Theorem 2lgslem3d 15898

Description: Lemma for 2lgslem3d1 15902. (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 6035	. . . . 5 |- ( P = ( ( 8 x. K ) + 7 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 7 ) - 1 ) )
3	2	oveq1d 6043	. . . 4 |- ( P = ( ( 8 x. K ) + 7 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) )
4		fvoveq1 6051	. . . 4 |- ( P = ( ( 8 x. K ) + 7 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) )
5	3, 4	oveq12d 6046	. . 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 2276	. 2 |- ( P = ( ( 8 x. K ) + 7 ) -> N = ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) )
7		8nn0 9467	. . . . . . . . . . 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 9504	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9501	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		7cn 9269	. . . . . . . . 9 |- 7 e. CC
13	12	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 7 e. CC )
14		1cnd 8238	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
15	11, 13, 14	addsubassd 8552	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( 8 x. K ) + ( 7 - 1 ) ) )
16		4t2e8 9344	. . . . . . . . . . . 12 |- ( 4 x. 2 ) = 8
17	16	eqcomi 2235	. . . . . . . . . . 11 |- 8 = ( 4 x. 2 )
18	17	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 = ( 4 x. 2 ) )
19	18	oveq1d 6043	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. 2 ) x. K ) )
20		4cn 9263	. . . . . . . . . . 11 |- 4 e. CC
21	20	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 4 e. CC )
22		2cn 9256	. . . . . . . . . . 11 |- 2 e. CC
23	22	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 2 e. CC )
24		nn0cn 9454	. . . . . . . . . 10 |- ( K e. NN0 -> K e. CC )
25	21, 23, 24	mul32d 8374	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 4 x. 2 ) x. K ) = ( ( 4 x. K ) x. 2 ) )
26	19, 25	eqtrd 2264	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. K ) x. 2 ) )
27		7m1e6 9309	. . . . . . . . 9 |- ( 7 - 1 ) = 6
28	27	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> ( 7 - 1 ) = 6 )
29	26, 28	oveq12d 6046	. . . . . . 7 |- ( K e. NN0 -> ( ( 8 x. K ) + ( 7 - 1 ) ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) )
30	15, 29	eqtrd 2264	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) )
31	30	oveq1d 6043	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) )
32		4nn0 9463	. . . . . . . . . 10 |- 4 e. NN0
33	32	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 4 e. NN0 )
34	33, 9	nn0mulcld 9504	. . . . . . . 8 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
35	34	nn0cnd 9501	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
36	35, 23	mulcld 8242	. . . . . 6 |- ( K e. NN0 -> ( ( 4 x. K ) x. 2 ) e. CC )
37		6cn 9267	. . . . . . 7 |- 6 e. CC
38	37	a1i 9	. . . . . 6 |- ( K e. NN0 -> 6 e. CC )
39		2rp 9937	. . . . . . . 8 |- 2 e. RR+
40	39	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 2 e. RR+ )
41	40	rpap0d 9981	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
42	36, 38, 23, 41	divdirapd 9051	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) )
43	35, 23, 41	divcanap4d 9018	. . . . . 6 |- ( K e. NN0 -> ( ( ( 4 x. K ) x. 2 ) / 2 ) = ( 4 x. K ) )
44		3t2e6 9342	. . . . . . . . . 10 |- ( 3 x. 2 ) = 6
45	44	eqcomi 2235	. . . . . . . . 9 |- 6 = ( 3 x. 2 )
46	45	oveq1i 6038	. . . . . . . 8 |- ( 6 / 2 ) = ( ( 3 x. 2 ) / 2 )
47		3cn 9260	. . . . . . . . 9 |- 3 e. CC
48		2ap0 9278	. . . . . . . . 9 |- 2 # 0
49	47, 22, 48	divcanap4i 8981	. . . . . . . 8 |- ( ( 3 x. 2 ) / 2 ) = 3
50	46, 49	eqtri 2252	. . . . . . 7 |- ( 6 / 2 ) = 3
51	50	a1i 9	. . . . . 6 |- ( K e. NN0 -> ( 6 / 2 ) = 3 )
52	43, 51	oveq12d 6046	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) = ( ( 4 x. K ) + 3 ) )
53	31, 42, 52	3eqtrd 2268	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( 4 x. K ) + 3 ) )
54		4ap0 9284	. . . . . . . . 9 |- 4 # 0
55	54	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
56	11, 13, 21, 55	divdirapd 9051	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) )
57		8cn 9271	. . . . . . . . . . 11 |- 8 e. CC
58	57	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
59	58, 24, 21, 55	div23apd 9050	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
60	17	oveq1i 6038	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
61	22, 20, 54	divcanap3i 8980	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
62	60, 61	eqtri 2252	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
63	62	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
64	63	oveq1d 6043	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
65	59, 64	eqtrd 2264	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
66	65	oveq1d 6043	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
67	56, 66	eqtrd 2264	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
68	67	fveq2d 5652	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) )
69		3lt4 9358	. . . . . 6 |- 3 < 4
70		2nn0 9461	. . . . . . . . . . . 12 |- 2 e. NN0
71	70	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> 2 e. NN0 )
72	71, 9	nn0mulcld 9504	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
73	72	nn0zd 9644	. . . . . . . . 9 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
74	73	peano2zd 9649	. . . . . . . 8 |- ( K e. NN0 -> ( ( 2 x. K ) + 1 ) e. ZZ )
75		3nn0 9462	. . . . . . . . 9 |- 3 e. NN0
76	75	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 3 e. NN0 )
77		4nn 9349	. . . . . . . . 9 |- 4 e. NN
78	77	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN )
79		adddivflid 10598	. . . . . . . 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 8242	. . . . . . . . . 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 9012	. . . . . . . . . 10 |- ( K e. NN0 -> ( 3 / 4 ) e. CC )
84	81, 14, 83	addassd 8244	. . . . . . . . 9 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) )
85		4p3e7 9330	. . . . . . . . . . . . . . 15 |- ( 4 + 3 ) = 7
86	85	eqcomi 2235	. . . . . . . . . . . . . 14 |- 7 = ( 4 + 3 )
87	86	oveq1i 6038	. . . . . . . . . . . . 13 |- ( 7 / 4 ) = ( ( 4 + 3 ) / 4 )
88	20, 47, 20, 54	divdirapi 8991	. . . . . . . . . . . . 13 |- ( ( 4 + 3 ) / 4 ) = ( ( 4 / 4 ) + ( 3 / 4 ) )
89	20, 54	dividapi 8967	. . . . . . . . . . . . . 14 |- ( 4 / 4 ) = 1
90	89	oveq1i 6038	. . . . . . . . . . . . 13 |- ( ( 4 / 4 ) + ( 3 / 4 ) ) = ( 1 + ( 3 / 4 ) )
91	87, 88, 90	3eqtri 2256	. . . . . . . . . . . 12 |- ( 7 / 4 ) = ( 1 + ( 3 / 4 ) )
92	91	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> ( 7 / 4 ) = ( 1 + ( 3 / 4 ) ) )
93	92	eqcomd 2237	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 + ( 3 / 4 ) ) = ( 7 / 4 ) )
94	93	oveq2d 6044	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
95	84, 94	eqtrd 2264	. . . . . . . 8 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) )
96	95	fveqeq2d 5656	. . . . . . 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 2264	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( ( 2 x. K ) + 1 ) )
100	53, 99	oveq12d 6046	. . 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 9501	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
102	35, 82, 101, 14	addsub4d 8579	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) + 3 ) - ( ( 2 x. K ) + 1 ) ) = ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) )
103		2t2e4 9340	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
104	103	eqcomi 2235	. . . . . . . . 9 |- 4 = ( 2 x. 2 )
105	104	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
106	105	oveq1d 6043	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
107	23, 23, 24	mulassd 8245	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
108	106, 107	eqtrd 2264	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
109	108	oveq1d 6043	. . . . 5 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
110		2txmxeqx 9317	. . . . . 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 2264	. . . 4 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( 2 x. K ) )
113		3m1e2 9305	. . . . 5 |- ( 3 - 1 ) = 2
114	113	a1i 9	. . . 4 |- ( K e. NN0 -> ( 3 - 1 ) = 2 )
115	112, 114	oveq12d 6046	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) = ( ( 2 x. K ) + 2 ) )
116	100, 102, 115	3eqtrd 2268	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) = ( ( 2 x. K ) + 2 ) )
117	6, 116	sylan9eqr 2286	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 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8256   - cmin 8392   # cap 8803   / cdiv 8894   NNcn 9185   2c2 9236   3c3 9237   4c4 9238   6c6 9240   7c7 9241   8c8 9242   NN0cn0 9444   ZZcz 9523   RR+crp 9932   |_cfl 10574

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-n0 9445  df-z 9524  df-q 9898  df-rp 9933  df-fl 10576

This theorem is referenced by:  2lgslem3d1  15902