Theorem mulgneg2 13909

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 8482	. . . . . . 7 |- ( x = 0 -> -u x = -u 0 )
2		neg0 8535	. . . . . . 7 |- -u 0 = 0
3	1, 2	eqtrdi 2283	. . . . . 6 |- ( x = 0 -> -u x = 0 )
4	3	oveq1d 6073	. . . . 5 |- ( x = 0 -> ( -u x .x. X ) = ( 0 .x. X ) )
5		oveq1 6065	. . . . 5 |- ( x = 0 -> ( x .x. ( I ` X ) ) = ( 0 .x. ( I ` X ) ) )
6	4, 5	eqeq12d 2249	. . . 4 |- ( x = 0 -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( 0 .x. X ) = ( 0 .x. ( I ` X ) ) ) )
7		negeq 8482	. . . . . 6 |- ( x = n -> -u x = -u n )
8	7	oveq1d 6073	. . . . 5 |- ( x = n -> ( -u x .x. X ) = ( -u n .x. X ) )
9		oveq1 6065	. . . . 5 |- ( x = n -> ( x .x. ( I ` X ) ) = ( n .x. ( I ` X ) ) )
10	8, 9	eqeq12d 2249	. . . 4 |- ( x = n -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u n .x. X ) = ( n .x. ( I ` X ) ) ) )
11		negeq 8482	. . . . . 6 |- ( x = ( n + 1 ) -> -u x = -u ( n + 1 ) )
12	11	oveq1d 6073	. . . . 5 |- ( x = ( n + 1 ) -> ( -u x .x. X ) = ( -u ( n + 1 ) .x. X ) )
13		oveq1 6065	. . . . 5 |- ( x = ( n + 1 ) -> ( x .x. ( I ` X ) ) = ( ( n + 1 ) .x. ( I ` X ) ) )
14	12, 13	eqeq12d 2249	. . . 4 |- ( x = ( n + 1 ) -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u ( n + 1 ) .x. X ) = ( ( n + 1 ) .x. ( I ` X ) ) ) )
15		negeq 8482	. . . . . 6 |- ( x = -u n -> -u x = -u -u n )
16	15	oveq1d 6073	. . . . 5 |- ( x = -u n -> ( -u x .x. X ) = ( -u -u n .x. X ) )
17		oveq1 6065	. . . . 5 |- ( x = -u n -> ( x .x. ( I ` X ) ) = ( -u n .x. ( I ` X ) ) )
18	16, 17	eqeq12d 2249	. . . 4 |- ( x = -u n -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u -u n .x. X ) = ( -u n .x. ( I ` X ) ) ) )
19		negeq 8482	. . . . . 6 |- ( x = N -> -u x = -u N )
20	19	oveq1d 6073	. . . . 5 |- ( x = N -> ( -u x .x. X ) = ( -u N .x. X ) )
21		oveq1 6065	. . . . 5 |- ( x = N -> ( x .x. ( I ` X ) ) = ( N .x. ( I ` X ) ) )
22	20, 21	eqeq12d 2249	. . . 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 2234	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
25		mulgneg2.m	. . . . . . 7 |- .x. = (.g ` G )
26	23, 24, 25	mulg0 13878	. . . . . 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 13803	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( I ` X ) e. B )
30	23, 24, 25	mulg0 13878	. . . . . 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 2270	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( 0 .x. X ) = ( 0 .x. ( I ` X ) ) )
33		oveq1 6065	. . . . . 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 9523	. . . . . . . . . . 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 8236	. . . . . . . . . 10 |- 1 e. CC
37		negdi 8546	. . . . . . . . . 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 6073	. . . . . . . 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 9628	. . . . . . . . . 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 9620	. . . . . . . . . 10 |- 1 e. ZZ
44		znegcl 9625	. . . . . . . . . 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 529	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN0 ) -> X e. B )
47		eqid 2234	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
48	23, 25, 47	mulgdir 13907	. . . . . . . . 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 1276	. . . . . . . 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 13895	. . . . . . . . . 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 6074	. . . . . . . 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 2271	. . . . . . 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 13762	. . . . . . . . 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 13886	. . . . . . . 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 1274	. . . . . . 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 2249	. . . . . 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 5675	. . . . . 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 9597	. . . . . . . . 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 529	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. B ) /\ n e. NN ) -> X e. B )
68	23, 25, 28	mulgneg 13893	. . . . . . . 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 1274	. . . . . . 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 13885	. . . . . . . 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 2249	. . . . . 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 9714	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( N e. ZZ -> ( -u N .x. X ) = ( N .x. ( I ` X ) ) ) )
77	76	3impia 1227	. 2 |- ( ( G e. Grp /\ X e. B /\ N e. ZZ ) -> ( -u N .x. X ) = ( N .x. ( I ` X ) ) )
78	77	3com23 1236	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 1005   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   -ucneg 8461   NNcn 9254   NN0cn0 9513   ZZcz 9594   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Mndcmnd 13677   Grpcgrp 13755   invgcminusg 13756  ._gcmg 13872

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-mulg 13873

This theorem is referenced by:  mulgass  13912