
67	56, 66	eqtrd 2229	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
68	67	fveq2d 5565	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) )
69		3lt4 9180	. . . . . 6 |- 3 < 4
70		2nn0 9283	. . . . . . . . . . . 12 |- 2 e. NN0
71	70	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> 2 e. NN0 )
72	71, 9	nn0mulcld 9324	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
73	72	nn0zd 9463	. . . . . . . . 9 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
74	73	peano2zd 9468	. . . . . . . 8 |- ( K e. NN0 -> ( ( 2 x. K ) + 1 ) e. ZZ )
75		3nn0 9284	. . . . . . . . 9 |- 3 e. NN0
76	75	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 3 e. NN0 )
77		4nn 9171	. . . . . . . . 9 |- 4 e. NN
78	77	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN )
79		adddivflid 10399	. . . . . . . 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 1249	. . . . . . 7 |- ( K e. NN0 -> ( 3 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
81	23, 24	mulcld 8064	. . . . . . . . . 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 8834	. . . . . . . . . 10 |- ( K e. NN0 -> ( 3 / 4 ) e. CC )
84	81, 14, 83	addassd 8066	. . . . . . . . 9 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) )
85		4p3e7 9152	. . . . . . . . . . . . . . 15 |- ( 4 + 3 ) = 7
86	85	eqcomi 2200	. . . . . . . . . . . . . 14 |- 7 = ( 4 + 3 )
87	86	oveq1i 5935	. . . . . . . . . . . . 13 |- ( 7 / 4 ) = ( ( 4 + 3 ) / 4 )
88	20, 47, 20, 54	divdirapi 8813	. . . . . . . . . . . . 13 |- ( ( 4 + 3 ) / 4 ) = ( ( 4 / 4 ) + ( 3 / 4 ) )
89	20, 54	dividapi 8789	. . . . . . . . . . . . . 14 |- ( 4 / 4 ) = 1
90	89	oveq1i 5935	. . . . . . . . . . . . 13 |- ( ( 4 / 4 ) + ( 3 / 4 ) ) = ( 1 + ( 3 / 4 ) )
91	87, 88, 90	3eqtri 2221	. . . . . . . . . . . 12 |- ( 7 / 4 ) = ( 1 + ( 3 / 4 ) )
92	91	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> ( 7 / 4 ) = ( 1 + ( 3 / 4 ) ) )
93	92	eqcomd 2202	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 + ( 3 / 4 ) ) = ( 7 / 4 ) )
94	93	oveq2d 5941	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
95	84, 94	eqtrd 2229	. . . . . . . 8 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
96	95	fveqeq2d 5569	. . . . . . 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 2229	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( ( 2 x. K ) + 1 ) )
100	53, 99	oveq12d 5943	. . 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 9321	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
102	35, 82, 101, 14	addsub4d 8401	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) + 3 ) - ( ( 2 x. K ) + 1 ) ) = ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) )
103		2t2e4 9162	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
104	103	eqcomi 2200	. . . . . . . . 9 |- 4 = ( 2 x. 2 )
105	104	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
106	105	oveq1d 5940	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
107	23, 23, 24	mulassd 8067	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
108	106, 107	eqtrd 2229	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
109	108	oveq1d 5940	. . . . 5 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
110		2txmxeqx 9139	. . . . . 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 2229	. . . 4 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( 2 x. K ) )
113		3m1e2 9127	. . . . 5 |- ( 3 - 1 ) = 2
114	113	a1i 9	. . . 4 |- ( K e. NN0 -> ( 3 - 1 ) = 2 )
115	112, 114	oveq12d 5943	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) = ( ( 2 x. K ) + 2 ) )
116	100, 102, 115	3eqtrd 2233	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) = ( ( 2 x. K ) + 2 ) )
117	6, 116	sylan9eqr 2251	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 7 ) ) -> N = ( ( 2 x. K ) + 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   CCcc 7894   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   < clt 8078   - cmin 8214   # cap 8625   / cdiv 8716   NNcn 9007   2c2 9058   3c3 9059   4c4 9060   6c6 9062   7c7 9063   8c8 9064   NN0cn0 9266   ZZcz 9343   RR+crp 9745   |_cfl 10375

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-n0 9267  df-z 9344  df-q 9711  df-rp 9746  df-fl 10377

This theorem is referenced by:  2lgslem3d1  15425