Theorem 4sqlem5 13161

Description: Lemma for 4sq 13189. (Contributed by Mario Carneiro, 15-Jul-2014.)

Hypotheses

Ref	Expression
4sqlem5.2	|- ( ph -> A e. ZZ )
4sqlem5.3	|- ( ph -> M e. NN )
4sqlem5.4	|- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )

Assertion

Ref	Expression
4sqlem5	|- ( ph -> ( B e. ZZ /\ ( ( A - B ) / M ) e. ZZ ) )

Proof of Theorem 4sqlem5

Step	Hyp	Ref	Expression
1		4sqlem5.2	. . . . 5 |- ( ph -> A e. ZZ )
2	1	zcnd 9769	. . . 4 |- ( ph -> A e. CC )
3		4sqlem5.4	. . . . 5 |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
4		zq 10026	. . . . . . . . . 10 |- ( A e. ZZ -> A e. QQ )
5	1, 4	syl 14	. . . . . . . . 9 |- ( ph -> A e. QQ )
6		4sqlem5.3	. . . . . . . . . . 11 |- ( ph -> M e. NN )
7	6	nnzd 9767	. . . . . . . . . 10 |- ( ph -> M e. ZZ )
8		2nn 9466	. . . . . . . . . 10 |- 2 e. NN
9		znq 10024	. . . . . . . . . 10 |- ( ( M e. ZZ /\ 2 e. NN ) -> ( M / 2 ) e. QQ )
10	7, 8, 9	sylancl 417	. . . . . . . . 9 |- ( ph -> ( M / 2 ) e. QQ )
11		qaddcl 10035	. . . . . . . . 9 |- ( ( A e. QQ /\ ( M / 2 ) e. QQ ) -> ( A + ( M / 2 ) ) e. QQ )
12	5, 10, 11	syl2anc 415	. . . . . . . 8 |- ( ph -> ( A + ( M / 2 ) ) e. QQ )
13		nnq 10033	. . . . . . . . 9 |- ( M e. NN -> M e. QQ )
14	6, 13	syl 14	. . . . . . . 8 |- ( ph -> M e. QQ )
15	6	nngt0d 9348	. . . . . . . 8 |- ( ph -> 0 < M )
16	12, 14, 15	modqcld 10765	. . . . . . 7 |- ( ph -> ( ( A + ( M / 2 ) ) mod M ) e. QQ )
17		qcn 10034	. . . . . . 7 |- ( ( ( A + ( M / 2 ) ) mod M ) e. QQ -> ( ( A + ( M / 2 ) ) mod M ) e. CC )
18	16, 17	syl 14	. . . . . 6 |- ( ph -> ( ( A + ( M / 2 ) ) mod M ) e. CC )
19	6	nnred 9317	. . . . . . . 8 |- ( ph -> M e. RR )
20	19	rehalfcld 9552	. . . . . . 7 |- ( ph -> ( M / 2 ) e. RR )
21	20	recnd 8354	. . . . . 6 |- ( ph -> ( M / 2 ) e. CC )
22	18, 21	subcld 8637	. . . . 5 |- ( ph -> ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) e. CC )
23	3, 22	eqeltrid 2325	. . . 4 |- ( ph -> B e. CC )
24	2, 23	nncand 8642	. . 3 |- ( ph -> ( A - ( A - B ) ) = B )
25	2, 23	subcld 8637	. . . . . 6 |- ( ph -> ( A - B ) e. CC )
26	19	recnd 8354	. . . . . 6 |- ( ph -> M e. CC )
27	6	nnap0d 9350	. . . . . 6 |- ( ph -> M # 0 )
28	25, 26, 27	divcanap1d 9121	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) = ( A - B ) )
29	3	oveq2i 6096	. . . . . . . . 9 |- ( A - B ) = ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) )
30	2, 18, 21	subsub3d 8667	. . . . . . . . 9 |- ( ph -> ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
31	29, 30	eqtrid 2283	. . . . . . . 8 |- ( ph -> ( A - B ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
32	31	oveq1d 6100	. . . . . . 7 |- ( ph -> ( ( A - B ) / M ) = ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) )
33		modqdifz 10773	. . . . . . . 8 |- ( ( ( A + ( M / 2 ) ) e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) e. ZZ )
34	12, 14, 15, 33	syl3anc 1278	. . . . . . 7 |- ( ph -> ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) e. ZZ )
35	32, 34	eqeltrd 2315	. . . . . 6 |- ( ph -> ( ( A - B ) / M ) e. ZZ )
36	35, 7	zmulcld 9774	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) e. ZZ )
37	28, 36	eqeltrrd 2316	. . . 4 |- ( ph -> ( A - B ) e. ZZ )
38	1, 37	zsubcld 9773	. . 3 |- ( ph -> ( A - ( A - B ) ) e. ZZ )
39	24, 38	eqeltrrd 2316	. 2 |- ( ph -> B e. ZZ )
40	39, 35	jca 306	1 |- ( ph -> ( B e. ZZ /\ ( ( A - B ) / M ) e. ZZ ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   0cc0 8179   + caddc 8182   x. cmul 8184   < clt 8360   - cmin 8497   / cdiv 9002   NNcn 9304   2c2 9355   ZZcz 9644   QQcq 10019   mod cmo 10759

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-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  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-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-q 10020  df-rp 10055  df-fl 10705  df-mod 10760

This theorem is used by:  4sqlem7  13163  4sqlem8  13164  4sqlem9  13165  4sqlem10  13166  4sqlem14  13183  4sqlem15  13184  4sqlem16  13185  4sqlem17  13186  2sqlem8a  16241  2sqlem8  16242