Theorem 2lgslem3c 15782

Description: Lemma for 2lgslem3c1 15786. (Contributed by AV, 16-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
2lgslem3c	|- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 5 ) ) -> N = ( ( 2 x. K ) + 1 ) )

Proof of Theorem 2lgslem3c

Step	Hyp	Ref	Expression
1		2lgslem2.n	. . 3 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
2		oveq1 6014	. . . . 5 |- ( P = ( ( 8 x. K ) + 5 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 5 ) - 1 ) )
3	2	oveq1d 6022	. . . 4 |- ( P = ( ( 8 x. K ) + 5 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) )
4		fvoveq1 6030	. . . 4 |- ( P = ( ( 8 x. K ) + 5 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) )
5	3, 4	oveq12d 6025	. . 3 |- ( P = ( ( 8 x. K ) + 5 ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) ) )
6	1, 5	eqtrid 2274	. 2 |- ( P = ( ( 8 x. K ) + 5 ) -> N = ( ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) ) )
7		8nn0 9400	. . . . . . . . . . 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 9435	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9432	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		5cn 9198	. . . . . . . . 9 |- 5 e. CC
13	12	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 5 e. CC )
14		1cnd 8170	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
15	11, 13, 14	addsubassd 8485	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) - 1 ) = ( ( 8 x. K ) + ( 5 - 1 ) ) )
16		4t2e8 9277	. . . . . . . . . . . 12 |- ( 4 x. 2 ) = 8
17	16	eqcomi 2233	. . . . . . . . . . 11 |- 8 = ( 4 x. 2 )
18	17	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 = ( 4 x. 2 ) )
19	18	oveq1d 6022	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. 2 ) x. K ) )
20		4cn 9196	. . . . . . . . . . 11 |- 4 e. CC
21	20	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 4 e. CC )
22		2cn 9189	. . . . . . . . . . 11 |- 2 e. CC
23	22	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 2 e. CC )
24		nn0cn 9387	. . . . . . . . . 10 |- ( K e. NN0 -> K e. CC )
25	21, 23, 24	mul32d 8307	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 4 x. 2 ) x. K ) = ( ( 4 x. K ) x. 2 ) )
26	19, 25	eqtrd 2262	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. K ) x. 2 ) )
27		5m1e4 9240	. . . . . . . . 9 |- ( 5 - 1 ) = 4
28	27	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> ( 5 - 1 ) = 4 )
29	26, 28	oveq12d 6025	. . . . . . 7 |- ( K e. NN0 -> ( ( 8 x. K ) + ( 5 - 1 ) ) = ( ( ( 4 x. K ) x. 2 ) + 4 ) )
30	15, 29	eqtrd 2262	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) - 1 ) = ( ( ( 4 x. K ) x. 2 ) + 4 ) )
31	30	oveq1d 6022	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) + 4 ) / 2 ) )
32		4nn0 9396	. . . . . . . . . 10 |- 4 e. NN0
33	32	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 4 e. NN0 )
34	33, 9	nn0mulcld 9435	. . . . . . . 8 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
35	34	nn0cnd 9432	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
36	35, 23	mulcld 8175	. . . . . 6 |- ( K e. NN0 -> ( ( 4 x. K ) x. 2 ) e. CC )
37		2rp 9862	. . . . . . . 8 |- 2 e. RR+
38	37	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 2 e. RR+ )
39	38	rpap0d 9906	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
40	36, 21, 23, 39	divdirapd 8984	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) + 4 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 4 / 2 ) ) )
41	35, 23, 39	divcanap4d 8951	. . . . . 6 |- ( K e. NN0 -> ( ( ( 4 x. K ) x. 2 ) / 2 ) = ( 4 x. K ) )
42		4d2e2 9279	. . . . . . 7 |- ( 4 / 2 ) = 2
43	42	a1i 9	. . . . . 6 |- ( K e. NN0 -> ( 4 / 2 ) = 2 )
44	41, 43	oveq12d 6025	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 4 / 2 ) ) = ( ( 4 x. K ) + 2 ) )
45	31, 40, 44	3eqtrd 2266	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) = ( ( 4 x. K ) + 2 ) )
46		4ap0 9217	. . . . . . . . 9 |- 4 # 0
47	46	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
48	11, 13, 21, 47	divdirapd 8984	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 5 / 4 ) ) )
49		8cn 9204	. . . . . . . . . . 11 |- 8 e. CC
50	49	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
51	50, 24, 21, 47	div23apd 8983	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
52	17	oveq1i 6017	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
53	22, 20, 46	divcanap3i 8913	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
54	52, 53	eqtri 2250	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
55	54	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
56	55	oveq1d 6022	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
57	51, 56	eqtrd 2262	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
58	57	oveq1d 6022	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 5 / 4 ) ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
59	48, 58	eqtrd 2262	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) / 4 ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
60	59	fveq2d 5633	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 5 / 4 ) ) ) )
61		1lt4 9293	. . . . . 6 |- 1 < 4
62		2nn0 9394	. . . . . . . . . . . 12 |- 2 e. NN0
63	62	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> 2 e. NN0 )
64	63, 9	nn0mulcld 9435	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
65	64	nn0zd 9575	. . . . . . . . 9 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
66	65	peano2zd 9580	. . . . . . . 8 |- ( K e. NN0 -> ( ( 2 x. K ) + 1 ) e. ZZ )
67		1nn0 9393	. . . . . . . . 9 |- 1 e. NN0
68	67	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 1 e. NN0 )
69		4nn 9282	. . . . . . . . 9 |- 4 e. NN
70	69	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN )
71		adddivflid 10520	. . . . . . . 8 |- ( ( ( ( 2 x. K ) + 1 ) e. ZZ /\ 1 e. NN0 /\ 4 e. NN ) -> ( 1 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
72	66, 68, 70, 71	syl3anc 1271	. . . . . . 7 |- ( K e. NN0 -> ( 1 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
73	23, 24	mulcld 8175	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
74	21, 47	recclapd 8936	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 / 4 ) e. CC )
75	73, 14, 74	addassd 8177	. . . . . . . . 9 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 1 + ( 1 / 4 ) ) ) )
76		df-5 9180	. . . . . . . . . . . . . 14 |- 5 = ( 4 + 1 )
77	76	oveq1i 6017	. . . . . . . . . . . . 13 |- ( 5 / 4 ) = ( ( 4 + 1 ) / 4 )
78		ax-1cn 8100	. . . . . . . . . . . . . 14 |- 1 e. CC
79	20, 78, 20, 46	divdirapi 8924	. . . . . . . . . . . . 13 |- ( ( 4 + 1 ) / 4 ) = ( ( 4 / 4 ) + ( 1 / 4 ) )
80	20, 46	dividapi 8900	. . . . . . . . . . . . . 14 |- ( 4 / 4 ) = 1
81	80	oveq1i 6017	. . . . . . . . . . . . 13 |- ( ( 4 / 4 ) + ( 1 / 4 ) ) = ( 1 + ( 1 / 4 ) )
82	77, 79, 81	3eqtri 2254	. . . . . . . . . . . 12 |- ( 5 / 4 ) = ( 1 + ( 1 / 4 ) )
83	82	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> ( 5 / 4 ) = ( 1 + ( 1 / 4 ) ) )
84	83	eqcomd 2235	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 + ( 1 / 4 ) ) = ( 5 / 4 ) )
85	84	oveq2d 6023	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 2 x. K ) + ( 1 + ( 1 / 4 ) ) ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
86	75, 85	eqtrd 2262	. . . . . . . 8 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
87	86	fveqeq2d 5637	. . . . . . 7 |- ( K e. NN0 -> ( ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) <-> ( |_ ` ( ( 2 x. K ) + ( 5 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
88	72, 87	bitrd 188	. . . . . 6 |- ( K e. NN0 -> ( 1 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 5 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
89	61, 88	mpbii 148	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( 2 x. K ) + ( 5 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) )
90	60, 89	eqtrd 2262	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) = ( ( 2 x. K ) + 1 ) )
91	45, 90	oveq12d 6025	. . 3 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) ) = ( ( ( 4 x. K ) + 2 ) - ( ( 2 x. K ) + 1 ) ) )
92	64	nn0cnd 9432	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
93	35, 23, 92, 14	addsub4d 8512	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) + 2 ) - ( ( 2 x. K ) + 1 ) ) = ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 2 - 1 ) ) )
94		2t2e4 9273	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
95	94	eqcomi 2233	. . . . . . . . 9 |- 4 = ( 2 x. 2 )
96	95	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
97	96	oveq1d 6022	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
98	23, 23, 24	mulassd 8178	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
99	97, 98	eqtrd 2262	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
100	99	oveq1d 6022	. . . . 5 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
101		2txmxeqx 9250	. . . . . 6 |- ( ( 2 x. K ) e. CC -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
102	92, 101	syl 14	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
103	100, 102	eqtrd 2262	. . . 4 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( 2 x. K ) )
104		2m1e1 9236	. . . . 5 |- ( 2 - 1 ) = 1
105	104	a1i 9	. . . 4 |- ( K e. NN0 -> ( 2 - 1 ) = 1 )
106	103, 105	oveq12d 6025	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 2 - 1 ) ) = ( ( 2 x. K ) + 1 ) )
107	91, 93, 106	3eqtrd 2266	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) ) = ( ( 2 x. K ) + 1 ) )
108	6, 107	sylan9eqr 2284	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 5 ) ) -> N = ( ( 2 x. K ) + 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8005   0cc0 8007   1c1 8008   + caddc 8010   x. cmul 8012   < clt 8189   - cmin 8325   # cap 8736   / cdiv 8827   NNcn 9118   2c2 9169   4c4 9171   5c5 9172   8c8 9175   NN0cn0 9377   ZZcz 9454   RR+crp 9857   |_cfl 10496

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-n0 9378  df-z 9455  df-q 9823  df-rp 9858  df-fl 10498

This theorem is referenced by:  2lgslem3c1  15786