Theorem lgsdir2lem1 15505

Description: Lemma for lgsdir2 15510. (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
lgsdir2lem1	|- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )

Proof of Theorem lgsdir2lem1

Step	Hyp	Ref	Expression
1		1nn 9047	. . . . 5 |- 1 e. NN
2		nnq 9754	. . . . 5 |- ( 1 e. NN -> 1 e. QQ )
3	1, 2	ax-mp 5	. . . 4 |- 1 e. QQ
4		8nn 9204	. . . . 5 |- 8 e. NN
5		nnq 9754	. . . . 5 |- ( 8 e. NN -> 8 e. QQ )
6	4, 5	ax-mp 5	. . . 4 |- 8 e. QQ
7		0le1 8554	. . . 4 |- 0 <_ 1
8		1lt8 9233	. . . 4 |- 1 < 8
9		modqid 10494	. . . 4 |- ( ( ( 1 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 1 /\ 1 < 8 ) ) -> ( 1 mod 8 ) = 1 )
10	3, 6, 7, 8, 9	mp4an 427	. . 3 |- ( 1 mod 8 ) = 1
11		8cn 9122	. . . . . . . 8 |- 8 e. CC
12	11	mullidi 8075	. . . . . . 7 |- ( 1 x. 8 ) = 8
13	12	oveq2i 5955	. . . . . 6 |- ( -u 1 + ( 1 x. 8 ) ) = ( -u 1 + 8 )
14		ax-1cn 8018	. . . . . . . 8 |- 1 e. CC
15	14	negcli 8340	. . . . . . 7 |- -u 1 e. CC
16	11, 14	negsubi 8350	. . . . . . . 8 |- ( 8 + -u 1 ) = ( 8 - 1 )
17		8m1e7 9161	. . . . . . . 8 |- ( 8 - 1 ) = 7
18	16, 17	eqtri 2226	. . . . . . 7 |- ( 8 + -u 1 ) = 7
19	11, 15, 18	addcomli 8217	. . . . . 6 |- ( -u 1 + 8 ) = 7
20	13, 19	eqtri 2226	. . . . 5 |- ( -u 1 + ( 1 x. 8 ) ) = 7
21	20	oveq1i 5954	. . . 4 |- ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( 7 mod 8 )
22		qnegcl 9757	. . . . . 6 |- ( 1 e. QQ -> -u 1 e. QQ )
23	3, 22	ax-mp 5	. . . . 5 |- -u 1 e. QQ
24		1z 9398	. . . . 5 |- 1 e. ZZ
25		8pos 9139	. . . . 5 |- 0 < 8
26		modqcyc 10504	. . . . 5 |- ( ( ( -u 1 e. QQ /\ 1 e. ZZ ) /\ ( 8 e. QQ /\ 0 < 8 ) ) -> ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( -u 1 mod 8 ) )
27	23, 24, 6, 25, 26	mp4an 427	. . . 4 |- ( ( -u 1 + ( 1 x. 8 ) ) mod 8 ) = ( -u 1 mod 8 )
28		7nn 9203	. . . . . 6 |- 7 e. NN
29		nnq 9754	. . . . . 6 |- ( 7 e. NN -> 7 e. QQ )
30	28, 29	ax-mp 5	. . . . 5 |- 7 e. QQ
31		0re 8072	. . . . . 6 |- 0 e. RR
32		7re 9119	. . . . . 6 |- 7 e. RR
33		7pos 9138	. . . . . 6 |- 0 < 7
34	31, 32, 33	ltleii 8175	. . . . 5 |- 0 <_ 7
35		7lt8 9227	. . . . 5 |- 7 < 8
36		modqid 10494	. . . . 5 |- ( ( ( 7 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 7 /\ 7 < 8 ) ) -> ( 7 mod 8 ) = 7 )
37	30, 6, 34, 35, 36	mp4an 427	. . . 4 |- ( 7 mod 8 ) = 7
38	21, 27, 37	3eqtr3i 2234	. . 3 |- ( -u 1 mod 8 ) = 7
39	10, 38	pm3.2i 272	. 2 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
40		3nn 9199	. . . . 5 |- 3 e. NN
41		nnq 9754	. . . . 5 |- ( 3 e. NN -> 3 e. QQ )
42	40, 41	ax-mp 5	. . . 4 |- 3 e. QQ
43		3re 9110	. . . . 5 |- 3 e. RR
44		3pos 9130	. . . . 5 |- 0 < 3
45	31, 43, 44	ltleii 8175	. . . 4 |- 0 <_ 3
46		3lt8 9231	. . . 4 |- 3 < 8
47		modqid 10494	. . . 4 |- ( ( ( 3 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 3 /\ 3 < 8 ) ) -> ( 3 mod 8 ) = 3 )
48	42, 6, 45, 46, 47	mp4an 427	. . 3 |- ( 3 mod 8 ) = 3
49	12	oveq2i 5955	. . . . . 6 |- ( -u 3 + ( 1 x. 8 ) ) = ( -u 3 + 8 )
50		3cn 9111	. . . . . . . 8 |- 3 e. CC
51	50	negcli 8340	. . . . . . 7 |- -u 3 e. CC
52	11, 50	negsubi 8350	. . . . . . . 8 |- ( 8 + -u 3 ) = ( 8 - 3 )
53		5cn 9116	. . . . . . . . 9 |- 5 e. CC
54		5p3e8 9184	. . . . . . . . . 10 |- ( 5 + 3 ) = 8
55	53, 50, 54	addcomli 8217	. . . . . . . . 9 |- ( 3 + 5 ) = 8
56	11, 50, 53, 55	subaddrii 8361	. . . . . . . 8 |- ( 8 - 3 ) = 5
57	52, 56	eqtri 2226	. . . . . . 7 |- ( 8 + -u 3 ) = 5
58	11, 51, 57	addcomli 8217	. . . . . 6 |- ( -u 3 + 8 ) = 5
59	49, 58	eqtri 2226	. . . . 5 |- ( -u 3 + ( 1 x. 8 ) ) = 5
60	59	oveq1i 5954	. . . 4 |- ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( 5 mod 8 )
61		qnegcl 9757	. . . . . 6 |- ( 3 e. QQ -> -u 3 e. QQ )
62	42, 61	ax-mp 5	. . . . 5 |- -u 3 e. QQ
63		modqcyc 10504	. . . . 5 |- ( ( ( -u 3 e. QQ /\ 1 e. ZZ ) /\ ( 8 e. QQ /\ 0 < 8 ) ) -> ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( -u 3 mod 8 ) )
64	62, 24, 6, 25, 63	mp4an 427	. . . 4 |- ( ( -u 3 + ( 1 x. 8 ) ) mod 8 ) = ( -u 3 mod 8 )
65		5nn 9201	. . . . . 6 |- 5 e. NN
66		nnq 9754	. . . . . 6 |- ( 5 e. NN -> 5 e. QQ )
67	65, 66	ax-mp 5	. . . . 5 |- 5 e. QQ
68		5re 9115	. . . . . 6 |- 5 e. RR
69		5pos 9136	. . . . . 6 |- 0 < 5
70	31, 68, 69	ltleii 8175	. . . . 5 |- 0 <_ 5
71		5lt8 9229	. . . . 5 |- 5 < 8
72		modqid 10494	. . . . 5 |- ( ( ( 5 e. QQ /\ 8 e. QQ ) /\ ( 0 <_ 5 /\ 5 < 8 ) ) -> ( 5 mod 8 ) = 5 )
73	67, 6, 70, 71, 72	mp4an 427	. . . 4 |- ( 5 mod 8 ) = 5
74	60, 64, 73	3eqtr3i 2234	. . 3 |- ( -u 3 mod 8 ) = 5
75	48, 74	pm3.2i 272	. 2 |- ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 )
76	39, 75	pm3.2i 272	1 |- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   0cc0 7925   1c1 7926   + caddc 7928   x. cmul 7930   < clt 8107   <_ cle 8108   - cmin 8243   -ucneg 8244   NNcn 9036   3c3 9088   5c5 9090   7c7 9092   8c8 9093   ZZcz 9372   QQcq 9740   mod cmo 10467

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-n0 9296  df-z 9373  df-q 9741  df-rp 9776  df-fl 10413  df-mod 10468

This theorem is referenced by:  lgsdir2lem4  15508  lgsdir2lem5  15509