Theorem gausslemma2dlem2 16398

Description: Lemma 2 for gausslemma2d 16405. (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 6095	. . . . . . 7 |- ( x = k -> ( x x. 2 ) = ( k x. 2 ) )
3	2	breq1d 4141	. . . . . 6 |- ( x = k -> ( ( x x. 2 ) < ( P / 2 ) <-> ( k x. 2 ) < ( P / 2 ) ) )
4	2	oveq2d 6104	. . . . . 6 |- ( x = k -> ( P - ( x x. 2 ) ) = ( P - ( k x. 2 ) ) )
5	3, 2, 4	ifbieq12d 3667	. . . . 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 10515	. . . . . . . 8 |- ( k e. ( 1 ... M ) <-> ( k e. NN /\ M e. NN /\ k <_ M ) )
8		nnre 9321	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
9	8	adantr 276	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> k e. RR )
10		nnre 9321	. . . . . . . . . . . 12 |- ( M e. NN -> M e. RR )
11	10	adantl 277	. . . . . . . . . . 11 |- ( ( k e. NN /\ M e. NN ) -> M e. RR )
12		2re 9384	. . . . . . . . . . . . 13 |- 2 e. RR
13		2pos 9405	. . . . . . . . . . . . 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 8930	. . . . . . . . . . 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 1278	. . . . . . . . . 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 16389	. . . . . . . . . . . . . 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 8363	. . . . . . . . . . . . . . 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 8363	. . . . . . . . . . . . . . 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 16385	. . . . . . . . . . . . . . . 16 |- ( ph -> P e. NN )
29	28	nnred 9327	. . . . . . . . . . . . . . 15 |- ( ph -> P e. RR )
30	29	rehalfcld 9564	. . . . . . . . . . . . . 14 |- ( ph -> ( P / 2 ) e. RR )
31		lelttr 8421	. . . . . . . . . . . . . 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 1339	. . . . . . . . . . . . 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 432	. . . . . . . . . . . 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 1231	. . . . . . . 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 3647	. . . 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 2271	. . 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 16388	. . . . . . 7 |- ( ph -> M e. NN0 )
44	43	nn0zd 9778	. . . . . 6 |- ( ph -> M e. ZZ )
45		gausslemma2d.h	. . . . . . . 8 |- H = ( ( P - 1 ) / 2 )
46	18, 45	gausslemma2dlem0b 16386	. . . . . . 7 |- ( ph -> H e. NN )
47	46	nnzd 9779	. . . . . 6 |- ( ph -> H e. ZZ )
48	18, 19, 45	gausslemma2dlem0g 16391	. . . . . 6 |- ( ph -> M <_ H )
49		eluz2 9944	. . . . . 6 |- ( H e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ H e. ZZ /\ M <_ H ) )
50	44, 47, 48, 49	syl3anbrc 1212	. . . . 5 |- ( ph -> H e. ( ZZ>= ` M ) )
51		fzss2 10488	. . . . 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 3248	. . 3 |- ( ( ph /\ k e. ( 1 ... M ) ) -> k e. ( 1 ... H ) )
54	53	elfzelzd 10447	. . . 4 |- ( ( ph /\ k e. ( 1 ... M ) ) -> k e. ZZ )
55		2z 9684	. . . . 5 |- 2 e. ZZ
56	55	a1i 9	. . . 4 |- ( ( ph /\ k e. ( 1 ... M ) ) -> 2 e. ZZ )
57	54, 56	zmulcld 9786	. . 3 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( k x. 2 ) e. ZZ )
58	1, 42, 53, 57	fvmptd2 5790	. 2 |- ( ( ph /\ k e. ( 1 ... M ) ) -> ( R ` k ) = ( k x. 2 ) )
59	58	ralrimiva 2623	1 |- ( ph -> A. k e. ( 1 ... M ) ( R ` k ) = ( k x. 2 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   \ cdif 3217   C_ wss 3220   ifcif 3638   {csn 3709   class class class wbr 4131   |-> cmpt 4193   ` cfv 5380  (class class class)co 6088   RRcr 8185   0cc0 8186   1c1 8187   x. cmul 8191   < clt 8367   <_ cle 8368   - cmin 8505   / cdiv 9011   NNcn 9314   2c2 9365   4c4 9367   ZZcz 9656   ZZ>=cuz 9938   ...cfz 10429   |_cfl 10721   Primecprime 12911

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-2o 6692  df-er 6811  df-en 7028  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-fz 10430  df-fl 10723  df-seqfrec 10907  df-exp 10998  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-dvds 12581  df-prm 12912

This theorem is used by:  gausslemma2dlem6  16403