Theorem 2lgslem3a 15334

Description: Lemma for 2lgslem3a1 15338. (Contributed by AV, 14-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
2lgslem3a	|- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 1 ) ) -> N = ( 2 x. K ) )

Proof of Theorem 2lgslem3a

Step	Hyp	Ref	Expression
1		2lgslem2.n	. . 3 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
2		oveq1 5929	. . . . 5 |- ( P = ( ( 8 x. K ) + 1 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 1 ) - 1 ) )
3	2	oveq1d 5937	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) )
4		fvoveq1 5945	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) )
5	3, 4	oveq12d 5940	. . 3 |- ( P = ( ( 8 x. K ) + 1 ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) )
6	1, 5	eqtrid 2241	. 2 |- ( P = ( ( 8 x. K ) + 1 ) -> N = ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) )
7		8nn0 9272	. . . . . . . . . 10 |- 8 e. NN0
8	7	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 8 e. NN0 )
9		id 19	. . . . . . . . 9 |- ( K e. NN0 -> K e. NN0 )
10	8, 9	nn0mulcld 9307	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9304	. . . . . . 7 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		pncan1 8403	. . . . . . 7 |- ( ( 8 x. K ) e. CC -> ( ( ( 8 x. K ) + 1 ) - 1 ) = ( 8 x. K ) )
13	11, 12	syl 14	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) - 1 ) = ( 8 x. K ) )
14	13	oveq1d 5937	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( ( 8 x. K ) / 2 ) )
15		4cn 9068	. . . . . . . . . . 11 |- 4 e. CC
16		2cn 9061	. . . . . . . . . . 11 |- 2 e. CC
17		4t2e8 9149	. . . . . . . . . . 11 |- ( 4 x. 2 ) = 8
18	15, 16, 17	mulcomli 8033	. . . . . . . . . 10 |- ( 2 x. 4 ) = 8
19	18	eqcomi 2200	. . . . . . . . 9 |- 8 = ( 2 x. 4 )
20	19	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 8 = ( 2 x. 4 ) )
21	20	oveq1d 5937	. . . . . . 7 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 2 x. 4 ) x. K ) )
22	16	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 2 e. CC )
23	15	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. CC )
24		nn0cn 9259	. . . . . . . 8 |- ( K e. NN0 -> K e. CC )
25	22, 23, 24	mulassd 8050	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 4 ) x. K ) = ( 2 x. ( 4 x. K ) ) )
26	21, 25	eqtrd 2229	. . . . . 6 |- ( K e. NN0 -> ( 8 x. K ) = ( 2 x. ( 4 x. K ) ) )
27	26	oveq1d 5937	. . . . 5 |- ( K e. NN0 -> ( ( 8 x. K ) / 2 ) = ( ( 2 x. ( 4 x. K ) ) / 2 ) )
28		4nn0 9268	. . . . . . . . 9 |- 4 e. NN0
29	28	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN0 )
30	29, 9	nn0mulcld 9307	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
31	30	nn0cnd 9304	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
32		2ap0 9083	. . . . . . 7 |- 2 # 0
33	32	a1i 9	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
34	31, 22, 33	divcanap3d 8822	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. ( 4 x. K ) ) / 2 ) = ( 4 x. K ) )
35	14, 27, 34	3eqtrd 2233	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( 4 x. K ) )
36		1cnd 8042	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
37		4ap0 9089	. . . . . . . . 9 |- 4 # 0
38	37	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
39	11, 36, 23, 38	divdirapd 8856	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) )
40		8cn 9076	. . . . . . . . . . 11 |- 8 e. CC
41	40	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
42	41, 24, 23, 38	div23apd 8855	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
43	17	eqcomi 2200	. . . . . . . . . . . . 13 |- 8 = ( 4 x. 2 )
44	43	oveq1i 5932	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
45	16, 15, 37	divcanap3i 8785	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
46	44, 45	eqtri 2217	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
47	46	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
48	47	oveq1d 5937	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
49	42, 48	eqtrd 2229	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
50	49	oveq1d 5937	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
51	39, 50	eqtrd 2229	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
52	51	fveq2d 5562	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) )
53		1lt4 9165	. . . . . 6 |- 1 < 4
54		2nn0 9266	. . . . . . . . . 10 |- 2 e. NN0
55	54	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 2 e. NN0 )
56	55, 9	nn0mulcld 9307	. . . . . . . 8 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
57	56	nn0zd 9446	. . . . . . 7 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
58		1nn0 9265	. . . . . . . 8 |- 1 e. NN0
59	58	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 1 e. NN0 )
60		4nn 9154	. . . . . . . 8 |- 4 e. NN
61	60	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 4 e. NN )
62		adddivflid 10382	. . . . . . 7 |- ( ( ( 2 x. K ) e. ZZ /\ 1 e. NN0 /\ 4 e. NN ) -> ( 1 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) = ( 2 x. K ) ) )
63	57, 59, 61, 62	syl3anc 1249	. . . . . 6 |- ( K e. NN0 -> ( 1 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) = ( 2 x. K ) ) )
64	53, 63	mpbii 148	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) = ( 2 x. K ) )
65	52, 64	eqtrd 2229	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( 2 x. K ) )
66	35, 65	oveq12d 5940	. . 3 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( ( 4 x. K ) - ( 2 x. K ) ) )
67		2t2e4 9145	. . . . . . . 8 |- ( 2 x. 2 ) = 4
68	67	eqcomi 2200	. . . . . . 7 |- 4 = ( 2 x. 2 )
69	68	a1i 9	. . . . . 6 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
70	69	oveq1d 5937	. . . . 5 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
71	22, 22, 24	mulassd 8050	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
72	70, 71	eqtrd 2229	. . . 4 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
73	72	oveq1d 5937	. . 3 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
74	56	nn0cnd 9304	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
75		2txmxeqx 9122	. . . 4 |- ( ( 2 x. K ) e. CC -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
76	74, 75	syl 14	. . 3 |- ( K e. NN0 -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
77	66, 73, 76	3eqtrd 2233	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( 2 x. K ) )
78	6, 77	sylan9eqr 2251	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 1 ) ) -> N = ( 2 x. K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   - cmin 8197   # cap 8608   / cdiv 8699   NNcn 8990   2c2 9041   4c4 9043   8c8 9047   NN0cn0 9249   ZZcz 9326   |_cfl 10358

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-n0 9250  df-z 9327  df-q 9694  df-rp 9729  df-fl 10360

This theorem is referenced by:  2lgslem3a1  15338