Theorem lgsdir2lem2 15581

Description: Lemma for lgsdir2 15585. (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 1009	. . . . . 6 |- K e. ZZ
5		peano2z 9428	. . . . . 6 |- ( K e. ZZ -> ( K + 1 ) e. ZZ )
6	4, 5	ax-mp 5	. . . . 5 |- ( K + 1 ) e. ZZ
7	2, 6	eqeltri 2279	. . . 4 |- M e. ZZ
8		peano2z 9428	. . . 4 |- ( M e. ZZ -> ( M + 1 ) e. ZZ )
9	7, 8	ax-mp 5	. . 3 |- ( M + 1 ) e. ZZ
10	1, 9	eqeltri 2279	. 2 |- N e. ZZ
11	3	simp2i 1010	. . . 4 |- 2 || ( K + 1 )
12		2z 9420	. . . . 5 |- 2 e. ZZ
13		dvdsadd 12222	. . . . 5 |- ( ( 2 e. ZZ /\ ( K + 1 ) e. ZZ ) -> ( 2 || ( K + 1 ) <-> 2 || ( 2 + ( K + 1 ) ) ) )
14	12, 6, 13	mp2an 426	. . . 4 |- ( 2 || ( K + 1 ) <-> 2 || ( 2 + ( K + 1 ) ) )
15	11, 14	mpbi 145	. . 3 |- 2 || ( 2 + ( K + 1 ) )
16		zcn 9397	. . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
17	4, 16	ax-mp 5	. . . . . . . . . 10 |- K e. CC
18		ax-1cn 8038	. . . . . . . . . 10 |- 1 e. CC
19	17, 18	addcomi 8236	. . . . . . . . 9 |- ( K + 1 ) = ( 1 + K )
20	2, 19	eqtri 2227	. . . . . . . 8 |- M = ( 1 + K )
21	20	oveq1i 5967	. . . . . . 7 |- ( M + 1 ) = ( ( 1 + K ) + 1 )
22	1, 21	eqtri 2227	. . . . . 6 |- N = ( ( 1 + K ) + 1 )
23		df-2 9115	. . . . . . . 8 |- 2 = ( 1 + 1 )
24	23	oveq1i 5967	. . . . . . 7 |- ( 2 + K ) = ( ( 1 + 1 ) + K )
25	18, 17, 18	add32i 8256	. . . . . . 7 |- ( ( 1 + K ) + 1 ) = ( ( 1 + 1 ) + K )
26	24, 25	eqtr4i 2230	. . . . . 6 |- ( 2 + K ) = ( ( 1 + K ) + 1 )
27	22, 26	eqtr4i 2230	. . . . 5 |- N = ( 2 + K )
28	27	oveq1i 5967	. . . 4 |- ( N + 1 ) = ( ( 2 + K ) + 1 )
29		2cn 9127	. . . . 5 |- 2 e. CC
30	29, 17, 18	addassi 8100	. . . 4 |- ( ( 2 + K ) + 1 ) = ( 2 + ( K + 1 ) )
31	28, 30	eqtri 2227	. . 3 |- ( N + 1 ) = ( 2 + ( K + 1 ) )
32	15, 31	breqtrri 4078	. 2 |- 2 || ( N + 1 )
33		elfzuz2 10171	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... N ) -> N e. ( ZZ>= ` 0 ) )
34		fzm1 10242	. . . . 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 14	. . . 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 176	. . 3 |- ( ( A mod 8 ) e. ( 0 ... N ) -> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) )
37		elfzuz2 10171	. . . . . . . 8 |- ( ( A mod 8 ) e. ( 0 ... M ) -> M e. ( ZZ>= ` 0 ) )
38		fzm1 10242	. . . . . . . 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 14	. . . . . . 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 176	. . . . . 6 |- ( ( A mod 8 ) e. ( 0 ... M ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) )
41		zcn 9397	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
42	7, 41	ax-mp 5	. . . . . . . 8 |- M e. CC
43	42, 18, 1	mvrraddi 8309	. . . . . . 7 |- ( N - 1 ) = M
44	43	oveq2i 5968	. . . . . 6 |- ( 0 ... ( N - 1 ) ) = ( 0 ... M )
45	40, 44	eleq2s 2301	. . . . 5 |- ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) )
46	17, 18, 2	mvrraddi 8309	. . . . . . . . 9 |- ( M - 1 ) = K
47	46	oveq2i 5968	. . . . . . . 8 |- ( 0 ... ( M - 1 ) ) = ( 0 ... K )
48	47	eleq2i 2273	. . . . . . 7 |- ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) <-> ( A mod 8 ) e. ( 0 ... K ) )
49	3	simp3i 1011	. . . . . . 7 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... K ) -> ( A mod 8 ) e. S ) )
50	48, 49	biimtrid 152	. . . . . 6 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) -> ( A mod 8 ) e. S ) )
51		2nn 9218	. . . . . . . . . . 11 |- 2 e. NN
52		8nn 9224	. . . . . . . . . . 11 |- 8 e. NN
53		4z 9422	. . . . . . . . . . . . . 14 |- 4 e. ZZ
54		dvdsmul2 12200	. . . . . . . . . . . . . 14 |- ( ( 4 e. ZZ /\ 2 e. ZZ ) -> 2 || ( 4 x. 2 ) )
55	53, 12, 54	mp2an 426	. . . . . . . . . . . . 13 |- 2 || ( 4 x. 2 )
56		4t2e8 9215	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
57	55, 56	breqtri 4076	. . . . . . . . . . . 12 |- 2 || 8
58		dvdsmod 12248	. . . . . . . . . . . 12 |- ( ( ( 2 e. NN /\ 8 e. NN /\ A e. ZZ ) /\ 2 || 8 ) -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
59	57, 58	mpan2 425	. . . . . . . . . . 11 |- ( ( 2 e. NN /\ 8 e. NN /\ A e. ZZ ) -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
60	51, 52, 59	mp3an12 1340	. . . . . . . . . 10 |- ( A e. ZZ -> ( 2 || ( A mod 8 ) <-> 2 || A ) )
61	60	notbid 669	. . . . . . . . 9 |- ( A e. ZZ -> ( -. 2 || ( A mod 8 ) <-> -. 2 || A ) )
62	61	biimpar 297	. . . . . . . 8 |- ( ( A e. ZZ /\ -. 2 || A ) -> -. 2 || ( A mod 8 ) )
63	11, 2	breqtrri 4078	. . . . . . . . 9 |- 2 || M
64		id 19	. . . . . . . . 9 |- ( ( A mod 8 ) = M -> ( A mod 8 ) = M )
65	63, 64	breqtrrid 4089	. . . . . . . 8 |- ( ( A mod 8 ) = M -> 2 || ( A mod 8 ) )
66	62, 65	nsyl 629	. . . . . . 7 |- ( ( A e. ZZ /\ -. 2 || A ) -> -. ( A mod 8 ) = M )
67	66	pm2.21d 620	. . . . . 6 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) = M -> ( A mod 8 ) e. S ) )
68	50, 67	jaod 719	. . . . 5 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( ( A mod 8 ) e. ( 0 ... ( M - 1 ) ) \/ ( A mod 8 ) = M ) -> ( A mod 8 ) e. S ) )
69	45, 68	syl5 32	. . . 4 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) -> ( A mod 8 ) e. S ) )
70		lgsdir2lem2.4	. . . . . 6 |- N e. S
71		eleq1 2269	. . . . . 6 |- ( ( A mod 8 ) = N -> ( ( A mod 8 ) e. S <-> N e. S ) )
72	70, 71	mpbiri 168	. . . . 5 |- ( ( A mod 8 ) = N -> ( A mod 8 ) e. S )
73	72	a1i 9	. . . 4 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) = N -> ( A mod 8 ) e. S ) )
74	69, 73	jaod 719	. . 3 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( ( A mod 8 ) e. ( 0 ... ( N - 1 ) ) \/ ( A mod 8 ) = N ) -> ( A mod 8 ) e. S ) )
75	36, 74	syl5 32	. 2 |- ( ( A e. ZZ /\ -. 2 || A ) -> ( ( A mod 8 ) e. ( 0 ... N ) -> ( A mod 8 ) e. S ) )
76	10, 32, 75	3pm3.2i 1178	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   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2177   class class class wbr 4051   ` cfv 5280  (class class class)co 5957   CCcc 7943   0cc0 7945   1c1 7946   + caddc 7948   x. cmul 7950   - cmin 8263   NNcn 9056   2c2 9107   4c4 9109   8c8 9113   ZZcz 9392   ZZ>=cuz 9668   ...cfz 10150   mod cmo 10489   || cdvds 12173

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-po 4351  df-iso 4352  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fl 10435  df-mod 10490  df-dvds 12174

This theorem is referenced by:  lgsdir2lem3  15582