Theorem 2lgslem1c 15777

Description: Lemma 3 for 2lgslem1 15778. (Contributed by AV, 19-Jun-2021.)

Assertion

Ref	Expression
2lgslem1c	|- ( ( P e. Prime /\ -. 2 || P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )

Proof of Theorem 2lgslem1c

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmnn 12640	. . . 4 |- ( P e. Prime -> P e. NN )
2		nnnn0 9384	. . . 4 |- ( P e. NN -> P e. NN0 )
3		oddnn02np1 12399	. . . 4 |- ( P e. NN0 -> ( -. 2 || P <-> E. n e. NN0 ( ( 2 x. n ) + 1 ) = P ) )
4	1, 2, 3	3syl 17	. . 3 |- ( P e. Prime -> ( -. 2 || P <-> E. n e. NN0 ( ( 2 x. n ) + 1 ) = P ) )
5		iftrue 3607	. . . . . . . . . 10 |- ( 2 || n -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) = ( n / 2 ) )
6	5	adantr 276	. . . . . . . . 9 |- ( ( 2 || n /\ n e. NN0 ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) = ( n / 2 ) )
7		2nn 9280	. . . . . . . . . . 11 |- 2 e. NN
8		nn0ledivnn 9971	. . . . . . . . . . 11 |- ( ( n e. NN0 /\ 2 e. NN ) -> ( n / 2 ) <_ n )
9	7, 8	mpan2 425	. . . . . . . . . 10 |- ( n e. NN0 -> ( n / 2 ) <_ n )
10	9	adantl 277	. . . . . . . . 9 |- ( ( 2 || n /\ n e. NN0 ) -> ( n / 2 ) <_ n )
11	6, 10	eqbrtrd 4105	. . . . . . . 8 |- ( ( 2 || n /\ n e. NN0 ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n )
12	11	expcom 116	. . . . . . 7 |- ( n e. NN0 -> ( 2 || n -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n ) )
13		iffalse 3610	. . . . . . . . . 10 |- ( -. 2 || n -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) = ( ( n - 1 ) / 2 ) )
14	13	adantr 276	. . . . . . . . 9 |- ( ( -. 2 || n /\ n e. NN0 ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) = ( ( n - 1 ) / 2 ) )
15		nn0re 9386	. . . . . . . . . . . 12 |- ( n e. NN0 -> n e. RR )
16		peano2rem 8421	. . . . . . . . . . . . 13 |- ( n e. RR -> ( n - 1 ) e. RR )
17	16	rehalfcld 9366	. . . . . . . . . . . 12 |- ( n e. RR -> ( ( n - 1 ) / 2 ) e. RR )
18	15, 17	syl 14	. . . . . . . . . . 11 |- ( n e. NN0 -> ( ( n - 1 ) / 2 ) e. RR )
19	15	rehalfcld 9366	. . . . . . . . . . 11 |- ( n e. NN0 -> ( n / 2 ) e. RR )
20	15	lem1d 9088	. . . . . . . . . . . 12 |- ( n e. NN0 -> ( n - 1 ) <_ n )
21	15, 16	syl 14	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( n - 1 ) e. RR )
22		2re 9188	. . . . . . . . . . . . . . 15 |- 2 e. RR
23		2pos 9209	. . . . . . . . . . . . . . 15 |- 0 < 2
24	22, 23	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( 2 e. RR /\ 0 < 2 )
25	24	a1i 9	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( 2 e. RR /\ 0 < 2 ) )
26		lediv1 9024	. . . . . . . . . . . . 13 |- ( ( ( n - 1 ) e. RR /\ n e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( n - 1 ) <_ n <-> ( ( n - 1 ) / 2 ) <_ ( n / 2 ) ) )
27	21, 15, 25, 26	syl3anc 1271	. . . . . . . . . . . 12 |- ( n e. NN0 -> ( ( n - 1 ) <_ n <-> ( ( n - 1 ) / 2 ) <_ ( n / 2 ) ) )
28	20, 27	mpbid 147	. . . . . . . . . . 11 |- ( n e. NN0 -> ( ( n - 1 ) / 2 ) <_ ( n / 2 ) )
29	18, 19, 15, 28, 9	letrd 8278	. . . . . . . . . 10 |- ( n e. NN0 -> ( ( n - 1 ) / 2 ) <_ n )
30	29	adantl 277	. . . . . . . . 9 |- ( ( -. 2 || n /\ n e. NN0 ) -> ( ( n - 1 ) / 2 ) <_ n )
31	14, 30	eqbrtrd 4105	. . . . . . . 8 |- ( ( -. 2 || n /\ n e. NN0 ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n )
32	31	expcom 116	. . . . . . 7 |- ( n e. NN0 -> ( -. 2 || n -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n ) )
33		nn0z 9474	. . . . . . . 8 |- ( n e. NN0 -> n e. ZZ )
34		zeo3 12387	. . . . . . . 8 |- ( n e. ZZ -> ( 2 || n \/ -. 2 || n ) )
35	33, 34	syl 14	. . . . . . 7 |- ( n e. NN0 -> ( 2 || n \/ -. 2 || n ) )
36	12, 32, 35	mpjaod 723	. . . . . 6 |- ( n e. NN0 -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n )
37	36	ad2antlr 489	. . . . 5 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n )
38	33	adantl 277	. . . . . 6 |- ( ( P e. Prime /\ n e. NN0 ) -> n e. ZZ )
39		eqcom 2231	. . . . . . 7 |- ( ( ( 2 x. n ) + 1 ) = P <-> P = ( ( 2 x. n ) + 1 ) )
40	39	biimpi 120	. . . . . 6 |- ( ( ( 2 x. n ) + 1 ) = P -> P = ( ( 2 x. n ) + 1 ) )
41		flodddiv4 12455	. . . . . 6 |- ( ( n e. ZZ /\ P = ( ( 2 x. n ) + 1 ) ) -> ( |_ ` ( P / 4 ) ) = if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) )
42	38, 40, 41	syl2an 289	. . . . 5 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( |_ ` ( P / 4 ) ) = if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) )
43		oveq1 6014	. . . . . . . . . 10 |- ( P = ( ( 2 x. n ) + 1 ) -> ( P - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) )
44	43	eqcoms 2232	. . . . . . . . 9 |- ( ( ( 2 x. n ) + 1 ) = P -> ( P - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) )
45	44	adantl 277	. . . . . . . 8 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( P - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) )
46		2nn0 9394	. . . . . . . . . . . . 13 |- 2 e. NN0
47	46	a1i 9	. . . . . . . . . . . 12 |- ( n e. NN0 -> 2 e. NN0 )
48		id 19	. . . . . . . . . . . 12 |- ( n e. NN0 -> n e. NN0 )
49	47, 48	nn0mulcld 9435	. . . . . . . . . . 11 |- ( n e. NN0 -> ( 2 x. n ) e. NN0 )
50	49	nn0cnd 9432	. . . . . . . . . 10 |- ( n e. NN0 -> ( 2 x. n ) e. CC )
51		pncan1 8531	. . . . . . . . . 10 |- ( ( 2 x. n ) e. CC -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) )
52	50, 51	syl 14	. . . . . . . . 9 |- ( n e. NN0 -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) )
53	52	ad2antlr 489	. . . . . . . 8 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) )
54	45, 53	eqtrd 2262	. . . . . . 7 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( P - 1 ) = ( 2 x. n ) )
55	54	oveq1d 6022	. . . . . 6 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( P - 1 ) / 2 ) = ( ( 2 x. n ) / 2 ) )
56		nn0cn 9387	. . . . . . . 8 |- ( n e. NN0 -> n e. CC )
57		2cnd 9191	. . . . . . . 8 |- ( n e. NN0 -> 2 e. CC )
58		2ap0 9211	. . . . . . . . 9 |- 2 # 0
59	58	a1i 9	. . . . . . . 8 |- ( n e. NN0 -> 2 # 0 )
60	56, 57, 59	divcanap3d 8950	. . . . . . 7 |- ( n e. NN0 -> ( ( 2 x. n ) / 2 ) = n )
61	60	ad2antlr 489	. . . . . 6 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( 2 x. n ) / 2 ) = n )
62	55, 61	eqtrd 2262	. . . . 5 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( P - 1 ) / 2 ) = n )
63	37, 42, 62	3brtr4d 4115	. . . 4 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )
64	63	rexlimdva2 2651	. . 3 |- ( P e. Prime -> ( E. n e. NN0 ( ( 2 x. n ) + 1 ) = P -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) ) )
65	4, 64	sylbid 150	. 2 |- ( P e. Prime -> ( -. 2 || P -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) ) )
66	65	imp 124	1 |- ( ( P e. Prime /\ -. 2 || P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   E.wrex 2509   ifcif 3602   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8005   RRcr 8006   0cc0 8007   1c1 8008   + caddc 8010   x. cmul 8012   < clt 8189   <_ cle 8190   - cmin 8325   # cap 8736   / cdiv 8827   NNcn 9118   2c2 9169   4c4 9171   NN0cn0 9377   ZZcz 9454   |_cfl 10496   || cdvds 12306   Primecprime 12637

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-z 9455  df-q 9823  df-rp 9858  df-fl 10498  df-dvds 12307  df-prm 12638

This theorem is referenced by:  2lgslem1  15778