Theorem mulgneg2 13693

Description: Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)

Hypotheses

Ref	Expression
mulgneg2.b	|- B = ( Base ` G )
mulgneg2.m	|- .x. = (.g ` G )
mulgneg2.i	|- I = ( invg ` G )

Assertion

Ref	Expression
mulgneg2	|- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( N .x. ( I ` X ) ) )

Proof of Theorem mulgneg2

Dummy variables x n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		negeq 8339	. . . . . . 7 |- ( x = 0 -> -u x = -u 0 )
2		neg0 8392	. . . . . . 7 |- -u 0 = 0
3	1, 2	eqtrdi 2278	. . . . . 6 |- ( x = 0 -> -u x = 0 )
4	3	oveq1d 6016	. . . . 5 |- ( x = 0 -> ( -u x .x. X ) = ( 0 .x. X ) )
5		oveq1 6008	. . . . 5 |- ( x = 0 -> ( x .x. ( I ` X ) ) = ( 0 .x. ( I ` X ) ) )
6	4, 5	eqeq12d 2244	. . . 4 |- ( x = 0 -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( 0 .x. X ) = ( 0 .x. ( I ` X ) ) ) )
7		negeq 8339	. . . . . 6 |- ( x = n -> -u x = -u n )
8	7	oveq1d 6016	. . . . 5 |- ( x = n -> ( -u x .x. X ) = ( -u n .x. X ) )
9		oveq1 6008	. . . . 5 |- ( x = n -> ( x .x. ( I ` X ) ) = ( n .x. ( I ` X ) ) )
10	8, 9	eqeq12d 2244	. . . 4 |- ( x = n -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u n .x. X ) = ( n .x. ( I ` X ) ) ) )
11		negeq 8339	. . . . . 6 |- ( x = ( n + 1 ) -> -u x = -u ( n + 1 ) )
12	11	oveq1d 6016	. . . . 5 |- ( x = ( n + 1 ) -> ( -u x .x. X ) = ( -u ( n + 1 ) .x. X ) )
13		oveq1 6008	. . . . 5 |- ( x = ( n + 1 ) -> ( x .x. ( I ` X ) ) = ( ( n + 1 ) .x. ( I ` X ) ) )
14	12, 13	eqeq12d 2244	. . . 4 |- ( x = ( n + 1 ) -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u ( n + 1 ) .x. X ) = ( ( n + 1 ) .x. ( I ` X ) ) ) )
15		negeq 8339	. . . . . 6 |- ( x = -u n -> -u x = -u -u n )
16	15	oveq1d 6016	. . . . 5 |- ( x = -u n -> ( -u x .x. X ) = ( -u -u n .x. X ) )
17		oveq1 6008	. . . . 5 |- ( x = -u n -> ( x .x. ( I ` X ) ) = ( -u n .x. ( I ` X ) ) )
18	16, 17	eqeq12d 2244	. . . 4 |- ( x = -u n -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u -u n .x. X ) = ( -u n .x. ( I ` X ) ) ) )
19		negeq 8339	. . . . . 6 |- ( x = N -> -u x = -u N )
20	19	oveq1d 6016	. . . . 5 |- ( x = N -> ( -u x .x. X ) = ( -u N .x. X ) )
21		oveq1 6008	. . . . 5 |- ( x = N -> ( x .x. ( I ` X ) ) = ( N .x. ( I ` X ) ) )
22	20, 21	eqeq12d 2244	. . . 4 |- ( x = N -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u N .x. X ) = ( N .x. ( I ` X ) ) ) )
23		mulgneg2.b	. . . . . . 7 |- B = ( Base ` G )
24		eqid 2229	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
25		mulgneg2.m	. . . . . . 7 |- .x. = (.g ` G )
26	23, 24, 25	mulg0 13662	. . . . . 6 |- ( X e. B -> ( 0 .x. X ) = ( 0g ` G ) )
27	26	adantl 277	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( 0 .x. X ) = ( 0g ` G ) )
28		mulgneg2.i	. . . . . . 7 |- I = ( invg ` G )
29	23, 28	grpinvcl 13581	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( I ` X ) e. B )
30	23, 24, 25	mulg0 13662	. . . . . 6 |- ( ( I ` X ) e. B -> ( 0 .x. ( I ` X ) ) = ( 0g ` G ) )
31	29, 30	syl 14	. . . . 5 |- ( ( G e. Grp /\ X e. B ) -> ( 0 .x. ( I ` X ) ) = ( 0g ` G ) )
32	27, 31	eqtr4d 2265	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( 0 .x. X ) = ( 0 .x. ( I ` X ) ) )
33		oveq1 6008	. . . . . 6 |- ( ( -u n .x. X ) = ( n .x. ( I ` X ) ) -> ( ( -u n .x. X ) ( +g ` G ) ( I ` X ) ) = ( ( n .x. ( I ` X ) ) ( +g ` G ) ( I ` X ) ) )
34		nn0cn 9379	. . . . . . . . . . 11 |- ( n e. NN0 -> n e. CC )
35	34	adantl 277	. . . . . . . . . 10 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> n e. CC )
36		ax-1cn 8092	. . . . . . . . . 10 |- 1 e. CC
37		negdi 8403	. . . . . . . . . 10 |- ( ( n e. CC /\ 1 e. CC ) -> -u ( n + 1 ) = ( -u n + -u 1 ) )
38	35, 36, 37	sylancl 413	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> -u ( n + 1 ) = ( -u n + -u 1 ) )
39	38	oveq1d 6016	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> ( -u ( n + 1 ) .x. X ) = ( ( -u n + -u 1 ) .x. X ) )
40		simpll 527	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> G e. Grp )
41		nn0negz 9480	. . . . . . . . . 10 |- ( n e. NN0 -> -u n e. ZZ )
42	41	adantl 277	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> -u n e. ZZ )
43		1z 9472	. . . . . . . . . 10 |- 1 e. ZZ
44		znegcl 9477	. . . . . . . . . 10 |- ( 1 e. ZZ -> -u 1 e. ZZ )
45	43, 44	mp1i 10	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> -u 1 e. ZZ )
46		simplr 528	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> X e. B )
47		eqid 2229	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
48	23, 25, 47	mulgdir 13691	. . . . . . . . 9 |- ( ( G e. Grp /\ ( -u n e. ZZ /\ -u 1 e. ZZ /\ X e. B ) ) -> ( ( -u n + -u 1 ) .x. X ) = ( ( -u n .x. X ) ( +g ` G ) ( -u 1 .x. X ) ) )
49	40, 42, 45, 46, 48	syl13anc 1273	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> ( ( -u n + -u 1 ) .x. X ) = ( ( -u n .x. X ) ( +g ` G ) ( -u 1 .x. X ) ) )
50	23, 25, 28	mulgm1 13679	. . . . . . . . . 10 |- ( ( G e. Grp /\ X e. B ) -> ( -u 1 .x. X ) = ( I ` X ) )
51	50	adantr 276	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> ( -u 1 .x. X ) = ( I ` X ) )
52	51	oveq2d 6017	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> ( ( -u n .x. X ) ( +g ` G ) ( -u 1 .x. X ) ) = ( ( -u n .x. X ) ( +g ` G ) ( I ` X ) ) )
53	39, 49, 52	3eqtrd 2266	. . . . . . 7 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> ( -u ( n + 1 ) .x. X ) = ( ( -u n .x. X ) ( +g ` G ) ( I ` X ) ) )
54		grpmnd 13540	. . . . . . . . 9 |- ( G e. Grp -> G e. Mnd )
55	54	ad2antrr 488	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> G e. Mnd )
56		simpr 110	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> n e. NN0 )
57	29	adantr 276	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> ( I ` X ) e. B )
58	23, 25, 47	mulgnn0p1 13670	. . . . . . . 8 |- ( ( G e. Mnd /\ n e. NN0 /\ ( I ` X ) e. B ) -> ( ( n + 1 ) .x. ( I ` X ) ) = ( ( n .x. ( I ` X ) ) ( +g ` G ) ( I ` X ) ) )
59	55, 56, 57, 58	syl3anc 1271	. . . . . . 7 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> ( ( n + 1 ) .x. ( I ` X ) ) = ( ( n .x. ( I ` X ) ) ( +g ` G ) ( I ` X ) ) )
60	53, 59	eqeq12d 2244	. . . . . 6 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> ( ( -u ( n + 1 ) .x. X ) = ( ( n + 1 ) .x. ( I ` X ) ) <-> ( ( -u n .x. X ) ( +g ` G ) ( I ` X ) ) = ( ( n .x. ( I ` X ) ) ( +g ` G ) ( I ` X ) ) ) )
61	33, 60	imbitrrid 156	. . . . 5 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> ( ( -u n .x. X ) = ( n .x. ( I ` X ) ) -> ( -u ( n + 1 ) .x. X ) = ( ( n + 1 ) .x. ( I ` X ) ) ) )
62	61	ex 115	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( n e. NN0 -> ( ( -u n .x. X ) = ( n .x. ( I ` X ) ) -> ( -u ( n + 1 ) .x. X ) = ( ( n + 1 ) .x. ( I ` X ) ) ) ) )
63		fveq2 5627	. . . . . 6 |- ( ( -u n .x. X ) = ( n .x. ( I ` X ) ) -> ( I ` ( -u n .x. X ) ) = ( I ` ( n .x. ( I ` X ) ) ) )
64		simpll 527	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN ) -> G e. Grp )
65		nnnegz 9449	. . . . . . . . 9 |- ( n e. NN -> -u n e. ZZ )
66	65	adantl 277	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN ) -> -u n e. ZZ )
67		simplr 528	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN ) -> X e. B )
68	23, 25, 28	mulgneg 13677	. . . . . . . 8 |- ( ( G e. Grp /\ -u n e. ZZ /\ X e. B ) -> ( -u -u n .x. X ) = ( I ` ( -u n .x. X ) ) )
69	64, 66, 67, 68	syl3anc 1271	. . . . . . 7 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN ) -> ( -u -u n .x. X ) = ( I ` ( -u n .x. X ) ) )
70		id 19	. . . . . . . 8 |- ( n e. NN -> n e. NN )
71	23, 25, 28	mulgnegnn 13669	. . . . . . . 8 |- ( ( n e. NN /\ ( I ` X ) e. B ) -> ( -u n .x. ( I ` X ) ) = ( I ` ( n .x. ( I ` X ) ) ) )
72	70, 29, 71	syl2anr 290	. . . . . . 7 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN ) -> ( -u n .x. ( I ` X ) ) = ( I ` ( n .x. ( I ` X ) ) ) )
73	69, 72	eqeq12d 2244	. . . . . 6 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN ) -> ( ( -u -u n .x. X ) = ( -u n .x. ( I ` X ) ) <-> ( I ` ( -u n .x. X ) ) = ( I ` ( n .x. ( I ` X ) ) ) ) )
74	63, 73	imbitrrid 156	. . . . 5 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN ) -> ( ( -u n .x. X ) = ( n .x. ( I ` X ) ) -> ( -u -u n .x. X ) = ( -u n .x. ( I ` X ) ) ) )
75	74	ex 115	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( n e. NN -> ( ( -u n .x. X ) = ( n .x. ( I ` X ) ) -> ( -u -u n .x. X ) = ( -u n .x. ( I ` X ) ) ) ) )
76	6, 10, 14, 18, 22, 32, 62, 75	zindd 9565	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( N e. ZZ -> ( -u N .x. X ) = ( N .x. ( I ` X ) ) ) )
77	76	3impia 1224	. 2 |- ( ( G e. Grp /\ X e. B /\ N e. ZZ ) -> ( -u N .x. X ) = ( N .x. ( I ` X ) ) )
78	77	3com23 1233	1 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( N .x. ( I ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   + caddc 8002   -ucneg 8318   NNcn 9110   NN0cn0 9369   ZZcz 9446   Basecbs 13032   +g cplusg 13110   0gc0g 13289   Mndcmnd 13449   Grpcgrp 13533   invgcminusg 13534  ._gcmg 13656

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-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  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-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  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-nul 3492  df-if 3603  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-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  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-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205  df-seqfrec 10670  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-mulg 13657

This theorem is referenced by:  mulgass  13696