Theorem 4sqlem5 12576

Description: Lemma for 4sq 12604. (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 9466	. . . 4 |- ( ph -> A e. CC )
3		4sqlem5.4	. . . . 5 |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
4		zq 9717	. . . . . . . . . 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 9464	. . . . . . . . . 10 |- ( ph -> M e. ZZ )
8		2nn 9169	. . . . . . . . . 10 |- 2 e. NN
9		znq 9715	. . . . . . . . . 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 9726	. . . . . . . . 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 9724	. . . . . . . . 9 |- ( M e. NN -> M e. QQ )
14	6, 13	syl 14	. . . . . . . 8 |- ( ph -> M e. QQ )
15	6	nngt0d 9051	. . . . . . . 8 |- ( ph -> 0 < M )
16	12, 14, 15	modqcld 10437	. . . . . . 7 |- ( ph -> ( ( A + ( M / 2 ) ) mod M ) e. QQ )
17		qcn 9725	. . . . . . 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 9020	. . . . . . . 8 |- ( ph -> M e. RR )
20	19	rehalfcld 9255	. . . . . . 7 |- ( ph -> ( M / 2 ) e. RR )
21	20	recnd 8072	. . . . . 6 |- ( ph -> ( M / 2 ) e. CC )
22	18, 21	subcld 8354	. . . . 5 |- ( ph -> ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) e. CC )
23	3, 22	eqeltrid 2283	. . . 4 |- ( ph -> B e. CC )
24	2, 23	nncand 8359	. . 3 |- ( ph -> ( A - ( A - B ) ) = B )
25	2, 23	subcld 8354	. . . . . 6 |- ( ph -> ( A - B ) e. CC )
26	19	recnd 8072	. . . . . 6 |- ( ph -> M e. CC )
27	6	nnap0d 9053	. . . . . 6 |- ( ph -> M # 0 )
28	25, 26, 27	divcanap1d 8835	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) = ( A - B ) )
29	3	oveq2i 5936	. . . . . . . . 9 |- ( A - B ) = ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) )
30	2, 18, 21	subsub3d 8384	. . . . . . . . 9 |- ( ph -> ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
31	29, 30	eqtrid 2241	. . . . . . . 8 |- ( ph -> ( A - B ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
32	31	oveq1d 5940	. . . . . . 7 |- ( ph -> ( ( A - B ) / M ) = ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) )
33		modqdifz 10445	. . . . . . . 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 1249	. . . . . . 7 |- ( ph -> ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) e. ZZ )
35	32, 34	eqeltrd 2273	. . . . . 6 |- ( ph -> ( ( A - B ) / M ) e. ZZ )
36	35, 7	zmulcld 9471	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) e. ZZ )
37	28, 36	eqeltrrd 2274	. . . 4 |- ( ph -> ( A - B ) e. ZZ )
38	1, 37	zsubcld 9470	. . 3 |- ( ph -> ( A - ( A - B ) ) e. ZZ )
39	24, 38	eqeltrrd 2274	. 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 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   0cc0 7896   + caddc 7899   x. cmul 7901   < clt 8078   - cmin 8214   / cdiv 8716   NNcn 9007   2c2 9058   ZZcz 9343   QQcq 9710   mod cmo 10431

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-q 9711  df-rp 9746  df-fl 10377  df-mod 10432

This theorem is referenced by:  4sqlem7  12578  4sqlem8  12579  4sqlem9  12580  4sqlem10  12581  4sqlem14  12598  4sqlem15  12599  4sqlem16  12600  4sqlem17  12601  2sqlem8a  15447  2sqlem8  15448