Theorem 2lgslem1 16193

Description: Lemma 1 for 2lgs 16206. (Contributed by AV, 19-Jun-2021.)

Assertion

Ref	Expression
2lgslem1	|- ( ( P e. Prime /\ -. 2 || P ) -> (♯ ` { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) )

Distinct variable group:   P, i, x

Proof of Theorem 2lgslem1

Dummy variables f y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2lgslem1a 16190	. . 3 |- ( ( P e. Prime /\ -. 2 || P ) -> { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } = { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } )
2	1	fveq2d 5697	. 2 |- ( ( P e. Prime /\ -. 2 || P ) -> (♯ ` { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } ) = (♯ ` { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } ) )
3		prmz 12872	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
4		4nn 9451	. . . . . . . 8 |- 4 e. NN
5		znq 10007	. . . . . . . 8 |- ( ( P e. ZZ /\ 4 e. NN ) -> ( P / 4 ) e. QQ )
6	3, 4, 5	sylancl 417	. . . . . . 7 |- ( P e. Prime -> ( P / 4 ) e. QQ )
7	6	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( P / 4 ) e. QQ )
8	7	flqcld 10695	. . . . 5 |- ( ( P e. Prime /\ -. 2 || P ) -> ( |_ ` ( P / 4 ) ) e. ZZ )
9	8	peano2zd 9754	. . . 4 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( |_ ` ( P / 4 ) ) + 1 ) e. ZZ )
10		nnoddn2prmb 13024	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ -. 2 || P ) )
11		oddprm 13021	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
12	10, 11	sylbir 135	. . . . 5 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( P - 1 ) / 2 ) e. NN )
13	12	nnzd 9750	. . . 4 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( P - 1 ) / 2 ) e. ZZ )
14	9, 13	fzfigd 10851	. . 3 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) e. Fin )
15	14	mptexd 5938	. . . 4 |- ( ( P e. Prime /\ -. 2 || P ) -> ( y e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) |-> ( y x. 2 ) ) e. _V )
16		eqid 2238	. . . . 5 |- ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) = ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) )
17		eqid 2238	. . . . 5 |- ( y e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) |-> ( y x. 2 ) ) = ( y e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) |-> ( y x. 2 ) )
18	16, 17	2lgslem1b 16191	. . . 4 |- ( y e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) |-> ( y x. 2 ) ) : ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) -1-1-onto-> { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) }
19		f1oeq1 5625	. . . . 5 |- ( f = ( y e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) |-> ( y x. 2 ) ) -> ( f : ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) -1-1-onto-> { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } <-> ( y e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) |-> ( y x. 2 ) ) : ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) -1-1-onto-> { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } ) )
20	19	spcegv 2913	. . . 4 |- ( ( y e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) |-> ( y x. 2 ) ) e. _V -> ( ( y e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) |-> ( y x. 2 ) ) : ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) -1-1-onto-> { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } -> E. f f : ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) -1-1-onto-> { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } ) )
21	15, 18, 20	mpisyl 1496	. . 3 |- ( ( P e. Prime /\ -. 2 || P ) -> E. f f : ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) -1-1-onto-> { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } )
22		fihasheqf1oi 11209	. . . . 5 |- ( ( ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) e. Fin /\ f : ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) -1-1-onto-> { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } ) -> (♯ ` ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) ) = (♯ ` { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } ) )
23	22	ex 115	. . . 4 |- ( ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) e. Fin -> ( f : ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) -1-1-onto-> { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } -> (♯ ` ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) ) = (♯ ` { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } ) ) )
24	23	exlimdv 1872	. . 3 |- ( ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) e. Fin -> ( E. f f : ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) -1-1-onto-> { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } -> (♯ ` ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) ) = (♯ ` { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } ) ) )
25	14, 21, 24	sylc 62	. 2 |- ( ( P e. Prime /\ -. 2 || P ) -> (♯ ` ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) ) = (♯ ` { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } ) )
26	6	flqcld 10695	. . . . 5 |- ( P e. Prime -> ( |_ ` ( P / 4 ) ) e. ZZ )
27	26	adantr 276	. . . 4 |- ( ( P e. Prime /\ -. 2 || P ) -> ( |_ ` ( P / 4 ) ) e. ZZ )
28		oddm1d2 12642	. . . . . 6 |- ( P e. ZZ -> ( -. 2 || P <-> ( ( P - 1 ) / 2 ) e. ZZ ) )
29	3, 28	syl 14	. . . . 5 |- ( P e. Prime -> ( -. 2 || P <-> ( ( P - 1 ) / 2 ) e. ZZ ) )
30	29	biimpa 296	. . . 4 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( P - 1 ) / 2 ) e. ZZ )
31		2lgslem1c 16192	. . . 4 |- ( ( P e. Prime /\ -. 2 || P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )
32		eluz2 9910	. . . 4 |- ( ( ( P - 1 ) / 2 ) e. ( ZZ>= ` ( |_ ` ( P / 4 ) ) ) <-> ( ( |_ ` ( P / 4 ) ) e. ZZ /\ ( ( P - 1 ) / 2 ) e. ZZ /\ ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) ) )
33	27, 30, 31, 32	syl3anbrc 1212	. . 3 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( P - 1 ) / 2 ) e. ( ZZ>= ` ( |_ ` ( P / 4 ) ) ) )
34		hashfzp1 11248	. . 3 |- ( ( ( P - 1 ) / 2 ) e. ( ZZ>= ` ( |_ ` ( P / 4 ) ) ) -> (♯ ` ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) )
35	33, 34	syl 14	. 2 |- ( ( P e. Prime /\ -. 2 || P ) -> (♯ ` ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) )
36	2, 25, 35	3eqtr2d 2277	1 |- ( ( P e. Prime /\ -. 2 || P ) -> (♯ ` { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   {crab 2532   _Vcvv 2821   \ cdif 3217   {csn 3708   class class class wbr 4128   |-> cmpt 4190   -1-1-onto->wf1o 5374   ` cfv 5375  (class class class)co 6079   Fincfn 7016   1c1 8174   + caddc 8176   x. cmul 8178   < clt 8354   <_ cle 8355   - cmin 8491   / cdiv 8996   NNcn 9287   2c2 9338   4c4 9340   ZZcz 9627   ZZ>=cuz 9904   QQcq 10002   ...cfz 10394   |_cfl 10686   mod cmo 10742  ♯chash 11197   || 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-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  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  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-dc 847  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-nul 3521  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-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  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-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-2o 6682  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  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-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-dvds 12538  df-prm 12869

This theorem is referenced by:  2lgs  16206