Theorem 2lgslem3a 15241

Description: Lemma for 2lgslem3a1 15245. (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 5926	. . . . 5 |- ( P = ( ( 8 x. K ) + 1 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 1 ) - 1 ) )
3	2	oveq1d 5934	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) )
4		fvoveq1 5942	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) )
5	3, 4	oveq12d 5937	. . 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 2238	. 2 |- ( P = ( ( 8 x. K ) + 1 ) -> N = ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) )
7		8nn0 9266	. . . . . . . . . 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 9301	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
11	10	nn0cnd 9298	. . . . . . 7 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
12		pncan1 8398	. . . . . . 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 5934	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( ( 8 x. K ) / 2 ) )
15		4cn 9062	. . . . . . . . . . 11 |- 4 e. CC
16		2cn 9055	. . . . . . . . . . 11 |- 2 e. CC
17		4t2e8 9143	. . . . . . . . . . 11 |- ( 4 x. 2 ) = 8
18	15, 16, 17	mulcomli 8028	. . . . . . . . . 10 |- ( 2 x. 4 ) = 8
19	18	eqcomi 2197	. . . . . . . . 9 |- 8 = ( 2 x. 4 )
20	19	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 8 = ( 2 x. 4 ) )
21	20	oveq1d 5934	. . . . . . 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 9253	. . . . . . . 8 |- ( K e. NN0 -> K e. CC )
25	22, 23, 24	mulassd 8045	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 4 ) x. K ) = ( 2 x. ( 4 x. K ) ) )
26	21, 25	eqtrd 2226	. . . . . 6 |- ( K e. NN0 -> ( 8 x. K ) = ( 2 x. ( 4 x. K ) ) )
27	26	oveq1d 5934	. . . . 5 |- ( K e. NN0 -> ( ( 8 x. K ) / 2 ) = ( ( 2 x. ( 4 x. K ) ) / 2 ) )
28		4nn0 9262	. . . . . . . . 9 |- 4 e. NN0
29	28	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN0 )
30	29, 9	nn0mulcld 9301	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
31	30	nn0cnd 9298	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
32		2ap0 9077	. . . . . . 7 |- 2 # 0
33	32	a1i 9	. . . . . 6 |- ( K e. NN0 -> 2 # 0 )
34	31, 22, 33	divcanap3d 8816	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. ( 4 x. K ) ) / 2 ) = ( 4 x. K ) )
35	14, 27, 34	3eqtrd 2230	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( 4 x. K ) )
36		1cnd 8037	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
37		4ap0 9083	. . . . . . . . 9 |- 4 # 0
38	37	a1i 9	. . . . . . . 8 |- ( K e. NN0 -> 4 # 0 )
39	11, 36, 23, 38	divdirapd 8850	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) )
40		8cn 9070	. . . . . . . . . . 11 |- 8 e. CC
41	40	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
42	41, 24, 23, 38	div23apd 8849	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
43	17	eqcomi 2197	. . . . . . . . . . . . 13 |- 8 = ( 4 x. 2 )
44	43	oveq1i 5929	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
45	16, 15, 37	divcanap3i 8779	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
46	44, 45	eqtri 2214	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
47	46	a1i 9	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
48	47	oveq1d 5934	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
49	42, 48	eqtrd 2226	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
50	49	oveq1d 5934	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
51	39, 50	eqtrd 2226	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
52	51	fveq2d 5559	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) )
53		1lt4 9159	. . . . . 6 |- 1 < 4
54		2nn0 9260	. . . . . . . . . 10 |- 2 e. NN0
55	54	a1i 9	. . . . . . . . 9 |- ( K e. NN0 -> 2 e. NN0 )
56	55, 9	nn0mulcld 9301	. . . . . . . 8 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
57	56	nn0zd 9440	. . . . . . 7 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
58		1nn0 9259	. . . . . . . 8 |- 1 e. NN0
59	58	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 1 e. NN0 )
60		4nn 9148	. . . . . . . 8 |- 4 e. NN
61	60	a1i 9	. . . . . . 7 |- ( K e. NN0 -> 4 e. NN )
62		adddivflid 10364	. . . . . . 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 2226	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( 2 x. K ) )
66	35, 65	oveq12d 5937	. . 3 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( ( 4 x. K ) - ( 2 x. K ) ) )
67		2t2e4 9139	. . . . . . . 8 |- ( 2 x. 2 ) = 4
68	67	eqcomi 2197	. . . . . . 7 |- 4 = ( 2 x. 2 )
69	68	a1i 9	. . . . . 6 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
70	69	oveq1d 5934	. . . . 5 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
71	22, 22, 24	mulassd 8045	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
72	70, 71	eqtrd 2226	. . . 4 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
73	72	oveq1d 5934	. . 3 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
74	56	nn0cnd 9298	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
75		2txmxeqx 9116	. . . 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 2230	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( 2 x. K ) )
78	6, 77	sylan9eqr 2248	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 2164   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   < clt 8056   - cmin 8192   # cap 8602   / cdiv 8693   NNcn 8984   2c2 9035   4c4 9037   8c8 9041   NN0cn0 9243   ZZcz 9320   |_cfl 10340

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-n0 9244  df-z 9321  df-q 9688  df-rp 9723  df-fl 10342

This theorem is referenced by:  2lgslem3a1  15245