Theorem 4sqlem5 12905

Description: Lemma for 4sq 12933. (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 9570	. . . 4 |- ( ph -> A e. CC )
3		4sqlem5.4	. . . . 5 |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
4		zq 9821	. . . . . . . . . 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 9568	. . . . . . . . . 10 |- ( ph -> M e. ZZ )
8		2nn 9272	. . . . . . . . . 10 |- 2 e. NN
9		znq 9819	. . . . . . . . . 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 9830	. . . . . . . . 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 9828	. . . . . . . . 9 |- ( M e. NN -> M e. QQ )
14	6, 13	syl 14	. . . . . . . 8 |- ( ph -> M e. QQ )
15	6	nngt0d 9154	. . . . . . . 8 |- ( ph -> 0 < M )
16	12, 14, 15	modqcld 10550	. . . . . . 7 |- ( ph -> ( ( A + ( M / 2 ) ) mod M ) e. QQ )
17		qcn 9829	. . . . . . 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 9123	. . . . . . . 8 |- ( ph -> M e. RR )
20	19	rehalfcld 9358	. . . . . . 7 |- ( ph -> ( M / 2 ) e. RR )
21	20	recnd 8175	. . . . . 6 |- ( ph -> ( M / 2 ) e. CC )
22	18, 21	subcld 8457	. . . . 5 |- ( ph -> ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) e. CC )
23	3, 22	eqeltrid 2316	. . . 4 |- ( ph -> B e. CC )
24	2, 23	nncand 8462	. . 3 |- ( ph -> ( A - ( A - B ) ) = B )
25	2, 23	subcld 8457	. . . . . 6 |- ( ph -> ( A - B ) e. CC )
26	19	recnd 8175	. . . . . 6 |- ( ph -> M e. CC )
27	6	nnap0d 9156	. . . . . 6 |- ( ph -> M # 0 )
28	25, 26, 27	divcanap1d 8938	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) = ( A - B ) )
29	3	oveq2i 6012	. . . . . . . . 9 |- ( A - B ) = ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) )
30	2, 18, 21	subsub3d 8487	. . . . . . . . 9 |- ( ph -> ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
31	29, 30	eqtrid 2274	. . . . . . . 8 |- ( ph -> ( A - B ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
32	31	oveq1d 6016	. . . . . . 7 |- ( ph -> ( ( A - B ) / M ) = ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) )
33		modqdifz 10558	. . . . . . . 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 1271	. . . . . . 7 |- ( ph -> ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) e. ZZ )
35	32, 34	eqeltrd 2306	. . . . . 6 |- ( ph -> ( ( A - B ) / M ) e. ZZ )
36	35, 7	zmulcld 9575	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) e. ZZ )
37	28, 36	eqeltrrd 2307	. . . 4 |- ( ph -> ( A - B ) e. ZZ )
38	1, 37	zsubcld 9574	. . 3 |- ( ph -> ( A - ( A - B ) ) e. ZZ )
39	24, 38	eqeltrrd 2307	. 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 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   CCcc 7997   0cc0 7999   + caddc 8002   x. cmul 8004   < clt 8181   - cmin 8317   / cdiv 8819   NNcn 9110   2c2 9161   ZZcz 9446   QQcq 9814   mod cmo 10544

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-q 9815  df-rp 9850  df-fl 10490  df-mod 10545

This theorem is referenced by:  4sqlem7  12907  4sqlem8  12908  4sqlem9  12909  4sqlem10  12910  4sqlem14  12927  4sqlem15  12928  4sqlem16  12929  4sqlem17  12930  2sqlem8a  15801  2sqlem8  15802