Theorem gausslemma2dlem2 25092

Description: Lemma 2 for gausslemma2d 25099. (Contributed by AV, 4-Jul-2021.)

Hypotheses

Ref	Expression
gausslemma2d.p	|- ( ph -> P e. ( Prime \ { 2 } ) )
gausslemma2d.h	|- H = ( ( P - 1 ) / 2 )
gausslemma2d.r	|- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )
gausslemma2d.m	|- M = ( |_ ` ( P / 4 ) )

Assertion

Ref	Expression
gausslemma2dlem2	|- ( ph -> A. k e. ( 1 ... M ) ( R ` k ) = ( k x. 2 ) )

Distinct variable groups:   x, H   x, P   ph, x   k, H   R, k   ph, k   x, M   x, k

Allowed substitution hints:   P( k)   R( x)   M( k)

Proof of Theorem gausslemma2dlem2

Step	Hyp	Ref	Expression
1		gausslemma2d.r	. . . 4 |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )
2	1	a1i 11	. . 3 |- ( ( ph /\ k e. ( 1 ... M ) ) -> R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) ) )
3		oveq1 6657	. . . . . . 7 |- ( x = k -> ( x x. 2 ) = ( k x. 2 ) )
4	3	breq1d 4663	. . . . . 6 |- ( x = k -> ( ( x x. 2 ) < ( P / 2 ) <-> ( k x. 2 ) < ( P / 2 ) ) )
5	3	oveq2d 6666	. . . . . 6 |- ( x = k -> ( P - ( x x. 2 ) ) = ( P - ( k x. 2 ) ) )
6	4, 3, 5	ifbieq12d 4113	. . . . 5 |- ( x = k -> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) = if ( ( k x. 2 ) < ( P / 2 ) , ( k x. 2 ) , ( P - ( k x. 2 ) ) ) )
7	6	adantl 482	. . . 4 |- ( ( ( ph /\ k e. ( 1 ... M ) ) /\ x = k ) -> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) = if ( ( k x. 2 ) < ( P / 2 ) , ( k x. 2 ) , ( P - ( k x. 2 ) ) ) )
8		elfz1b 12409	. . . . . . . 8 |- ( k e. ( 1 ... M ) <-> ( k e. NN /\ M e. NN /\ k <_ M ) )
9		nnre 11027	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
10	9	adantr 481	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> k e. RR )
11		nnre 11027	. . . . . . . . . . . 12 |- ( M e. NN -> M e. RR )
12	11	adantl 482	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> M e. RR )
13		2re 11090	. . . . . . . . . . . . 13 |- 2 e. RR
14		2pos 11112	. . . . . . . . . . . . 13 |- 0 < 2
15	13, 14	pm3.2i 471	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
16	15	a1i 11	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> ( 2 e. RR /\ 0 < 2 ) )
17		lemul1 10875	. . . . . . . . . . 11 |- ( ( k e. RR /\ M e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( k <_ M <-> ( k x. 2 ) <_ ( M x. 2 ) ) )
18	10, 12, 16, 17	syl3anc 1326	. . . . . . . . . 10 |- ( ( k e. NN /\ M e. NN ) -> ( k <_ M <-> ( k x. 2 ) <_ ( M x. 2 ) ) )
19		gausslemma2d.p	. . . . . . . . . . . . . . 15 |- ( ph -> P e. ( Prime \ { 2 } ) )
20		gausslemma2d.m	. . . . . . . . . . . . . . 15 |- M = ( |_ ` ( P / 4 ) )
21	19, 20	gausslemma2dlem0e 25085	. . . . . . . . . . . . . 14 |- ( ph -> ( M x. 2 ) < ( P / 2 ) )
22	21	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( k e. NN /\ M e. NN ) /\ ph ) -> ( M x. 2 ) < ( P / 2 ) )
23	13	a1i 11	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> 2 e. RR )
24	9, 23	remulcld 10070	. . . . . . . . . . . . . . 15 |- ( k e. NN -> ( k x. 2 ) e. RR )
25	24	adantr 481	. . . . . . . . . . . . . 14 |- ( ( k e. NN /\ M e. NN ) -> ( k x. 2 ) e. RR )
26	13	a1i 11	. . . . . . . . . . . . . . . 16 |- ( M e. NN -> 2 e. RR )
27	11, 26	remulcld 10070	. . . . . . . . . . . . . . 15 |- ( M e. NN -> ( M x. 2 ) e. RR )
28	27	adantl 482	. . . . . . . . . . . . . 14 |- ( ( k e. NN /\ M e. NN ) -> ( M x. 2 ) e. RR )
29	19	gausslemma2dlem0a 25081	. . . . . . . . . . . . . . . 16 |- ( ph -> P e. NN )
30	29	nnred 11035	. . . . . . . . . . . . . . 15 |- ( ph -> P e. RR )
31	30	rehalfcld 11279	. . . . . . . . . . . . . 14 |- ( ph -> ( P / 2 ) e. RR )
32		lelttr 10128	. . . . . . . . . . . . . 14 |- ( ( ( k x. 2 ) e. RR /\ ( M x. 2 ) e. RR /\ ( P / 2 ) e. RR ) -> ( ( ( k x. 2 ) <_ ( M x. 2 ) /\ ( M x. 2 ) < ( P / 2 ) ) -> ( k x. 2 ) < ( P / 2 ) ) )
33	25, 28, 31, 32	syl2an3an 1386	. . . . . . . . . . . . 13 |- ( ( ( k e. NN /\ M e. NN ) /\ ph ) -> ( ( ( k x. 2 ) <_ ( M x. 2 ) /\ ( M x. 2 ) < ( P / 2 ) ) -> ( k x. 2 ) < ( P / 2 ) ) )
34	22, 33	mpan2d 710	. . . . . . . . . . . 12 |- ( ( ( k e. NN /\ M e. NN ) /\ ph ) -> ( ( k x. 2 ) <_ ( M x. 2 ) -> ( k x. 2 ) < ( P / 2 ) ) )
35	34	ex 450	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> ( ph -> ( ( k x. 2 ) <_ ( M x. 2 ) -> ( k x. 2 ) < ( P / 2 ) ) ) )
36	35	com23 86	. . . . . . . . . 10 |- ( ( k e. NN /\ M e. NN ) -> ( ( k x. 2 ) <_ ( M x. 2 ) -> ( ph -> ( k x. 2 ) < ( P / 2 ) ) ) )
37	18, 36	sylbid 230	. . . . . . . . 9 |- ( ( k e. NN /\ M e. NN ) -> ( k <_ M -> ( ph -> ( k x. 2 ) < ( P / 2 ) ) ) )
38	37	3impia 1261	. . . . . . . 8 |- ( ( k e. NN /\ M e. NN /\ k <_ M ) -> ( ph -> ( k x. 2 ) < ( P / 2 ) ) )
39	8, 38	sylbi 207	. . . . . . 7 |- ( k e. ( 1 ... M ) -> ( ph -> ( k x. 2 ) < ( P / 2 ) ) )
40	39	impcom 446	. . . . . 6 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( k x. 2 ) < ( P / 2 ) )
41	40	adantr 481	. . . . 5 |- ( ( ( ph /\ k e. ( 1 ... M ) ) /\ x = k ) -> ( k x. 2 ) < ( P / 2 ) )
42	41	iftrued 4094	. . . 4 |- ( ( ( ph /\ k e. ( 1 ... M ) ) /\ x = k ) -> if ( ( k x. 2 ) < ( P / 2 ) , ( k x. 2 ) , ( P - ( k x. 2 ) ) ) = ( k x. 2 ) )
43	7, 42	eqtrd 2656	. . 3 |- ( ( ( ph /\ k e. ( 1 ... M ) ) /\ x = k ) -> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) = ( k x. 2 ) )
44	19, 20	gausslemma2dlem0d 25084	. . . . . . 7 |- ( ph -> M e. NN0 )
45	44	nn0zd 11480	. . . . . 6 |- ( ph -> M e. ZZ )
46		gausslemma2d.h	. . . . . . . 8 |- H = ( ( P - 1 ) / 2 )
47	19, 46	gausslemma2dlem0b 25082	. . . . . . 7 |- ( ph -> H e. NN )
48	47	nnzd 11481	. . . . . 6 |- ( ph -> H e. ZZ )
49	19, 20, 46	gausslemma2dlem0g 25087	. . . . . 6 |- ( ph -> M <_ H )
50		eluz2 11693	. . . . . 6 |- ( H e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ H e. ZZ /\ M <_ H ) )
51	45, 48, 49, 50	syl3anbrc 1246	. . . . 5 |- ( ph -> H e. ( ZZ>= ` M ) )
52		fzss2 12381	. . . . 5 |- ( H e. ( ZZ>= ` M ) -> ( 1 ... M ) C_ ( 1 ... H ) )
53	51, 52	syl 17	. . . 4 |- ( ph -> ( 1 ... M ) C_ ( 1 ... H ) )
54	53	sselda 3603	. . 3 |- ( ( ph /\ k e. ( 1 ... M ) ) -> k e. ( 1 ... H ) )
55		ovexd 6680	. . 3 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( k x. 2 ) e. _V )
56	2, 43, 54, 55	fvmptd 6288	. 2 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( R ` k ) = ( k x. 2 ) )
57	56	ralrimiva 2966	1 |- ( ph -> A. k e. ( 1 ... M ) ( R ` k ) = ( k x. 2 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   4c4 11072   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   |_cfl 12591   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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386

This theorem is referenced by:  gausslemma2dlem6  25097