Theorem mulgneg2 13525

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 8267	. . . . . . 7 |- ( x = 0 -> -u x = -u 0 )
2		neg0 8320	. . . . . . 7 |- -u 0 = 0
3	1, 2	eqtrdi 2254	. . . . . 6 |- ( x = 0 -> -u x = 0 )
4	3	oveq1d 5961	. . . . 5 |- ( x = 0 -> ( -u x .x. X ) = ( 0 .x. X ) )
5		oveq1 5953	. . . . 5 |- ( x = 0 -> ( x .x. ( I ` X ) ) = ( 0 .x. ( I ` X ) ) )
6	4, 5	eqeq12d 2220	. . . 4 |- ( x = 0 -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( 0 .x. X ) = ( 0 .x. ( I ` X ) ) ) )
7		negeq 8267	. . . . . 6 |- ( x = n -> -u x = -u n )
8	7	oveq1d 5961	. . . . 5 |- ( x = n -> ( -u x .x. X ) = ( -u n .x. X ) )
9		oveq1 5953	. . . . 5 |- ( x = n -> ( x .x. ( I ` X ) ) = ( n .x. ( I ` X ) ) )
10	8, 9	eqeq12d 2220	. . . 4 |- ( x = n -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u n .x. X ) = ( n .x. ( I ` X ) ) ) )
11		negeq 8267	. . . . . 6 |- ( x = ( n + 1 ) -> -u x = -u ( n + 1 ) )
12	11	oveq1d 5961	. . . . 5 |- ( x = ( n + 1 ) -> ( -u x .x. X ) = ( -u ( n + 1 ) .x. X ) )
13		oveq1 5953	. . . . 5 |- ( x = ( n + 1 ) -> ( x .x. ( I ` X ) ) = ( ( n + 1 ) .x. ( I ` X ) ) )
14	12, 13	eqeq12d 2220	. . . 4 |- ( x = ( n + 1 ) -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u ( n + 1 ) .x. X ) = ( ( n + 1 ) .x. ( I ` X ) ) ) )
15		negeq 8267	. . . . . 6 |- ( x = -u n -> -u x = -u -u n )
16	15	oveq1d 5961	. . . . 5 |- ( x = -u n -> ( -u x .x. X ) = ( -u -u n .x. X ) )
17		oveq1 5953	. . . . 5 |- ( x = -u n -> ( x .x. ( I ` X ) ) = ( -u n .x. ( I ` X ) ) )
18	16, 17	eqeq12d 2220	. . . 4 |- ( x = -u n -> ( ( -u x .x. X ) = ( x .x. ( I ` X ) ) <-> ( -u -u n .x. X ) = ( -u n .x. ( I ` X ) ) ) )
19		negeq 8267	. . . . . 6 |- ( x = N -> -u x = -u N )
20	19	oveq1d 5961	. . . . 5 |- ( x = N -> ( -u x .x. X ) = ( -u N .x. X ) )
21		oveq1 5953	. . . . 5 |- ( x = N -> ( x .x. ( I ` X ) ) = ( N .x. ( I ` X ) ) )
22	20, 21	eqeq12d 2220	. . . 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 2205	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
25		mulgneg2.m	. . . . . . 7 |- .x. = (.g ` G )
26	23, 24, 25	mulg0 13494	. . . . . 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 13413	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( I ` X ) e. B )
30	23, 24, 25	mulg0 13494	. . . . . 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 2241	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( 0 .x. X ) = ( 0 .x. ( I ` X ) ) )
33		oveq1 5953	. . . . . 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 9307	. . . . . . . . . . 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 8020	. . . . . . . . . 10 |- 1 e. CC
37		negdi 8331	. . . . . . . . . 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 5961	. . . . . . . 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 9408	. . . . . . . . . 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 9400	. . . . . . . . . 10 |- 1 e. ZZ
44		znegcl 9405	. . . . . . . . . 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 2205	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
48	23, 25, 47	mulgdir 13523	. . . . . . . . 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 13511	. . . . . . . . . 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 5962	. . . . . . . 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 2242	. . . . . . 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 13372	. . . . . . . . 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 13502	. . . . . . . 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 2220	. . . . . 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 5578	. . . . . 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 9377	. . . . . . . . 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 13509	. . . . . . . 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 13501	. . . . . . . 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 2220	. . . . . 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 9493	. . 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 2176   ` cfv 5272  (class class class)co 5946   CCcc 7925   0cc0 7927   1c1 7928   + caddc 7930   -ucneg 8246   NNcn 9038   NN0cn0 9297   ZZcz 9374   Basecbs 12865   +g cplusg 12942   0gc0g 13121   Mndcmnd 13281   Grpcgrp 13365   invgcminusg 13366  ._gcmg 13488

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-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-ltadd 8043

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 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-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-2 9097  df-n0 9298  df-z 9375  df-uz 9651  df-fz 10133  df-seqfrec 10595  df-ndx 12868  df-slot 12869  df-base 12871  df-plusg 12955  df-0g 13123  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-grp 13368  df-minusg 13369  df-mulg 13489

This theorem is referenced by:  mulgass  13528