Theorem 4sqlem5 12334

Description: Lemma for 4sq (not yet proved here). (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 9335	. . . 4 |- ( ph -> A e. CC )
3		4sqlem5.4	. . . . 5 |- B = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
4		zq 9585	. . . . . . . . . 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 9333	. . . . . . . . . 10 |- ( ph -> M e. ZZ )
8		2nn 9039	. . . . . . . . . 10 |- 2 e. NN
9		znq 9583	. . . . . . . . . 10 |- ( ( M e. ZZ /\ 2 e. NN ) -> ( M / 2 ) e. QQ )
10	7, 8, 9	sylancl 411	. . . . . . . . 9 |- ( ph -> ( M / 2 ) e. QQ )
11		qaddcl 9594	. . . . . . . . 9 |- ( ( A e. QQ /\ ( M / 2 ) e. QQ ) -> ( A + ( M / 2 ) ) e. QQ )
12	5, 10, 11	syl2anc 409	. . . . . . . 8 |- ( ph -> ( A + ( M / 2 ) ) e. QQ )
13		nnq 9592	. . . . . . . . 9 |- ( M e. NN -> M e. QQ )
14	6, 13	syl 14	. . . . . . . 8 |- ( ph -> M e. QQ )
15	6	nngt0d 8922	. . . . . . . 8 |- ( ph -> 0 < M )
16	12, 14, 15	modqcld 10284	. . . . . . 7 |- ( ph -> ( ( A + ( M / 2 ) ) mod M ) e. QQ )
17		qcn 9593	. . . . . . 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 8891	. . . . . . . 8 |- ( ph -> M e. RR )
20	19	rehalfcld 9124	. . . . . . 7 |- ( ph -> ( M / 2 ) e. RR )
21	20	recnd 7948	. . . . . 6 |- ( ph -> ( M / 2 ) e. CC )
22	18, 21	subcld 8230	. . . . 5 |- ( ph -> ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) e. CC )
23	3, 22	eqeltrid 2257	. . . 4 |- ( ph -> B e. CC )
24	2, 23	nncand 8235	. . 3 |- ( ph -> ( A - ( A - B ) ) = B )
25	2, 23	subcld 8230	. . . . . 6 |- ( ph -> ( A - B ) e. CC )
26	19	recnd 7948	. . . . . 6 |- ( ph -> M e. CC )
27	6	nnap0d 8924	. . . . . 6 |- ( ph -> M # 0 )
28	25, 26, 27	divcanap1d 8708	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) = ( A - B ) )
29	3	oveq2i 5864	. . . . . . . . 9 |- ( A - B ) = ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) )
30	2, 18, 21	subsub3d 8260	. . . . . . . . 9 |- ( ph -> ( A - ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
31	29, 30	eqtrid 2215	. . . . . . . 8 |- ( ph -> ( A - B ) = ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) )
32	31	oveq1d 5868	. . . . . . 7 |- ( ph -> ( ( A - B ) / M ) = ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) )
33		modqdifz 10292	. . . . . . . 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 1233	. . . . . . 7 |- ( ph -> ( ( ( A + ( M / 2 ) ) - ( ( A + ( M / 2 ) ) mod M ) ) / M ) e. ZZ )
35	32, 34	eqeltrd 2247	. . . . . 6 |- ( ph -> ( ( A - B ) / M ) e. ZZ )
36	35, 7	zmulcld 9340	. . . . 5 |- ( ph -> ( ( ( A - B ) / M ) x. M ) e. ZZ )
37	28, 36	eqeltrrd 2248	. . . 4 |- ( ph -> ( A - B ) e. ZZ )
38	1, 37	zsubcld 9339	. . 3 |- ( ph -> ( A - ( A - B ) ) e. ZZ )
39	24, 38	eqeltrrd 2248	. 2 |- ( ph -> B e. ZZ )
40	39, 35	jca 304	1 |- ( ph -> ( B e. ZZ /\ ( ( A - B ) / M ) e. ZZ ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   + caddc 7777   x. cmul 7779   < clt 7954   - cmin 8090   / cdiv 8589   NNcn 8878   2c2 8929   ZZcz 9212   QQcq 9578   mod cmo 10278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-q 9579  df-rp 9611  df-fl 10226  df-mod 10279

This theorem is referenced by:  4sqlem7  12336  4sqlem8  12337  4sqlem9  12338  4sqlem10  12339  2sqlem8a  13752  2sqlem8  13753