Theorem gausslemma2dlem2 15778

Description: Lemma 2 for gausslemma2d 15785. (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	. . 3 |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )
2		oveq1 6018	. . . . . . 7 |- ( x = k -> ( x x. 2 ) = ( k x. 2 ) )
3	2	breq1d 4094	. . . . . 6 |- ( x = k -> ( ( x x. 2 ) < ( P / 2 ) <-> ( k x. 2 ) < ( P / 2 ) ) )
4	2	oveq2d 6027	. . . . . 6 |- ( x = k -> ( P - ( x x. 2 ) ) = ( P - ( k x. 2 ) ) )
5	3, 2, 4	ifbieq12d 3630	. . . . 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 ) ) ) )
6	5	adantl 277	. . . 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 ) ) ) )
7		elfz1b 10313	. . . . . . . 8 |- ( k e. ( 1 ... M ) <-> ( k e. NN /\ M e. NN /\ k <_ M ) )
8		nnre 9138	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
9	8	adantr 276	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> k e. RR )
10		nnre 9138	. . . . . . . . . . . 12 |- ( M e. NN -> M e. RR )
11	10	adantl 277	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> M e. RR )
12		2re 9201	. . . . . . . . . . . . 13 |- 2 e. RR
13		2pos 9222	. . . . . . . . . . . . 13 |- 0 < 2
14	12, 13	pm3.2i 272	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
15	14	a1i 9	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> ( 2 e. RR /\ 0 < 2 ) )
16		lemul1 8761	. . . . . . . . . . 11 |- ( ( k e. RR /\ M e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( k <_ M <-> ( k x. 2 ) <_ ( M x. 2 ) ) )
17	9, 11, 15, 16	syl3anc 1271	. . . . . . . . . 10 |- ( ( k e. NN /\ M e. NN ) -> ( k <_ M <-> ( k x. 2 ) <_ ( M x. 2 ) ) )
18		gausslemma2d.p	. . . . . . . . . . . . . . 15 |- ( ph -> P e. ( Prime \ { 2 } ) )
19		gausslemma2d.m	. . . . . . . . . . . . . . 15 |- M = ( |_ ` ( P / 4 ) )
20	18, 19	gausslemma2dlem0e 15769	. . . . . . . . . . . . . 14 |- ( ph -> ( M x. 2 ) < ( P / 2 ) )
21	20	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( k e. NN /\ M e. NN ) /\ ph ) -> ( M x. 2 ) < ( P / 2 ) )
22	12	a1i 9	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> 2 e. RR )
23	8, 22	remulcld 8198	. . . . . . . . . . . . . . 15 |- ( k e. NN -> ( k x. 2 ) e. RR )
24	23	adantr 276	. . . . . . . . . . . . . 14 |- ( ( k e. NN /\ M e. NN ) -> ( k x. 2 ) e. RR )
25	12	a1i 9	. . . . . . . . . . . . . . . 16 |- ( M e. NN -> 2 e. RR )
26	10, 25	remulcld 8198	. . . . . . . . . . . . . . 15 |- ( M e. NN -> ( M x. 2 ) e. RR )
27	26	adantl 277	. . . . . . . . . . . . . 14 |- ( ( k e. NN /\ M e. NN ) -> ( M x. 2 ) e. RR )
28	18	gausslemma2dlem0a 15765	. . . . . . . . . . . . . . . 16 |- ( ph -> P e. NN )
29	28	nnred 9144	. . . . . . . . . . . . . . 15 |- ( ph -> P e. RR )
30	29	rehalfcld 9379	. . . . . . . . . . . . . 14 |- ( ph -> ( P / 2 ) e. RR )
31		lelttr 8256	. . . . . . . . . . . . . 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 ) ) )
32	24, 27, 30, 31	syl2an3an 1332	. . . . . . . . . . . . 13 |- ( ( ( k e. NN /\ M e. NN ) /\ ph ) -> ( ( ( k x. 2 ) <_ ( M x. 2 ) /\ ( M x. 2 ) < ( P / 2 ) ) -> ( k x. 2 ) < ( P / 2 ) ) )
33	21, 32	mpan2d 428	. . . . . . . . . . . 12 |- ( ( ( k e. NN /\ M e. NN ) /\ ph ) -> ( ( k x. 2 ) <_ ( M x. 2 ) -> ( k x. 2 ) < ( P / 2 ) ) )
34	33	ex 115	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> ( ph -> ( ( k x. 2 ) <_ ( M x. 2 ) -> ( k x. 2 ) < ( P / 2 ) ) ) )
35	34	com23 78	. . . . . . . . . 10 |- ( ( k e. NN /\ M e. NN ) -> ( ( k x. 2 ) <_ ( M x. 2 ) -> ( ph -> ( k x. 2 ) < ( P / 2 ) ) ) )
36	17, 35	sylbid 150	. . . . . . . . 9 |- ( ( k e. NN /\ M e. NN ) -> ( k <_ M -> ( ph -> ( k x. 2 ) < ( P / 2 ) ) ) )
37	36	3impia 1224	. . . . . . . 8 |- ( ( k e. NN /\ M e. NN /\ k <_ M ) -> ( ph -> ( k x. 2 ) < ( P / 2 ) ) )
38	7, 37	sylbi 121	. . . . . . 7 |- ( k e. ( 1 ... M ) -> ( ph -> ( k x. 2 ) < ( P / 2 ) ) )
39	38	impcom 125	. . . . . 6 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( k x. 2 ) < ( P / 2 ) )
40	39	adantr 276	. . . . 5 |- ( ( ( ph /\ k e. ( 1 ... M ) ) /\ x = k ) -> ( k x. 2 ) < ( P / 2 ) )
41	40	iftrued 3610	. . . 4 |- ( ( ( ph /\ k e. ( 1 ... M ) ) /\ x = k ) -> if ( ( k x. 2 ) < ( P / 2 ) , ( k x. 2 ) , ( P - ( k x. 2 ) ) ) = ( k x. 2 ) )
42	6, 41	eqtrd 2262	. . 3 |- ( ( ( ph /\ k e. ( 1 ... M ) ) /\ x = k ) -> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) = ( k x. 2 ) )
43	18, 19	gausslemma2dlem0d 15768	. . . . . . 7 |- ( ph -> M e. NN0 )
44	43	nn0zd 9588	. . . . . 6 |- ( ph -> M e. ZZ )
45		gausslemma2d.h	. . . . . . . 8 |- H = ( ( P - 1 ) / 2 )
46	18, 45	gausslemma2dlem0b 15766	. . . . . . 7 |- ( ph -> H e. NN )
47	46	nnzd 9589	. . . . . 6 |- ( ph -> H e. ZZ )
48	18, 19, 45	gausslemma2dlem0g 15771	. . . . . 6 |- ( ph -> M <_ H )
49		eluz2 9749	. . . . . 6 |- ( H e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ H e. ZZ /\ M <_ H ) )
50	44, 47, 48, 49	syl3anbrc 1205	. . . . 5 |- ( ph -> H e. ( ZZ>= ` M ) )
51		fzss2 10287	. . . . 5 |- ( H e. ( ZZ>= ` M ) -> ( 1 ... M ) C_ ( 1 ... H ) )
52	50, 51	syl 14	. . . 4 |- ( ph -> ( 1 ... M ) C_ ( 1 ... H ) )
53	52	sselda 3225	. . 3 |- ( ( ph /\ k e. ( 1 ... M ) ) -> k e. ( 1 ... H ) )
54	53	elfzelzd 10249	. . . 4 |- ( ( ph /\ k e. ( 1 ... M ) ) -> k e. ZZ )
55		2z 9495	. . . . 5 |- 2 e. ZZ
56	55	a1i 9	. . . 4 |- ( ( ph /\ k e. ( 1 ... M ) ) -> 2 e. ZZ )
57	54, 56	zmulcld 9596	. . 3 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( k x. 2 ) e. ZZ )
58	1, 42, 53, 57	fvmptd2 5722	. 2 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( R ` k ) = ( k x. 2 ) )
59	58	ralrimiva 2603	1 |- ( ph -> A. k e. ( 1 ... M ) ( R ` k ) = ( k x. 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   \ cdif 3195   C_ wss 3198   ifcif 3603   {csn 3667   class class class wbr 4084   |-> cmpt 4146   ` cfv 5322  (class class class)co 6011   RRcr 8019   0cc0 8020   1c1 8021   x. cmul 8025   < clt 8202   <_ cle 8203   - cmin 8338   / cdiv 8840   NNcn 9131   2c2 9182   4c4 9184   ZZcz 9467   ZZ>=cuz 9743   ...cfz 10231   |_cfl 10516   Primecprime 12666

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137  ax-pre-mulext 8138  ax-arch 8139  ax-caucvg 8140

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-po 4389  df-iso 4390  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-2o 6576  df-er 6695  df-en 6903  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-div 8841  df-inn 9132  df-2 9190  df-3 9191  df-4 9192  df-n0 9391  df-z 9468  df-uz 9744  df-q 9842  df-rp 9877  df-fz 10232  df-fl 10518  df-seqfrec 10698  df-exp 10789  df-cj 11390  df-re 11391  df-im 11392  df-rsqrt 11546  df-abs 11547  df-dvds 12336  df-prm 12667

This theorem is referenced by:  gausslemma2dlem6  15783