Theorem 2lgslem1c 25118

Description: Lemma 3 for 2lgslem1 25119. (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 15388	. . . 4 |- ( P e. Prime -> P e. NN )
2		nnnn0 11299	. . . 4 |- ( P e. NN -> P e. NN0 )
3		oddnn02np1 15072	. . . 4 |- ( P e. NN0 -> ( -. 2 || P <-> E. n e. NN0 ( ( 2 x. n ) + 1 ) = P ) )
4	1, 2, 3	3syl 18	. . 3 |- ( P e. Prime -> ( -. 2 || P <-> E. n e. NN0 ( ( 2 x. n ) + 1 ) = P ) )
5		iftrue 4092	. . . . . . . . . 10 |- ( 2 || n -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) = ( n / 2 ) )
6	5	adantr 481	. . . . . . . . 9 |- ( ( 2 || n /\ n e. NN0 ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) = ( n / 2 ) )
7		2nn 11185	. . . . . . . . . . 11 |- 2 e. NN
8		nn0ledivnn 11941	. . . . . . . . . . 11 |- ( ( n e. NN0 /\ 2 e. NN ) -> ( n / 2 ) <_ n )
9	7, 8	mpan2 707	. . . . . . . . . 10 |- ( n e. NN0 -> ( n / 2 ) <_ n )
10	9	adantl 482	. . . . . . . . 9 |- ( ( 2 || n /\ n e. NN0 ) -> ( n / 2 ) <_ n )
11	6, 10	eqbrtrd 4675	. . . . . . . 8 |- ( ( 2 || n /\ n e. NN0 ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n )
12		iffalse 4095	. . . . . . . . . 10 |- ( -. 2 || n -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) = ( ( n - 1 ) / 2 ) )
13	12	adantr 481	. . . . . . . . 9 |- ( ( -. 2 || n /\ n e. NN0 ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) = ( ( n - 1 ) / 2 ) )
14		nn0re 11301	. . . . . . . . . . . 12 |- ( n e. NN0 -> n e. RR )
15		peano2rem 10348	. . . . . . . . . . . . 13 |- ( n e. RR -> ( n - 1 ) e. RR )
16	15	rehalfcld 11279	. . . . . . . . . . . 12 |- ( n e. RR -> ( ( n - 1 ) / 2 ) e. RR )
17	14, 16	syl 17	. . . . . . . . . . 11 |- ( n e. NN0 -> ( ( n - 1 ) / 2 ) e. RR )
18	14	rehalfcld 11279	. . . . . . . . . . 11 |- ( n e. NN0 -> ( n / 2 ) e. RR )
19	14	lem1d 10957	. . . . . . . . . . . 12 |- ( n e. NN0 -> ( n - 1 ) <_ n )
20	14, 15	syl 17	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( n - 1 ) e. RR )
21		2re 11090	. . . . . . . . . . . . . . 15 |- 2 e. RR
22		2pos 11112	. . . . . . . . . . . . . . 15 |- 0 < 2
23	21, 22	pm3.2i 471	. . . . . . . . . . . . . 14 |- ( 2 e. RR /\ 0 < 2 )
24	23	a1i 11	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( 2 e. RR /\ 0 < 2 ) )
25		lediv1 10888	. . . . . . . . . . . . 13 |- ( ( ( n - 1 ) e. RR /\ n e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( n - 1 ) <_ n <-> ( ( n - 1 ) / 2 ) <_ ( n / 2 ) ) )
26	20, 14, 24, 25	syl3anc 1326	. . . . . . . . . . . 12 |- ( n e. NN0 -> ( ( n - 1 ) <_ n <-> ( ( n - 1 ) / 2 ) <_ ( n / 2 ) ) )
27	19, 26	mpbid 222	. . . . . . . . . . 11 |- ( n e. NN0 -> ( ( n - 1 ) / 2 ) <_ ( n / 2 ) )
28	17, 18, 14, 27, 9	letrd 10194	. . . . . . . . . 10 |- ( n e. NN0 -> ( ( n - 1 ) / 2 ) <_ n )
29	28	adantl 482	. . . . . . . . 9 |- ( ( -. 2 || n /\ n e. NN0 ) -> ( ( n - 1 ) / 2 ) <_ n )
30	13, 29	eqbrtrd 4675	. . . . . . . 8 |- ( ( -. 2 || n /\ n e. NN0 ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n )
31	11, 30	pm2.61ian 831	. . . . . . 7 |- ( n e. NN0 -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n )
32	31	ad2antlr 763	. . . . . 6 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n )
33		nn0z 11400	. . . . . . . 8 |- ( n e. NN0 -> n e. ZZ )
34	33	adantl 482	. . . . . . 7 |- ( ( P e. Prime /\ n e. NN0 ) -> n e. ZZ )
35		eqcom 2629	. . . . . . . 8 |- ( ( ( 2 x. n ) + 1 ) = P <-> P = ( ( 2 x. n ) + 1 ) )
36	35	biimpi 206	. . . . . . 7 |- ( ( ( 2 x. n ) + 1 ) = P -> P = ( ( 2 x. n ) + 1 ) )
37		flodddiv4 15137	. . . . . . 7 |- ( ( n e. ZZ /\ P = ( ( 2 x. n ) + 1 ) ) -> ( |_ ` ( P / 4 ) ) = if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) )
38	34, 36, 37	syl2an 494	. . . . . 6 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( |_ ` ( P / 4 ) ) = if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) )
39		oveq1 6657	. . . . . . . . . . 11 |- ( P = ( ( 2 x. n ) + 1 ) -> ( P - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) )
40	39	eqcoms 2630	. . . . . . . . . 10 |- ( ( ( 2 x. n ) + 1 ) = P -> ( P - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) )
41	40	adantl 482	. . . . . . . . 9 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( P - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) )
42		2nn0 11309	. . . . . . . . . . . . . 14 |- 2 e. NN0
43	42	a1i 11	. . . . . . . . . . . . 13 |- ( n e. NN0 -> 2 e. NN0 )
44		id 22	. . . . . . . . . . . . 13 |- ( n e. NN0 -> n e. NN0 )
45	43, 44	nn0mulcld 11356	. . . . . . . . . . . 12 |- ( n e. NN0 -> ( 2 x. n ) e. NN0 )
46	45	nn0cnd 11353	. . . . . . . . . . 11 |- ( n e. NN0 -> ( 2 x. n ) e. CC )
47		pncan1 10454	. . . . . . . . . . 11 |- ( ( 2 x. n ) e. CC -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) )
48	46, 47	syl 17	. . . . . . . . . 10 |- ( n e. NN0 -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) )
49	48	ad2antlr 763	. . . . . . . . 9 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) )
50	41, 49	eqtrd 2656	. . . . . . . 8 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( P - 1 ) = ( 2 x. n ) )
51	50	oveq1d 6665	. . . . . . 7 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( P - 1 ) / 2 ) = ( ( 2 x. n ) / 2 ) )
52		nn0cn 11302	. . . . . . . . 9 |- ( n e. NN0 -> n e. CC )
53		2cnd 11093	. . . . . . . . 9 |- ( n e. NN0 -> 2 e. CC )
54		2ne0 11113	. . . . . . . . . 10 |- 2 =/= 0
55	54	a1i 11	. . . . . . . . 9 |- ( n e. NN0 -> 2 =/= 0 )
56	52, 53, 55	divcan3d 10806	. . . . . . . 8 |- ( n e. NN0 -> ( ( 2 x. n ) / 2 ) = n )
57	56	ad2antlr 763	. . . . . . 7 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( 2 x. n ) / 2 ) = n )
58	51, 57	eqtrd 2656	. . . . . 6 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( P - 1 ) / 2 ) = n )
59	32, 38, 58	3brtr4d 4685	. . . . 5 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )
60	59	ex 450	. . . 4 |- ( ( P e. Prime /\ n e. NN0 ) -> ( ( ( 2 x. n ) + 1 ) = P -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) ) )
61	60	rexlimdva 3031	. . 3 |- ( P e. Prime -> ( E. n e. NN0 ( ( 2 x. n ) + 1 ) = P -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) ) )
62	4, 61	sylbid 230	. 2 |- ( P e. Prime -> ( -. 2 || P -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) ) )
63	62	imp 445	1 |- ( ( P e. Prime /\ -. 2 || P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   4c4 11072   NN0cn0 11292   ZZcz 11377   |_cfl 12591   || cdvds 14983   Primecprime 15385

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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-dvds 14984  df-prm 15386

This theorem is referenced by:  2lgslem1  25119