Theorem 2lgslem3a 25121

Description: Lemma for 2lgslem3a1 25125. (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 6657	. . . . 5 |- ( P = ( ( 8 x. K ) + 1 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 1 ) - 1 ) )
3	2	oveq1d 6665	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) )
4		oveq1 6657	. . . . 5 |- ( P = ( ( 8 x. K ) + 1 ) -> ( P / 4 ) = ( ( ( 8 x. K ) + 1 ) / 4 ) )
5	4	fveq2d 6195	. . . 4 |- ( P = ( ( 8 x. K ) + 1 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) )
6	3, 5	oveq12d 6668	. . 3 |- ( P = ( ( 8 x. K ) + 1 ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) )
7	1, 6	syl5eq 2668	. 2 |- ( P = ( ( 8 x. K ) + 1 ) -> N = ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) )
8		8nn0 11315	. . . . . . . . . 10 |- 8 e. NN0
9	8	a1i 11	. . . . . . . . 9 |- ( K e. NN0 -> 8 e. NN0 )
10		id 22	. . . . . . . . 9 |- ( K e. NN0 -> K e. NN0 )
11	9, 10	nn0mulcld 11356	. . . . . . . 8 |- ( K e. NN0 -> ( 8 x. K ) e. NN0 )
12	11	nn0cnd 11353	. . . . . . 7 |- ( K e. NN0 -> ( 8 x. K ) e. CC )
13		pncan1 10454	. . . . . . 7 |- ( ( 8 x. K ) e. CC -> ( ( ( 8 x. K ) + 1 ) - 1 ) = ( 8 x. K ) )
14	12, 13	syl 17	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) - 1 ) = ( 8 x. K ) )
15	14	oveq1d 6665	. . . . 5 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( ( 8 x. K ) / 2 ) )
16		4cn 11098	. . . . . . . . . . 11 |- 4 e. CC
17		2cn 11091	. . . . . . . . . . 11 |- 2 e. CC
18		4t2e8 11181	. . . . . . . . . . 11 |- ( 4 x. 2 ) = 8
19	16, 17, 18	mulcomli 10047	. . . . . . . . . 10 |- ( 2 x. 4 ) = 8
20	19	eqcomi 2631	. . . . . . . . 9 |- 8 = ( 2 x. 4 )
21	20	a1i 11	. . . . . . . 8 |- ( K e. NN0 -> 8 = ( 2 x. 4 ) )
22	21	oveq1d 6665	. . . . . . 7 |- ( K e. NN0 -> ( 8 x. K ) = ( ( 2 x. 4 ) x. K ) )
23	17	a1i 11	. . . . . . . 8 |- ( K e. NN0 -> 2 e. CC )
24	16	a1i 11	. . . . . . . 8 |- ( K e. NN0 -> 4 e. CC )
25		nn0cn 11302	. . . . . . . 8 |- ( K e. NN0 -> K e. CC )
26	23, 24, 25	mulassd 10063	. . . . . . 7 |- ( K e. NN0 -> ( ( 2 x. 4 ) x. K ) = ( 2 x. ( 4 x. K ) ) )
27	22, 26	eqtrd 2656	. . . . . 6 |- ( K e. NN0 -> ( 8 x. K ) = ( 2 x. ( 4 x. K ) ) )
28	27	oveq1d 6665	. . . . 5 |- ( K e. NN0 -> ( ( 8 x. K ) / 2 ) = ( ( 2 x. ( 4 x. K ) ) / 2 ) )
29		4nn0 11311	. . . . . . . . 9 |- 4 e. NN0
30	29	a1i 11	. . . . . . . 8 |- ( K e. NN0 -> 4 e. NN0 )
31	30, 10	nn0mulcld 11356	. . . . . . 7 |- ( K e. NN0 -> ( 4 x. K ) e. NN0 )
32	31	nn0cnd 11353	. . . . . 6 |- ( K e. NN0 -> ( 4 x. K ) e. CC )
33		2ne0 11113	. . . . . . 7 |- 2 =/= 0
34	33	a1i 11	. . . . . 6 |- ( K e. NN0 -> 2 =/= 0 )
35	32, 23, 34	divcan3d 10806	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. ( 4 x. K ) ) / 2 ) = ( 4 x. K ) )
36	15, 28, 35	3eqtrd 2660	. . . 4 |- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) = ( 4 x. K ) )
37		1cnd 10056	. . . . . . . 8 |- ( K e. NN0 -> 1 e. CC )
38		4ne0 11117	. . . . . . . . . 10 |- 4 =/= 0
39	16, 38	pm3.2i 471	. . . . . . . . 9 |- ( 4 e. CC /\ 4 =/= 0 )
40	39	a1i 11	. . . . . . . 8 |- ( K e. NN0 -> ( 4 e. CC /\ 4 =/= 0 ) )
41		divdir 10710	. . . . . . . 8 |- ( ( ( 8 x. K ) e. CC /\ 1 e. CC /\ ( 4 e. CC /\ 4 =/= 0 ) ) -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) )
42	12, 37, 40, 41	syl3anc 1326	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) )
43		8cn 11106	. . . . . . . . . . 11 |- 8 e. CC
44	43	a1i 11	. . . . . . . . . 10 |- ( K e. NN0 -> 8 e. CC )
45		div23 10704	. . . . . . . . . 10 |- ( ( 8 e. CC /\ K e. CC /\ ( 4 e. CC /\ 4 =/= 0 ) ) -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
46	44, 25, 40, 45	syl3anc 1326	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) )
47	18	eqcomi 2631	. . . . . . . . . . . . 13 |- 8 = ( 4 x. 2 )
48	47	oveq1i 6660	. . . . . . . . . . . 12 |- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 )
49	17, 16, 38	divcan3i 10771	. . . . . . . . . . . 12 |- ( ( 4 x. 2 ) / 4 ) = 2
50	48, 49	eqtri 2644	. . . . . . . . . . 11 |- ( 8 / 4 ) = 2
51	50	a1i 11	. . . . . . . . . 10 |- ( K e. NN0 -> ( 8 / 4 ) = 2 )
52	51	oveq1d 6665	. . . . . . . . 9 |- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) )
53	46, 52	eqtrd 2656	. . . . . . . 8 |- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) )
54	53	oveq1d 6665	. . . . . . 7 |- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 1 / 4 ) ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
55	42, 54	eqtrd 2656	. . . . . 6 |- ( K e. NN0 -> ( ( ( 8 x. K ) + 1 ) / 4 ) = ( ( 2 x. K ) + ( 1 / 4 ) ) )
56	55	fveq2d 6195	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) )
57		1lt4 11199	. . . . . 6 |- 1 < 4
58		2nn0 11309	. . . . . . . . . 10 |- 2 e. NN0
59	58	a1i 11	. . . . . . . . 9 |- ( K e. NN0 -> 2 e. NN0 )
60	59, 10	nn0mulcld 11356	. . . . . . . 8 |- ( K e. NN0 -> ( 2 x. K ) e. NN0 )
61	60	nn0zd 11480	. . . . . . 7 |- ( K e. NN0 -> ( 2 x. K ) e. ZZ )
62		1nn0 11308	. . . . . . . 8 |- 1 e. NN0
63	62	a1i 11	. . . . . . 7 |- ( K e. NN0 -> 1 e. NN0 )
64		4nn 11187	. . . . . . . 8 |- 4 e. NN
65	64	a1i 11	. . . . . . 7 |- ( K e. NN0 -> 4 e. NN )
66		adddivflid 12619	. . . . . . 7 |- ( ( ( 2 x. K ) e. ZZ /\ 1 e. NN0 /\ 4 e. NN ) -> ( 1 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) = ( 2 x. K ) ) )
67	61, 63, 65, 66	syl3anc 1326	. . . . . 6 |- ( K e. NN0 -> ( 1 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) = ( 2 x. K ) ) )
68	57, 67	mpbii 223	. . . . 5 |- ( K e. NN0 -> ( |_ ` ( ( 2 x. K ) + ( 1 / 4 ) ) ) = ( 2 x. K ) )
69	56, 68	eqtrd 2656	. . . 4 |- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) = ( 2 x. K ) )
70	36, 69	oveq12d 6668	. . 3 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( ( 4 x. K ) - ( 2 x. K ) ) )
71		2t2e4 11177	. . . . . . . 8 |- ( 2 x. 2 ) = 4
72	71	eqcomi 2631	. . . . . . 7 |- 4 = ( 2 x. 2 )
73	72	a1i 11	. . . . . 6 |- ( K e. NN0 -> 4 = ( 2 x. 2 ) )
74	73	oveq1d 6665	. . . . 5 |- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) )
75	23, 23, 25	mulassd 10063	. . . . 5 |- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) )
76	74, 75	eqtrd 2656	. . . 4 |- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) )
77	76	oveq1d 6665	. . 3 |- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) )
78	60	nn0cnd 11353	. . . 4 |- ( K e. NN0 -> ( 2 x. K ) e. CC )
79		2txmxeqx 11149	. . . 4 |- ( ( 2 x. K ) e. CC -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
80	78, 79	syl 17	. . 3 |- ( K e. NN0 -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) )
81	70, 77, 80	3eqtrd 2660	. 2 |- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 1 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 1 ) / 4 ) ) ) = ( 2 x. K ) )
82	7, 81	sylan9eqr 2678	1 |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 1 ) ) -> N = ( 2 x. K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   4c4 11072   8c8 11076   NN0cn0 11292   ZZcz 11377   |_cfl 12591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593

This theorem is referenced by:  2lgslem3a1  25125