Theorem 2lgslem1c 15822

Description: Lemma 3 for 2lgslem1 15823. (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 12684	. . . 4 |- ( P e. Prime -> P e. NN )
2		nnnn0 9409	. . . 4 |- ( P e. NN -> P e. NN0 )
3		oddnn02np1 12443	. . . 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 3610	. . . . . . . . . 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 9305	. . . . . . . . . . 11 |- 2 e. NN
8		nn0ledivnn 10002	. . . . . . . . . . 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 4110	. . . . . . . 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 3613	. . . . . . . . . 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 9411	. . . . . . . . . . . 12 |- ( n e. NN0 -> n e. RR )
16		peano2rem 8446	. . . . . . . . . . . . 13 |- ( n e. RR -> ( n - 1 ) e. RR )
17	16	rehalfcld 9391	. . . . . . . . . . . 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 9391	. . . . . . . . . . 11 |- ( n e. NN0 -> ( n / 2 ) e. RR )
20	15	lem1d 9113	. . . . . . . . . . . 12 |- ( n e. NN0 -> ( n - 1 ) <_ n )
21	15, 16	syl 14	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( n - 1 ) e. RR )
22		2re 9213	. . . . . . . . . . . . . . 15 |- 2 e. RR
23		2pos 9234	. . . . . . . . . . . . . . 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 9049	. . . . . . . . . . . . 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 1273	. . . . . . . . . . . 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 8303	. . . . . . . . . 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 4110	. . . . . . . 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 9499	. . . . . . . 8 |- ( n e. NN0 -> n e. ZZ )
34		zeo3 12431	. . . . . . . 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 725	. . . . . 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 2233	. . . . . . 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 12499	. . . . . 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 6025	. . . . . . . . . 10 |- ( P = ( ( 2 x. n ) + 1 ) -> ( P - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) )
44	43	eqcoms 2234	. . . . . . . . 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 9419	. . . . . . . . . . . . 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 9460	. . . . . . . . . . 11 |- ( n e. NN0 -> ( 2 x. n ) e. NN0 )
50	49	nn0cnd 9457	. . . . . . . . . 10 |- ( n e. NN0 -> ( 2 x. n ) e. CC )
51		pncan1 8556	. . . . . . . . . 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 2264	. . . . . . 7 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( P - 1 ) = ( 2 x. n ) )
55	54	oveq1d 6033	. . . . . 6 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( P - 1 ) / 2 ) = ( ( 2 x. n ) / 2 ) )
56		nn0cn 9412	. . . . . . . 8 |- ( n e. NN0 -> n e. CC )
57		2cnd 9216	. . . . . . . 8 |- ( n e. NN0 -> 2 e. CC )
58		2ap0 9236	. . . . . . . . 9 |- 2 # 0
59	58	a1i 9	. . . . . . . 8 |- ( n e. NN0 -> 2 # 0 )
60	56, 57, 59	divcanap3d 8975	. . . . . . 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 2264	. . . . 5 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( P - 1 ) / 2 ) = n )
63	37, 42, 62	3brtr4d 4120	. . . 4 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )
64	63	rexlimdva2 2653	. . 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 715   = wceq 1397   e. wcel 2202   E.wrex 2511   ifcif 3605   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   < clt 8214   <_ cle 8215   - cmin 8350   # cap 8761   / cdiv 8852   NNcn 9143   2c2 9194   4c4 9196   NN0cn0 9402   ZZcz 9479   |_cfl 10529   || cdvds 12350   Primecprime 12681

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-q 9854  df-rp 9889  df-fl 10531  df-dvds 12351  df-prm 12682

This theorem is referenced by:  2lgslem1  15823