Theorem gausslemma2dlem2 15861

Description: Lemma 2 for gausslemma2d 15868. (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 6035	. . . . . . 7 |- ( x = k -> ( x x. 2 ) = ( k x. 2 ) )
3	2	breq1d 4103	. . . . . 6 |- ( x = k -> ( ( x x. 2 ) < ( P / 2 ) <-> ( k x. 2 ) < ( P / 2 ) ) )
4	2	oveq2d 6044	. . . . . 6 |- ( x = k -> ( P - ( x x. 2 ) ) = ( P - ( k x. 2 ) ) )
5	3, 2, 4	ifbieq12d 3636	. . . . 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 10368	. . . . . . . 8 |- ( k e. ( 1 ... M ) <-> ( k e. NN /\ M e. NN /\ k <_ M ) )
8		nnre 9193	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
9	8	adantr 276	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> k e. RR )
10		nnre 9193	. . . . . . . . . . . 12 |- ( M e. NN -> M e. RR )
11	10	adantl 277	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> M e. RR )
12		2re 9256	. . . . . . . . . . . . 13 |- 2 e. RR
13		2pos 9277	. . . . . . . . . . . . 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 8816	. . . . . . . . . . 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 1274	. . . . . . . . . 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 15852	. . . . . . . . . . . . . 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 8253	. . . . . . . . . . . . . . 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 8253	. . . . . . . . . . . . . . 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 15848	. . . . . . . . . . . . . . . 16 |- ( ph -> P e. NN )
29	28	nnred 9199	. . . . . . . . . . . . . . 15 |- ( ph -> P e. RR )
30	29	rehalfcld 9434	. . . . . . . . . . . . . 14 |- ( ph -> ( P / 2 ) e. RR )
31		lelttr 8311	. . . . . . . . . . . . . 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 1335	. . . . . . . . . . . . 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 1227	. . . . . . . 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 3616	. . . 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 2264	. . 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 15851	. . . . . . 7 |- ( ph -> M e. NN0 )
44	43	nn0zd 9643	. . . . . 6 |- ( ph -> M e. ZZ )
45		gausslemma2d.h	. . . . . . . 8 |- H = ( ( P - 1 ) / 2 )
46	18, 45	gausslemma2dlem0b 15849	. . . . . . 7 |- ( ph -> H e. NN )
47	46	nnzd 9644	. . . . . 6 |- ( ph -> H e. ZZ )
48	18, 19, 45	gausslemma2dlem0g 15854	. . . . . 6 |- ( ph -> M <_ H )
49		eluz2 9804	. . . . . 6 |- ( H e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ H e. ZZ /\ M <_ H ) )
50	44, 47, 48, 49	syl3anbrc 1208	. . . . 5 |- ( ph -> H e. ( ZZ>= ` M ) )
51		fzss2 10342	. . . . 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 3228	. . 3 |- ( ( ph /\ k e. ( 1 ... M ) ) -> k e. ( 1 ... H ) )
54	53	elfzelzd 10304	. . . 4 |- ( ( ph /\ k e. ( 1 ... M ) ) -> k e. ZZ )
55		2z 9550	. . . . 5 |- 2 e. ZZ
56	55	a1i 9	. . . 4 |- ( ( ph /\ k e. ( 1 ... M ) ) -> 2 e. ZZ )
57	54, 56	zmulcld 9651	. . 3 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( k x. 2 ) e. ZZ )
58	1, 42, 53, 57	fvmptd2 5737	. 2 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( R ` k ) = ( k x. 2 ) )
59	58	ralrimiva 2606	1 |- ( ph -> A. k e. ( 1 ... M ) ( R ` k ) = ( k x. 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   \ cdif 3198   C_ wss 3201   ifcif 3607   {csn 3673   class class class wbr 4093   |-> cmpt 4155   ` cfv 5333  (class class class)co 6028   RRcr 8074   0cc0 8075   1c1 8076   x. cmul 8080   < clt 8257   <_ cle 8258   - cmin 8393   / cdiv 8895   NNcn 9186   2c2 9237   4c4 9239   ZZcz 9522   ZZ>=cuz 9798   ...cfz 10286   |_cfl 10572   Primecprime 12740

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-n0 9446  df-z 9523  df-uz 9799  df-q 9897  df-rp 9932  df-fz 10287  df-fl 10574  df-seqfrec 10754  df-exp 10845  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-dvds 12410  df-prm 12741

This theorem is referenced by:  gausslemma2dlem6  15866