Theorem lgsdir2lem2 25051

Description: Lemma for lgsdir2 25055. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypotheses

Ref	Expression
lgsdir2lem2.1	|- ( K e. ZZ /\ 2 || ( K + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... K ) -> ( A mod 8 ) e. S ) ) )
lgsdir2lem2.2	|- M = ( K + 1 )
lgsdir2lem2.3	|- N = ( M + 1 )
lgsdir2lem2.4	|- N e. S

Assertion

Ref	Expression
lgsdir2lem2	|- ( N e. ZZ /\ 2 || ( N + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... N ) -> ( A mod 8 ) e. S ) ) )

Proof of Theorem lgsdir2lem2

Step	Hyp	Ref	Expression
1		lgsdir2lem2.3	. . 3 |- N = ( M + 1 )
2		lgsdir2lem2.2	. . . . 5 |- M = ( K + 1 )
3		lgsdir2lem2.1	. . . . . . 7 |- ( K e. ZZ /\ 2 || ( K + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... K ) -> ( A mod 8 ) e. S ) ) )
4	3	simp1i 1070	. . . . . 6 |- K e. ZZ
5		peano2z 11418	. . . . . 6 |- ( K e. ZZ -> ( K + 1 ) e. ZZ )
6	4, 5	ax-mp 5	. . . . 5 |- ( K + 1 ) e. ZZ
7	2, 6	eqeltri 2697	. . . 4 |- M e. ZZ
8		peano2z 11418	. . . 4 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
9	7, 8	ax-mp 5	. . 3 |- ( M + 1 ) e. ZZ
10	1, 9	eqeltri 2697	. 2 |- N e. ZZ
11	3	simp2i 1071	. . . 4 |- 2 || ( K + 1 )
12		2z 11409	. . . . 5 |- 2 e. ZZ
13		dvdsadd 15024	. . . . 5 |- ( ( 2 e. ZZ /\ ( K + 1 ) e. ZZ ) -> ( 2 || ( K + 1 ) <-> 2 || ( 2 + ( K + 1 ) ) ) )
14	12, 6, 13	mp2an 708	. . . 4 |- ( 2 || ( K + 1 ) <-> 2 || ( 2 + ( K + 1 ) ) )
15	11, 14	mpbi 220	. . 3 |- 2 || ( 2 + ( K + 1 ) )
16		zcn 11382	. . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
17	4, 16	ax-mp 5	. . . . . . . . . 10 |- K e. CC
18		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
19	17, 18	addcomi 10227	. . . . . . . . 9 |- ( K + 1 ) = ( 1 + K )
20	2, 19	eqtri 2644	. . . . . . . 8 |- M = ( 1 + K )
21	20	oveq1i 6660	. . . . . . 7 |- ( M + 1 ) = ( ( 1 + K ) + 1 )
22	1, 21	eqtri 2644	. . . . . 6 |- N = ( ( 1 + K ) + 1 )
23		df-2 11079	. . . . . . . 8 |- 2 = ( 1 + 1 )
24	23	oveq1i 6660	. . . . . . 7 |- ( 2 + K ) = ( ( 1 + 1 ) + K )
25	18, 17, 18	add32i 10259	. . . . . . 7 |- ( ( 1 + K ) + 1 ) = ( ( 1 + 1 ) + K )
26	24, 25	eqtr4i 2647	. . . . . 6 |- ( 2 + K ) = ( ( 1 + K ) + 1 )
27	22, 26	eqtr4i 2647	. . . . 5 |- N = ( 2 + K )
28	27	oveq1i 6660	. . . 4 |- ( N + 1 ) = ( ( 2 + K ) + 1 )
29		2cn 11091	. . . . 5 |- 2 e. CC
30	29, 17, 18	addassi 10048	. . . 4 |- ( ( 2 + K ) + 1 ) = ( 2 + ( K + 1 ) )
31	28, 30	eqtri 2644	. . 3 |- ( N + 1 ) = ( 2 + ( K + 1 ) )
32	15, 31	breqtrri 4680	. 2 |- 2 || ( N + 1 )
33		elfzuz2 12346	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... N ) -> N e. ( ZZ>= ` 0 ) )
34		fzm1 12420	. . . . 5 |- ( N e. ( ZZ>= ` 0 ) -> ( ( A mod 8 ) e. ( 0 ... N ) <-> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) ) )
35	33, 34	syl 17	. . . 4 |- ( ( A mod 8 ) e. ( 0 ... N ) -> ( ( A mod 8 ) e. ( 0 ... N ) <-> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) ) )
36	35	ibi 256	. . 3 |- ( ( A mod 8 ) e. ( 0 ... N ) -> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) )
37		elfzuz2 12346	. . . . . . . 8 |- ( ( A mod 8 ) e. ( 0 ... M ) -> M e. ( ZZ>= ` 0 ) )
38		fzm1 12420	. . . . . . . 8 |- ( M e. ( ZZ>= ` 0 ) -> ( ( A mod 8 ) e. ( 0 ... M ) <-> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) ) )
39	37, 38	syl 17	. . . . . . 7 |- ( ( A mod 8 ) e. ( 0 ... M ) -> ( ( A mod 8 ) e. ( 0 ... M ) <-> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) ) )
40	39	ibi 256	. . . . . 6 |- ( ( A mod 8 ) e. ( 0 ... M ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) )
41		zcn 11382	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
42	10, 41	ax-mp 5	. . . . . . . 8 |- N e. CC
43		zcn 11382	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
44	7, 43	ax-mp 5	. . . . . . . 8 |- M e. CC
45	18, 44	addcomi 10227	. . . . . . . . 9 |- ( 1 + M ) = ( M + 1 )
46	45, 1	eqtr4i 2647	. . . . . . . 8 |- ( 1 + M ) = N
47	42, 18, 44, 46	subaddrii 10370	. . . . . . 7 |- ( N - 1 ) = M
48	47	oveq2i 6661	. . . . . 6 |- ( 0 ... ( N - 1 ) ) = ( 0 ... M )
49	40, 48	eleq2s 2719	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) )
50	20	eqcomi 2631	. . . . . . . . . 10 |- ( 1 + K ) = M
51	44, 18, 17, 50	subaddrii 10370	. . . . . . . . 9 |- ( M - 1 ) = K
52	51	oveq2i 6661	. . . . . . . 8 |- ( 0 ... ( M - 1 ) ) = ( 0 ... K )
53	52	eleq2i 2693	. . . . . . 7 |- ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) <-> ( A mod 8 ) e. ( 0 ... K ) )
54	3	simp3i 1072	. . . . . . 7 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... K ) -> ( A mod 8 ) e. S ) )
55	53, 54	syl5bi 232	. . . . . 6 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) -> ( A mod 8 ) e. S ) )
56		2nn 11185	. . . . . . . . . . 11 |- 2 e. NN
57		8nn 11191	. . . . . . . . . . 11 |- 8 e. NN
58		4z 11411	. . . . . . . . . . . . . 14 |- 4 e. ZZ
59		dvdsmul2 15004	. . . . . . . . . . . . . 14 |- ( ( 4 e. ZZ /\ 2 e. ZZ ) -> 2 || ( 4 x. 2 ) )
60	58, 12, 59	mp2an 708	. . . . . . . . . . . . 13 |- 2 || ( 4 x. 2 )
61		4t2e8 11181	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
62	60, 61	breqtri 4678	. . . . . . . . . . . 12 |- 2 || 8
63		dvdsmod 15050	. . . . . . . . . . . 12 |- ( ( ( 2 e. NN /\ 8 e. NN /\ A e. ZZ ) /\ 2 || 8 ) -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
64	62, 63	mpan2 707	. . . . . . . . . . 11 |- ( ( 2 e. NN /\ 8 e. NN /\ A e. ZZ ) -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
65	56, 57, 64	mp3an12 1414	. . . . . . . . . 10 |- ( A e. ZZ -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
66	65	notbid 308	. . . . . . . . 9 |- ( A e. ZZ -> ( -. 2 || ( A mod 8 ) <-> -. 2 || A ) )
67	66	biimpar 502	. . . . . . . 8 |- ( ( A e. ZZ /\ -. 2 || A ) -> -. 2 || ( A mod 8 ) )
68	11, 2	breqtrri 4680	. . . . . . . . 9 |- 2 || M
69		id 22	. . . . . . . . 9 |- ( ( A mod 8 ) = M -> ( A mod 8 ) = M )
70	68, 69	syl5breqr 4691	. . . . . . . 8 |- ( ( A mod 8 ) = M -> 2 || ( A mod 8 ) )
71	67, 70	nsyl 135	. . . . . . 7 |- ( ( A e. ZZ /\ -. 2 || A ) -> -. ( A mod 8 ) = M )
72	71	pm2.21d 118	. . . . . 6 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) = M -> ( A mod 8 ) e. S ) )
73	55, 72	jaod 395	. . . . 5 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) -> ( A mod 8 ) e. S ) )
74	49, 73	syl5 34	. . . 4 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) -> ( A mod 8 ) e. S ) )
75		lgsdir2lem2.4	. . . . . 6 |- N e. S
76		eleq1 2689	. . . . . 6 |- ( ( A mod 8 ) = N -> ( ( A mod 8 ) e. S <-> N e. S ) )
77	75, 76	mpbiri 248	. . . . 5 |- ( ( A mod 8 ) = N -> ( A mod 8 ) e. S )
78	77	a1i 11	. . . 4 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) = N -> ( A mod 8 ) e. S ) )
79	74, 78	jaod 395	. . 3 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) -> ( A mod 8 ) e. S ) )
80	36, 79	syl5 34	. 2 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... N ) -> ( A mod 8 ) e. S ) )
81	10, 32, 80	3pm3.2i 1239	1 |- ( N e. ZZ /\ 2 || ( N + 1 ) /\ ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... N ) -> ( A mod 8 ) e. S ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   4c4 11072   8c8 11076   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   mod cmo 12668   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-dvds 14984

This theorem is referenced by:  lgsdir2lem3  25052