Theorem 2lgslem3c 15616

Description: Lemma for 2lgslem3c1 15620. (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 5958	. . . . 5 |- ( P = ( ( 8 x. K ) + 5 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 5 ) - 1 ) )
3	2	oveq1d 5966	. . . 4 |- ( P = ( ( 8 x. K ) + 5 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) )
4		fvoveq1 5974	. . . 4 |- ( P = ( ( 8 x. K ) + 5 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) )
5	3, 4	oveq12d 5969	. . 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 2251	. 2 |- ( P = ( ( 8 x. K ) + 5 ) -> N = ( ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) ) )
7		8nn0 9325	. . . . . . . . . . 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 9360	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9357	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		5cn 9123	. . . . . . . . 9 |- 5 e. CC
13	12	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 5 e. CC )
14		1cnd 8095	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
15	11, 13, 14	addsubassd 8410	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) - 1 ) = ( ( 8 x. K ) + ( 5 - 1 ) ) )
16		4t2e8 9202	. . . . . . . . . . . 12 |- ( 4 x. 2 ) = 8
17	16	eqcomi 2210	. . . . . . . . . . 11 |- 8 = ( 4 x. 2 )
18	17	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 = ( 4 x. 2 ) )
19	18	oveq1d 5966	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. 2 ) x. K ) )
20		4cn 9121	. . . . . . . . . . 11 |- 4 e. CC
21	20	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 4 e. CC )
22		2cn 9114	. . . . . . . . . . 11 |- 2 e. CC
23	22	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 2 e. CC )
24		nn0cn 9312	. . . . . . . . . 10 |- ( K e. NN0 -> K e. CC )
25	21, 23, 24	mul32d 8232	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 4 x. 2 ) x. K ) = ( ( 4 x. K ) x. 2 ) )
26	19, 25	eqtrd 2239	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. K ) x. 2 ) )
27		5m1e4 9165	. . . . . . . . 9 |- ( 5 - 1 ) = 4
28	27	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> ( 5 - 1 ) = 4 )
29	26, 28	oveq12d 5969	. . . . . . 7 |- ( K e. NN0 -> ( ( 8 x. K ) + ( 5 - 1 ) ) = ( ( ( 4 x. K ) x. 2 ) + 4 ) )
30	15, 29	eqtrd 2239	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) - 1 ) = ( ( ( 4 x. K ) x. 2 ) + 4 ) )
31	30	oveq1d 5966	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) + 4 ) / 2 ) )
32		4nn0 9321	. . . . . . . . . 10 |- 4 e. NN0
33	32	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 4 e. NN0 )
34	33, 9	nn0mulcld 9360	. . . . . . . 8 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
35	34	nn0cnd 9357	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
36	35, 23	mulcld 8100	. . . . . 6 |- ( K e. NN0 -> ( ( 4 x. K ) x. 2 ) e. CC )
37		2rp 9787	. . . . . . . 8 |- 2 e. RR+
38	37	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 2 e. RR+ )
39	38	rpap0d 9831	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
40	36, 21, 23, 39	divdirapd 8909	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) + 4 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 4 / 2 ) ) )
41	35, 23, 39	divcanap4d 8876	. . . . . 6 |- ( K e. NN0 -> ( ( ( 4 x. K ) x. 2 ) / 2 ) = ( 4 x. K ) )
42		4d2e2 9204	. . . . . . 7 |- ( 4 / 2 ) = 2
43	42	a1i 9	. . . . . 6 |- ( K e. NN0 -> ( 4 / 2 ) = 2 )
44	41, 43	oveq12d 5969	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 4 / 2 ) ) = ( ( 4 x. K ) + 2 ) )
45	31, 40, 44	3eqtrd 2243	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) = ( ( 4 x. K ) + 2 ) )
46		4ap0 9142	. . . . . . . . 9 |- 4 # 0
47	46	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
48	11, 13, 21, 47	divdirapd 8909	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 5 / 4 ) ) )
49		8cn 9129	. . . . . . . . . . 11 |- 8 e. CC
50	49	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
51	50, 24, 21, 47	div23apd 8908	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
52	17	oveq1i 5961	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
53	22, 20, 46	divcanap3i 8838	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
54	52, 53	eqtri 2227	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
55	54	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
56	55	oveq1d 5966	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
57	51, 56	eqtrd 2239	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
58	57	oveq1d 5966	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 5 / 4 ) ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
59	48, 58	eqtrd 2239	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) / 4 ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
60	59	fveq2d 5587	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 5 / 4 ) ) ) )
61		1lt4 9218	. . . . . 6 |- 1 < 4
62		2nn0 9319	. . . . . . . . . . . 12 |- 2 e. NN0
63	62	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> 2 e. NN0 )
64	63, 9	nn0mulcld 9360	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
65	64	nn0zd 9500	. . . . . . . . 9 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
66	65	peano2zd 9505	. . . . . . . 8 |- ( K e. NN0 -> ( ( 2 x. K ) + 1 ) e. ZZ )
67		1nn0 9318	. . . . . . . . 9 |- 1 e. NN0
68	67	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 1 e. NN0 )
69		4nn 9207	. . . . . . . . 9 |- 4 e. NN
70	69	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN )
71		adddivflid 10442	. . . . . . . 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 1250	. . . . . . 7 |- ( K e. NN0 -> ( 1 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
73	23, 24	mulcld 8100	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
74	21, 47	recclapd 8861	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 / 4 ) e. CC )
75	73, 14, 74	addassd 8102	. . . . . . . . 9 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 1 + ( 1 / 4 ) ) ) )
76		df-5 9105	. . . . . . . . . . . . . 14 |- 5 = ( 4 + 1 )
77	76	oveq1i 5961	. . . . . . . . . . . . 13 |- ( 5 / 4 ) = ( ( 4 + 1 ) / 4 )
78		ax-1cn 8025	. . . . . . . . . . . . . 14 |- 1 e. CC
79	20, 78, 20, 46	divdirapi 8849	. . . . . . . . . . . . 13 |- ( ( 4 + 1 ) / 4 ) = ( ( 4 / 4 ) + ( 1 / 4 ) )
80	20, 46	dividapi 8825	. . . . . . . . . . . . . 14 |- ( 4 / 4 ) = 1
81	80	oveq1i 5961	. . . . . . . . . . . . 13 |- ( ( 4 / 4 ) + ( 1 / 4 ) ) = ( 1 + ( 1 / 4 ) )
82	77, 79, 81	3eqtri 2231	. . . . . . . . . . . 12 |- ( 5 / 4 ) = ( 1 + ( 1 / 4 ) )
83	82	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> ( 5 / 4 ) = ( 1 + ( 1 / 4 ) ) )
84	83	eqcomd 2212	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 + ( 1 / 4 ) ) = ( 5 / 4 ) )
85	84	oveq2d 5967	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 2 x. K ) + ( 1 + ( 1 / 4 ) ) ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
86	75, 85	eqtrd 2239	. . . . . . . 8 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
87	86	fveqeq2d 5591	. . . . . . 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 2239	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) = ( ( 2 x. K ) + 1 ) )
91	45, 90	oveq12d 5969	. . 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 9357	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
93	35, 23, 92, 14	addsub4d 8437	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) + 2 ) - ( ( 2 x. K ) + 1 ) ) = ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 2 - 1 ) ) )
94		2t2e4 9198	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
95	94	eqcomi 2210	. . . . . . . . 9 |- 4 = ( 2 x. 2 )
96	95	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
97	96	oveq1d 5966	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
98	23, 23, 24	mulassd 8103	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
99	97, 98	eqtrd 2239	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
100	99	oveq1d 5966	. . . . 5 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
101		2txmxeqx 9175	. . . . . 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 2239	. . . 4 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( 2 x. K ) )
104		2m1e1 9161	. . . . 5 |- ( 2 - 1 ) = 1
105	104	a1i 9	. . . 4 |- ( K e. NN0 -> ( 2 - 1 ) = 1 )
106	103, 105	oveq12d 5969	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 2 - 1 ) ) = ( ( 2 x. K ) + 1 ) )
107	91, 93, 106	3eqtrd 2243	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) ) = ( ( 2 x. K ) + 1 ) )
108	6, 107	sylan9eqr 2261	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 5 ) ) -> N = ( ( 2 x. K ) + 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   class class class wbr 4047   ` cfv 5276  (class class class)co 5951   CCcc 7930   0cc0 7932   1c1 7933   + caddc 7935   x. cmul 7937   < clt 8114   - cmin 8250   # cap 8661   / cdiv 8752   NNcn 9043   2c2 9094   4c4 9096   5c5 9097   8c8 9100   NN0cn0 9302   ZZcz 9379   RR+crp 9782   |_cfl 10418

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-po 4347  df-iso 4348  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-n0 9303  df-z 9380  df-q 9748  df-rp 9783  df-fl 10420

This theorem is referenced by:  2lgslem3c1  15620