Theorem 4sqlem5 12973

Description: Lemma for 4sq 13001. (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 9603	. . . 4 |- ( ph -> A e. CC )
3		4sqlem5.4	. . . . 5 |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
4		zq 9860	. . . . . . . . . 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 9601	. . . . . . . . . 10 |- ( ph -> M e. ZZ )
8		2nn 9305	. . . . . . . . . 10 |- 2 e. NN
9		znq 9858	. . . . . . . . . 10 |- ( ( M e. ZZ /\ 2 e. NN ) -> ( M / 2 ) e. QQ )
10	7, 8, 9	sylancl 413	. . . . . . . . 9 |- ( ph -> ( M / 2 ) e. QQ )
11		qaddcl 9869	. . . . . . . . 9 |- ( ( A e. QQ /\ ( M / 2 ) e. QQ ) -> ( A + ( M / 2 ) ) e. QQ )
12	5, 10, 11	syl2anc 411	. . . . . . . 8 |- ( ph -> ( A + ( M / 2 ) ) e. QQ )
13		nnq 9867	. . . . . . . . 9 |- ( M e. NN -> M e. QQ )
14	6, 13	syl 14	. . . . . . . 8 |- ( ph -> M e. QQ )
15	6	nngt0d 9187	. . . . . . . 8 |- ( ph -> 0 < M )
16	12, 14, 15	modqcld 10591	. . . . . . 7 |- ( ph -> ( ( A + ( M / 2 ) ) mod M ) e. QQ )
17		qcn 9868	. . . . . . 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 9156	. . . . . . . 8 |- ( ph -> M e. RR )
20	19	rehalfcld 9391	. . . . . . 7 |- ( ph -> ( M / 2 ) e. RR )
21	20	recnd 8208	. . . . . 6 |- ( ph -> ( M / 2 ) e. CC )
22	18, 21	subcld 8490	. . . . 5 |- ( ph -> ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) e. CC )
23	3, 22	eqeltrid 2318	. . . 4 |- ( ph -> B e. CC )
24	2, 23	nncand 8495	. . 3 |- ( ph -> ( A - ( A - B ) ) = B )
25	2, 23	subcld 8490	. . . . . 6 |- ( ph -> ( A - B ) e. CC )
26	19	recnd 8208	. . . . . 6 |- ( ph -> M e. CC )
27	6	nnap0d 9189	. . . . . 6 |- ( ph -> M # 0 )
28	25, 26, 27	divcanap1d 8971	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) = ( A - B ) )
29	3	oveq2i 6029	. . . . . . . . 9 |- ( A - B ) = ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) )
30	2, 18, 21	subsub3d 8520	. . . . . . . . 9 |- ( ph -> ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
31	29, 30	eqtrid 2276	. . . . . . . 8 |- ( ph -> ( A - B ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
32	31	oveq1d 6033	. . . . . . 7 |- ( ph -> ( ( A - B ) / M ) = ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) )
33		modqdifz 10599	. . . . . . . 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 1273	. . . . . . 7 |- ( ph -> ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) e. ZZ )
35	32, 34	eqeltrd 2308	. . . . . 6 |- ( ph -> ( ( A - B ) / M ) e. ZZ )
36	35, 7	zmulcld 9608	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) e. ZZ )
37	28, 36	eqeltrrd 2309	. . . 4 |- ( ph -> ( A - B ) e. ZZ )
38	1, 37	zsubcld 9607	. . 3 |- ( ph -> ( A - ( A - B ) ) e. ZZ )
39	24, 38	eqeltrrd 2309	. 2 |- ( ph -> B e. ZZ )
40	39, 35	jca 306	1 |- ( ph -> ( B e. ZZ /\ ( ( A - B ) / M ) e. ZZ ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6018   CCcc 8030   0cc0 8032   + caddc 8035   x. cmul 8037   < clt 8214   - cmin 8350   / cdiv 8852   NNcn 9143   2c2 9194   ZZcz 9479   QQcq 9853   mod cmo 10585

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-q 9854  df-rp 9889  df-fl 10531  df-mod 10586

This theorem is referenced by:  4sqlem7  12975  4sqlem8  12976  4sqlem9  12977  4sqlem10  12978  4sqlem14  12995  4sqlem15  12996  4sqlem16  12997  4sqlem17  12998  2sqlem8a  15870  2sqlem8  15871