Theorem 2lgslem3c 16197

Description: Lemma for 2lgslem3c1 16201. (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 6086	. . . . 5 |- ( P = ( ( 8 x. K ) + 5 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 5 ) - 1 ) )
3	2	oveq1d 6094	. . . 4 |- ( P = ( ( 8 x. K ) + 5 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) )
4		fvoveq1 6102	. . . 4 |- ( P = ( ( 8 x. K ) + 5 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) )
5	3, 4	oveq12d 6097	. . 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 2283	. 2 |- ( P = ( ( 8 x. K ) + 5 ) -> N = ( ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) ) )
7		8nn0 9569	. . . . . . . . . . 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 9608	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9605	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		5cn 9367	. . . . . . . . 9 |- 5 e. CC
13	12	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 5 e. CC )
14		1cnd 8336	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
15	11, 13, 14	addsubassd 8651	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) - 1 ) = ( ( 8 x. K ) + ( 5 - 1 ) ) )
16		4t2e8 9446	. . . . . . . . . . . 12 |- ( 4 x. 2 ) = 8
17	16	eqcomi 2242	. . . . . . . . . . 11 |- 8 = ( 4 x. 2 )
18	17	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 = ( 4 x. 2 ) )
19	18	oveq1d 6094	. . . . . . . . 9 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. 2 ) x. K ) )
20		4cn 9365	. . . . . . . . . . 11 |- 4 e. CC
21	20	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 4 e. CC )
22		2cn 9358	. . . . . . . . . . 11 |- 2 e. CC
23	22	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 2 e. CC )
24		nn0cn 9556	. . . . . . . . . 10 |- ( K e. NN0 -> K e. CC )
25	21, 23, 24	mul32d 8473	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 4 x. 2 ) x. K ) = ( ( 4 x. K ) x. 2 ) )
26	19, 25	eqtrd 2271	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. K ) x. 2 ) )
27		5m1e4 9409	. . . . . . . . 9 |- ( 5 - 1 ) = 4
28	27	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> ( 5 - 1 ) = 4 )
29	26, 28	oveq12d 6097	. . . . . . 7 |- ( K e. NN0 -> ( ( 8 x. K ) + ( 5 - 1 ) ) = ( ( ( 4 x. K ) x. 2 ) + 4 ) )
30	15, 29	eqtrd 2271	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) - 1 ) = ( ( ( 4 x. K ) x. 2 ) + 4 ) )
31	30	oveq1d 6094	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) + 4 ) / 2 ) )
32		4nn0 9565	. . . . . . . . . 10 |- 4 e. NN0
33	32	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 4 e. NN0 )
34	33, 9	nn0mulcld 9608	. . . . . . . 8 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
35	34	nn0cnd 9605	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
36	35, 23	mulcld 8340	. . . . . 6 |- ( K e. NN0 -> ( ( 4 x. K ) x. 2 ) e. CC )
37		2rp 10042	. . . . . . . 8 |- 2 e. RR+
38	37	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 2 e. RR+ )
39	38	rpap0d 10086	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
40	36, 21, 23, 39	divdirapd 9153	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) + 4 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 4 / 2 ) ) )
41	35, 23, 39	divcanap4d 9120	. . . . . 6 |- ( K e. NN0 -> ( ( ( 4 x. K ) x. 2 ) / 2 ) = ( 4 x. K ) )
42		4d2e2 9448	. . . . . . 7 |- ( 4 / 2 ) = 2
43	42	a1i 9	. . . . . 6 |- ( K e. NN0 -> ( 4 / 2 ) = 2 )
44	41, 43	oveq12d 6097	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 4 / 2 ) ) = ( ( 4 x. K ) + 2 ) )
45	31, 40, 44	3eqtrd 2275	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) = ( ( 4 x. K ) + 2 ) )
46		4ap0 9386	. . . . . . . . 9 |- 4 # 0
47	46	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
48	11, 13, 21, 47	divdirapd 9153	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 5 / 4 ) ) )
49		8cn 9373	. . . . . . . . . . 11 |- 8 e. CC
50	49	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
51	50, 24, 21, 47	div23apd 9152	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
52	17	oveq1i 6089	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
53	22, 20, 46	divcanap3i 9082	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
54	52, 53	eqtri 2259	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
55	54	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
56	55	oveq1d 6094	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
57	51, 56	eqtrd 2271	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
58	57	oveq1d 6094	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 5 / 4 ) ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
59	48, 58	eqtrd 2271	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 5 ) / 4 ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
60	59	fveq2d 5697	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 5 / 4 ) ) ) )
61		1lt4 9462	. . . . . 6 |- 1 < 4
62		2nn0 9563	. . . . . . . . . . . 12 |- 2 e. NN0
63	62	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> 2 e. NN0 )
64	63, 9	nn0mulcld 9608	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
65	64	nn0zd 9749	. . . . . . . . 9 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
66	65	peano2zd 9754	. . . . . . . 8 |- ( K e. NN0 -> ( ( 2 x. K ) + 1 ) e. ZZ )
67		1nn0 9562	. . . . . . . . 9 |- 1 e. NN0
68	67	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 1 e. NN0 )
69		4nn 9451	. . . . . . . . 9 |- 4 e. NN
70	69	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN )
71		adddivflid 10710	. . . . . . . 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 1278	. . . . . . 7 |- ( K e. NN0 -> ( 1 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) )
73	23, 24	mulcld 8340	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
74	21, 47	recclapd 9105	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 / 4 ) e. CC )
75	73, 14, 74	addassd 8342	. . . . . . . . 9 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 1 + ( 1 / 4 ) ) ) )
76		df-5 9349	. . . . . . . . . . . . . 14 |- 5 = ( 4 + 1 )
77	76	oveq1i 6089	. . . . . . . . . . . . 13 |- ( 5 / 4 ) = ( ( 4 + 1 ) / 4 )
78		ax-1cn 8266	. . . . . . . . . . . . . 14 |- 1 e. CC
79	20, 78, 20, 46	divdirapi 9093	. . . . . . . . . . . . 13 |- ( ( 4 + 1 ) / 4 ) = ( ( 4 / 4 ) + ( 1 / 4 ) )
80	20, 46	dividapi 9069	. . . . . . . . . . . . . 14 |- ( 4 / 4 ) = 1
81	80	oveq1i 6089	. . . . . . . . . . . . 13 |- ( ( 4 / 4 ) + ( 1 / 4 ) ) = ( 1 + ( 1 / 4 ) )
82	77, 79, 81	3eqtri 2263	. . . . . . . . . . . 12 |- ( 5 / 4 ) = ( 1 + ( 1 / 4 ) )
83	82	a1i 9	. . . . . . . . . . 11 |- ( K e. NN0 -> ( 5 / 4 ) = ( 1 + ( 1 / 4 ) ) )
84	83	eqcomd 2244	. . . . . . . . . 10 |- ( K e. NN0 -> ( 1 + ( 1 / 4 ) ) = ( 5 / 4 ) )
85	84	oveq2d 6095	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 2 x. K ) + ( 1 + ( 1 / 4 ) ) ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
86	75, 85	eqtrd 2271	. . . . . . . 8 |- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 5 / 4 ) ) )
87	86	fveqeq2d 5701	. . . . . . 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 2271	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) = ( ( 2 x. K ) + 1 ) )
91	45, 90	oveq12d 6097	. . 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 9605	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
93	35, 23, 92, 14	addsub4d 8678	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) + 2 ) - ( ( 2 x. K ) + 1 ) ) = ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 2 - 1 ) ) )
94		2t2e4 9442	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
95	94	eqcomi 2242	. . . . . . . . 9 |- 4 = ( 2 x. 2 )
96	95	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
97	96	oveq1d 6094	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
98	23, 23, 24	mulassd 8343	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
99	97, 98	eqtrd 2271	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
100	99	oveq1d 6094	. . . . 5 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
101		2txmxeqx 9419	. . . . . 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 2271	. . . 4 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( 2 x. K ) )
104		2m1e1 9405	. . . . 5 |- ( 2 - 1 ) = 1
105	104	a1i 9	. . . 4 |- ( K e. NN0 -> ( 2 - 1 ) = 1 )
106	103, 105	oveq12d 6097	. . 3 |- ( K e. NN0 -> ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 2 - 1 ) ) = ( ( 2 x. K ) + 1 ) )
107	91, 93, 106	3eqtrd 2275	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 5 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 5 ) / 4 ) ) ) = ( ( 2 x. K ) + 1 ) )
108	6, 107	sylan9eqr 2293	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 5 ) ) -> N = ( ( 2 x. K ) + 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   CCcc 8171   0cc0 8173   1c1 8174   + caddc 8176   x. cmul 8178   < clt 8354   - cmin 8491   # cap 8903   / cdiv 8996   NNcn 9287   2c2 9338   4c4 9340   5c5 9341   8c8 9344   NN0cn0 9546   ZZcz 9627   RR+crp 10037   |_cfl 10686

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-n0 9547  df-z 9628  df-q 10003  df-rp 10038  df-fl 10688

This theorem is referenced by:  2lgslem3c1  16201