Theorem 2lgslem1c 16192

Description: Lemma 3 for 2lgslem1 16193. (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 12871	. . . 4 |- ( P e. Prime -> P e. NN )
2		nnnn0 9553	. . . 4 |- ( P e. NN -> P e. NN0 )
3		oddnn02np1 12630	. . . 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 3645	. . . . . . . . . 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 9449	. . . . . . . . . . 11 |- 2 e. NN
8		nn0ledivnn 10151	. . . . . . . . . . 11 |- ( ( n e. NN0 /\ 2 e. NN ) -> ( n / 2 ) <_ n )
9	7, 8	mpan2 429	. . . . . . . . . 10 |- ( n e. NN0 -> ( n / 2 ) <_ n )
10	9	adantl 277	. . . . . . . . 9 |- ( ( 2 || n /\ n e. NN0 ) -> ( n / 2 ) <_ n )
11	6, 10	eqbrtrd 4150	. . . . . . . 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 3648	. . . . . . . . . 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 9555	. . . . . . . . . . . 12 |- ( n e. NN0 -> n e. RR )
16		peano2rem 8587	. . . . . . . . . . . . 13 |- ( n e. RR -> ( n - 1 ) e. RR )
17	16	rehalfcld 9535	. . . . . . . . . . . 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 9535	. . . . . . . . . . 11 |- ( n e. NN0 -> ( n / 2 ) e. RR )
20	15	lem1d 9257	. . . . . . . . . . . 12 |- ( n e. NN0 -> ( n - 1 ) <_ n )
21	15, 16	syl 14	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( n - 1 ) e. RR )
22		2re 9357	. . . . . . . . . . . . . . 15 |- 2 e. RR
23		2pos 9378	. . . . . . . . . . . . . . 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 9193	. . . . . . . . . . . . 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 1278	. . . . . . . . . . . 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 8444	. . . . . . . . . 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 4150	. . . . . . . 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 9647	. . . . . . . 8 |- ( n e. NN0 -> n e. ZZ )
34		zeo3 12618	. . . . . . . 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 730	. . . . . 6 |- ( n e. NN0 -> if ( 2 || n , ( n / 2 ) , ( ( n - 1 ) / 2 ) ) <_ n )
37	36	ad2antlr 493	. . . . 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 2240	. . . . . . 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 12686	. . . . . 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 6086	. . . . . . . . . 10 |- ( P = ( ( 2 x. n ) + 1 ) -> ( P - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) )
44	43	eqcoms 2241	. . . . . . . . 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 9563	. . . . . . . . . . . . 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 9608	. . . . . . . . . . 11 |- ( n e. NN0 -> ( 2 x. n ) e. NN0 )
50	49	nn0cnd 9605	. . . . . . . . . 10 |- ( n e. NN0 -> ( 2 x. n ) e. CC )
51		pncan1 8698	. . . . . . . . . 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 493	. . . . . . . 8 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) )
54	45, 53	eqtrd 2271	. . . . . . 7 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( P - 1 ) = ( 2 x. n ) )
55	54	oveq1d 6094	. . . . . 6 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( P - 1 ) / 2 ) = ( ( 2 x. n ) / 2 ) )
56		nn0cn 9556	. . . . . . . 8 |- ( n e. NN0 -> n e. CC )
57		2cnd 9360	. . . . . . . 8 |- ( n e. NN0 -> 2 e. CC )
58		2ap0 9380	. . . . . . . . 9 |- 2 # 0
59	58	a1i 9	. . . . . . . 8 |- ( n e. NN0 -> 2 # 0 )
60	56, 57, 59	divcanap3d 9119	. . . . . . 7 |- ( n e. NN0 -> ( ( 2 x. n ) / 2 ) = n )
61	60	ad2antlr 493	. . . . . 6 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( 2 x. n ) / 2 ) = n )
62	55, 61	eqtrd 2271	. . . . 5 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( ( P - 1 ) / 2 ) = n )
63	37, 42, 62	3brtr4d 4160	. . . 4 |- ( ( ( P e. Prime /\ n e. NN0 ) /\ ( ( 2 x. n ) + 1 ) = P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )
64	63	rexlimdva2 2671	. . 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 720   = wceq 1402   e. wcel 2209   E.wrex 2529   ifcif 3638   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   CCcc 8171   RRcr 8172   0cc0 8173   1c1 8174   + caddc 8176   x. cmul 8178   < clt 8354   <_ cle 8355   - cmin 8491   # cap 8903   / cdiv 8996   NNcn 9287   2c2 9338   4c4 9340   NN0cn0 9546   ZZcz 9627   |_cfl 10686   || cdvds 12537   Primecprime 12868

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-q 10003  df-rp 10038  df-fl 10688  df-dvds 12538  df-prm 12869

This theorem is referenced by:  2lgslem1  16193