Theorem mulgneg2 13607

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 8300	. . . . . . 7 |- ( x = 0 -> -u x = -u 0 )
2		neg0 8353	. . . . . . 7 |- -u 0 = 0
3	1, 2	eqtrdi 2256	. . . . . 6 |- ( x = 0 -> -u x = 0 )
4	3	oveq1d 5982	. . . . 5 |- ( x = 0 -> ( -u x .x. X ) = ( 0 .x. X ) )
5		oveq1 5974	. . . . 5 |- ( x = 0 -> ( x .x. ( I ` X ) ) = ( 0 .x. ( I ` X ) ) )
6	4, 5	eqeq12d 2222	. . . 4 |- ( x = 0 -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( 0 .x. X ) = ( 0 .x. ( I ` X ) ) ) )
7		negeq 8300	. . . . . 6 |- ( x = n -> -u x = -u n )
8	7	oveq1d 5982	. . . . 5 |- ( x = n -> ( -u x .x. X ) = ( -u n .x. X ) )
9		oveq1 5974	. . . . 5 |- ( x = n -> ( x .x. ( I ` X ) ) = ( n .x. ( I ` X ) ) )
10	8, 9	eqeq12d 2222	. . . 4 |- ( x = n -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u n .x. X ) = ( n .x. ( I ` X ) ) ) )
11		negeq 8300	. . . . . 6 |- ( x = ( n + 1 ) -> -u x = -u ( n + 1 ) )
12	11	oveq1d 5982	. . . . 5 |- ( x = ( n + 1 ) -> ( -u x .x. X ) = ( -u ( n + 1 ) .x. X ) )
13		oveq1 5974	. . . . 5 |- ( x = ( n + 1 ) -> ( x .x. ( I ` X ) ) = ( ( n + 1 ) .x. ( I ` X ) ) )
14	12, 13	eqeq12d 2222	. . . 4 |- ( x = ( n + 1 ) -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u ( n + 1 ) .x. X ) = ( ( n + 1 ) .x. ( I ` X ) ) ) )
15		negeq 8300	. . . . . 6 |- ( x = -u n -> -u x = -u -u n )
16	15	oveq1d 5982	. . . . 5 |- ( x = -u n -> ( -u x .x. X ) = ( -u -u n .x. X ) )
17		oveq1 5974	. . . . 5 |- ( x = -u n -> ( x .x. ( I ` X ) ) = ( -u n .x. ( I ` X ) ) )
18	16, 17	eqeq12d 2222	. . . 4 |- ( x = -u n -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u -u n .x. X ) = ( -u n .x. ( I ` X ) ) ) )
19		negeq 8300	. . . . . 6 |- ( x = N -> -u x = -u N )
20	19	oveq1d 5982	. . . . 5 |- ( x = N -> ( -u x .x. X ) = ( -u N .x. X ) )
21		oveq1 5974	. . . . 5 |- ( x = N -> ( x .x. ( I ` X ) ) = ( N .x. ( I ` X ) ) )
22	20, 21	eqeq12d 2222	. . . 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 2207	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
25		mulgneg2.m	. . . . . . 7 |- .x. = (.g ` G )
26	23, 24, 25	mulg0 13576	. . . . . 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 13495	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( I ` X ) e. B )
30	23, 24, 25	mulg0 13576	. . . . . 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 2243	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( 0 .x. X ) = ( 0 .x. ( I ` X ) ) )
33		oveq1 5974	. . . . . 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 9340	. . . . . . . . . . 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 8053	. . . . . . . . . 10 |- 1 e. CC
37		negdi 8364	. . . . . . . . . 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 5982	. . . . . . . 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 9441	. . . . . . . . . 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 9433	. . . . . . . . . 10 |- 1 e. ZZ
44		znegcl 9438	. . . . . . . . . 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 2207	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
48	23, 25, 47	mulgdir 13605	. . . . . . . . 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 1252	. . . . . . . 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 13593	. . . . . . . . . 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 5983	. . . . . . . 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 2244	. . . . . . 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 13454	. . . . . . . . 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 13584	. . . . . . . 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 1250	. . . . . . 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 2222	. . . . . 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 5599	. . . . . 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 9410	. . . . . . . . 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 13591	. . . . . . . 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 1250	. . . . . . 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 13583	. . . . . . . 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 2222	. . . . . 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 9526	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( N e. ZZ -> ( -u N .x. X ) = ( N .x. ( I ` X ) ) ) )
77	76	3impia 1203	. 2 |- ( ( G e. Grp /\ X e. B /\ N e. ZZ ) -> ( -u N .x. X ) = ( N .x. ( I ` X ) ) )
78	77	3com23 1212	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 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   CCcc 7958   0cc0 7960   1c1 7961   + caddc 7963   -ucneg 8279   NNcn 9071   NN0cn0 9330   ZZcz 9407   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Mndcmnd 13363   Grpcgrp 13447   invgcminusg 13448  ._gcmg 13570

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 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-seqfrec 10630  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-mulg 13571

This theorem is referenced by:  mulgass  13610