Theorem 2lgslem3b 15816

Description: Lemma for 2lgslem3b1 15820. (Contributed by AV, 16-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
2lgslem3b	|- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 3 ) ) -> N = ( ( 2 x. K ) + 1 ) )

Proof of Theorem 2lgslem3b

Step	Hyp	Ref	Expression
1		2lgslem2.n	. . 3 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
2		oveq1 6020	. . . . 5 |- ( P = ( ( 8 x. K ) + 3 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 3 ) - 1 ) )
3	2	oveq1d 6028	. . . 4 |- ( P = ( ( 8 x. K ) + 3 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 3 ) - 1 ) / 2 ) )
4		fvoveq1 6036	. . . 4 |- ( P = ( ( 8 x. K ) + 3 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 3 ) / 4 ) ) )
5	3, 4	oveq12d 6031	. . 3 |- ( P = ( ( 8 x. K ) + 3 ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( ( ( 8 x. K ) + 3 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 3 ) / 4 ) ) ) )
6	1, 5	eqtrid 2274	. 2 |- ( P = ( ( 8 x. K ) + 3 ) -> N = ( ( ( ( ( 8 x. K ) + 3 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 3 ) / 4 ) ) ) )
7		8nn0 9418	. . . . . . . . . . 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 9453	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9450	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		3cn 9211	. . . . . . . . 9 |- 3 e. CC
13	12	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 3 e. CC )
14		1cnd 8188	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
15	11, 13, 14	addsubassd 8503	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 3 ) - 1 ) = ( ( 8 x. K ) + ( 3 - 1 ) ) )
16		4t2e8 9295	. . . . . . . . . . . 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 6028	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. 2 ) x. K ) )
20		4cn 9214	. . . . . . . . . . 11 |- 4 e. CC
21	20	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 4 e. CC )
22		2cn 9207	. . . . . . . . . . 11 |- 2 e. CC
23	22	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 2 e. CC )
24		nn0cn 9405	. . . . . . . . . 10 |- ( K e. NN0 -> K e. CC )
25	21, 23, 24	mul32d 8325	. . . . . . . . 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		3m1e2 9256	. . . . . . . . 9 |- ( 3 - 1 ) = 2
28	27	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> ( 3 - 1 ) = 2 )
29	26, 28	oveq12d 6031	. . . . . . 7 |- ( K e. NN0 -> ( ( 8 x. K ) + ( 3 - 1 ) ) = ( ( ( 4 x. K ) x. 2 ) + 2 ) )
30	15, 29	eqtrd 2262	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 3 ) - 1 ) = ( ( ( 4 x. K ) x. 2 ) + 2 ) )
31	30	oveq1d 6028	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 3 ) - 1 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) + 2 ) / 2 ) )
32		4nn0 9414	. . . . . . . . . 10 |- 4 e. NN0
33	32	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 4 e. NN0 )
34	33, 9	nn0mulcld 9453	. . . . . . . 8 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
35	34	nn0cnd 9450	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
36	35, 23	mulcld 8193	. . . . . 6 |- ( K e. NN0 -> ( ( 4 x. K ) x. 2 ) e. CC )
37		2rp 9886	. . . . . . . 8 |- 2 e. RR+
38	37	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 2 e. RR+ )
39	38	rpap0d 9930	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
40	36, 23, 23, 39	divdirapd 9002	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) + 2 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 2 / 2 ) ) )
41		2ap0 9229	. . . . . . . 8 |- 2 # 0
42	41	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 2 # 0 )
43	35, 23, 42	divcanap4d 8969	. . . . . 6 |- ( K e. NN0 -> ( ( ( 4 x. K ) x. 2 ) / 2 ) = ( 4 x. K ) )
44		2div2e1 9269	. . . . . . 7 |- ( 2 / 2 ) = 1
45	44	a1i 9	. . . . . 6 |- ( K e. NN0 -> ( 2 / 2 ) = 1 )
46	43, 45	oveq12d 6031	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 2 / 2 ) ) = ( ( 4 x. K ) + 1 ) )
47	31, 40, 46	3eqtrd 2266	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 3 ) - 1 ) / 2 ) = ( ( 4 x. K ) + 1 ) )
48		4ap0 9235	. . . . . . . . 9 |- 4 # 0
49	48	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
50	11, 13, 21, 49	divdirapd 9002	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 3 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 3 / 4 ) ) )
51		8cn 9222	. . . . . . . . . . 11 |- 8 e. CC
52	51	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
53	52, 24, 21, 49	div23apd 9001	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
54	17	oveq1i 6023	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
55	22, 20, 48	divcanap3i 8931	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
56	54, 55	eqtri 2250	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
57	56	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
58	57	oveq1d 6028	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
59	53, 58	eqtrd 2262	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
60	59	oveq1d 6028	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 3 / 4 ) ) )
61	50, 60	eqtrd 2262	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 3 ) / 4 ) = ( ( 2 x. K ) + ( 3 / 4 ) ) )
62	61	fveq2d 5639	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 3 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 3 / 4 ) ) ) )
63		3lt4 9309	. . . . . 6 |- 3 < 4
64		2nn0 9412	. . . . . . . . . 10 |- 2 e. NN0
65	64	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 2 e. NN0 )
66	65, 9	nn0mulcld 9453	. . . . . . . 8 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
67	66	nn0zd 9593	. . . . . . 7 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
68		3nn0 9413	. . . . . . . 8 |- 3 e. NN0
69	68	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 3 e. NN0 )
70		4nn 9300	. . . . . . . 8 |- 4 e. NN
71	70	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 4 e. NN )
72		adddivflid 10545	. . . . . . 7 |- ( ( ( 2 x. K ) e. ZZ /\ 3 e. NN0 /\ 4 e. NN ) -> ( 3 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 3 / 4 ) ) ) = ( 2 x. K ) ) )
73	67, 69, 71, 72	syl3anc 1271	. . . . . 6 |- ( K e. NN0 -> ( 3 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 3 / 4 ) ) ) = ( 2 x. K ) ) )
74	63, 73	mpbii 148	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( 2 x. K ) + ( 3 / 4 ) ) ) = ( 2 x. K ) )
75	62, 74	eqtrd 2262	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 3 ) / 4 ) ) = ( 2 x. K ) )
76	47, 75	oveq12d 6031	. . 3 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 3 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 3 ) / 4 ) ) ) = ( ( ( 4 x. K ) + 1 ) - ( 2 x. K ) ) )
77	66	nn0cnd 9450	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
78	35, 14, 77	addsubd 8504	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) + 1 ) - ( 2 x. K ) ) = ( ( ( 4 x. K ) - ( 2 x. K ) ) + 1 ) )
79		2t2e4 9291	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
80	79	eqcomi 2233	. . . . . . . . 9 |- 4 = ( 2 x. 2 )
81	80	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
82	81	oveq1d 6028	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
83	23, 23, 24	mulassd 8196	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
84	82, 83	eqtrd 2262	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
85	84	oveq1d 6028	. . . . 5 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
86		2txmxeqx 9268	. . . . . 6 |- ( ( 2 x. K ) e. CC -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
87	77, 86	syl 14	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
88	85, 87	eqtrd 2262	. . . 4 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( 2 x. K ) )
89	88	oveq1d 6028	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) - ( 2 x. K ) ) + 1 ) = ( ( 2 x. K ) + 1 ) )
90	76, 78, 89	3eqtrd 2266	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 3 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 3 ) / 4 ) ) ) = ( ( 2 x. K ) + 1 ) )
91	6, 90	sylan9eqr 2284	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 3 ) ) -> N = ( ( 2 x. K ) + 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8023   0cc0 8025   1c1 8026   + caddc 8028   x. cmul 8030   < clt 8207   - cmin 8343   # cap 8754   / cdiv 8845   NNcn 9136   2c2 9187   3c3 9188   4c4 9189   8c8 9193   NN0cn0 9395   ZZcz 9472   RR+crp 9881   |_cfl 10521

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-n0 9396  df-z 9473  df-q 9847  df-rp 9882  df-fl 10523

This theorem is referenced by:  2lgslem3b1  15820