Theorem mulgval 13699

Description: Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)

Hypotheses

Ref	Expression
mulgval.b	|- B = ( Base ` G )
mulgval.p	|- .+ = ( +g ` G )
mulgval.o	|- .0. = ( 0g ` G )
mulgval.i	|- I = ( invg ` G )
mulgval.t	|- .x. = (.g ` G )
mulgval.s	|- S = seq 1 ( .+ , ( NN X. { X } ) )

Assertion

Ref	Expression
mulgval	|- ( ( N e. ZZ /\ X e. B ) -> ( N .x. X ) = if ( N = 0 , .0. , if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) ) )

Proof of Theorem mulgval

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

Step	Hyp	Ref	Expression
1		mulgval.b	. . . 4 |- B = ( Base ` G )
2	1	basmex 13132	. . 3 |- ( X e. B -> G e. _V )
3	2	adantl 277	. 2 |- ( ( N e. ZZ /\ X e. B ) -> G e. _V )
4		mulgval.p	. . . . 5 |- .+ = ( +g ` G )
5		mulgval.o	. . . . 5 |- .0. = ( 0g ` G )
6		mulgval.i	. . . . 5 |- I = ( invg ` G )
7		mulgval.t	. . . . 5 |- .x. = (.g ` G )
8	1, 4, 5, 6, 7	mulgfvalg 13698	. . . 4 |- ( G e. _V -> .x. = ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) )
9	8	adantl 277	. . 3 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> .x. = ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) )
10		simpl 109	. . . . . 6 |- ( ( n = N /\ x = X ) -> n = N )
11	10	eqeq1d 2238	. . . . 5 |- ( ( n = N /\ x = X ) -> ( n = 0 <-> N = 0 ) )
12	10	breq2d 4098	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( 0 < n <-> 0 < N ) )
13		simpr 110	. . . . . . . . . . 11 |- ( ( n = N /\ x = X ) -> x = X )
14	13	sneqd 3680	. . . . . . . . . 10 |- ( ( n = N /\ x = X ) -> { x } = { X } )
15	14	xpeq2d 4747	. . . . . . . . 9 |- ( ( n = N /\ x = X ) -> ( NN X. { x } ) = ( NN X. { X } ) )
16	15	seqeq3d 10707	. . . . . . . 8 |- ( ( n = N /\ x = X ) -> seq 1 ( .+ , ( NN X. { x } ) ) = seq 1 ( .+ , ( NN X. { X } ) ) )
17		mulgval.s	. . . . . . . 8 |- S = seq 1 ( .+ , ( NN X. { X } ) )
18	16, 17	eqtr4di 2280	. . . . . . 7 |- ( ( n = N /\ x = X ) -> seq 1 ( .+ , ( NN X. { x } ) ) = S )
19	18, 10	fveq12d 5642	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = ( S ` N ) )
20	10	negeqd 8364	. . . . . . . 8 |- ( ( n = N /\ x = X ) -> -u n = -u N )
21	18, 20	fveq12d 5642	. . . . . . 7 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) = ( S ` -u N ) )
22	21	fveq2d 5639	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) = ( I ` ( S ` -u N ) ) )
23	12, 19, 22	ifbieq12d 3630	. . . . 5 |- ( ( n = N /\ x = X ) -> if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) = if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) )
24	11, 23	ifbieq2d 3628	. . . 4 |- ( ( n = N /\ x = X ) -> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) = if ( N = 0 , .0. , if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) ) )
25	24	adantl 277	. . 3 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ ( n = N /\ x = X ) ) -> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) = if ( N = 0 , .0. , if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) ) )
26		simpll 527	. . 3 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> N e. ZZ )
27		simplr 528	. . 3 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> X e. B )
28		fn0g 13448	. . . . . . 7 |- 0g Fn _V
29		funfvex 5652	. . . . . . . 8 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
30	29	funfni 5429	. . . . . . 7 |- ( ( 0g Fn _V /\ G e. _V ) -> ( 0g ` G ) e. _V )
31	28, 30	mpan 424	. . . . . 6 |- ( G e. _V -> ( 0g ` G ) e. _V )
32	5, 31	eqeltrid 2316	. . . . 5 |- ( G e. _V -> .0. e. _V )
33	32	ad2antlr 489	. . . 4 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ N = 0 ) -> .0. e. _V )
34		nnuz 9782	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
35		1zzd 9496	. . . . . . . . 9 |- ( ( X e. B /\ G e. _V ) -> 1 e. ZZ )
36		fvconst2g 5863	. . . . . . . . . . . 12 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) = X )
37		simpl 109	. . . . . . . . . . . 12 |- ( ( X e. B /\ u e. NN ) -> X e. B )
38	36, 37	eqeltrd 2306	. . . . . . . . . . 11 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. B )
39	38	elexd 2814	. . . . . . . . . 10 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
40	39	adantlr 477	. . . . . . . . 9 |- ( ( ( X e. B /\ G e. _V ) /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
41		simprl 529	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> u e. _V )
42		plusgslid 13185	. . . . . . . . . . . . 13 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13099	. . . . . . . . . . . 12 |- ( G e. _V -> ( +g ` G ) e. _V )
44	4, 43	eqeltrid 2316	. . . . . . . . . . 11 |- ( G e. _V -> .+ e. _V )
45	44	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> .+ e. _V )
46		simprr 531	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> v e. _V )
47		ovexg 6047	. . . . . . . . . 10 |- ( ( u e. _V /\ .+ e. _V /\ v e. _V ) -> ( u .+ v ) e. _V )
48	41, 45, 46, 47	syl3anc 1271	. . . . . . . . 9 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> ( u .+ v ) e. _V )
49	34, 35, 40, 48	seqf 10716	. . . . . . . 8 |- ( ( X e. B /\ G e. _V ) -> seq 1 ( .+ , ( NN X. { X } ) ) : NN --> _V )
50	17	feq1i 5472	. . . . . . . 8 |- ( S : NN --> _V <-> seq 1 ( .+ , ( NN X. { X } ) ) : NN --> _V )
51	49, 50	sylibr 134	. . . . . . 7 |- ( ( X e. B /\ G e. _V ) -> S : NN --> _V )
52	51	ad5ant23 522	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> S : NN --> _V )
53		simp-4l 541	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> N e. ZZ )
54		simpr 110	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> 0 < N )
55		elnnz 9479	. . . . . . 7 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
56	53, 54, 55	sylanbrc 417	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> N e. NN )
57	52, 56	ffvelcdmd 5779	. . . . 5 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> ( S ` N ) e. _V )
58	1, 6	grpinvfng 13617	. . . . . . . 8 |- ( G e. _V -> I Fn B )
59		basfn 13131	. . . . . . . . . 10 |- Base Fn _V
60		funfvex 5652	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
61	60	funfni 5429	. . . . . . . . . 10 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
62	59, 61	mpan 424	. . . . . . . . 9 |- ( G e. _V -> ( Base ` G ) e. _V )
63	1, 62	eqeltrid 2316	. . . . . . . 8 |- ( G e. _V -> B e. _V )
64		fnex 5871	. . . . . . . 8 |- ( ( I Fn B /\ B e. _V ) -> I e. _V )
65	58, 63, 64	syl2anc 411	. . . . . . 7 |- ( G e. _V -> I e. _V )
66	65	ad3antlr 493	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> I e. _V )
67	51	ad5ant23 522	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> S : NN --> _V )
68		znegcl 9500	. . . . . . . . 9 |- ( N e. ZZ -> -u N e. ZZ )
69	68	ad4antr 494	. . . . . . . 8 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. ZZ )
70		simplr 528	. . . . . . . . . 10 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. N = 0 )
71		simpr 110	. . . . . . . . . 10 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 < N )
72		ztri3or0 9511	. . . . . . . . . . 11 |- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
73	72	ad4antr 494	. . . . . . . . . 10 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( N < 0 \/ N = 0 \/ 0 < N ) )
74	70, 71, 73	ecase23d 1384	. . . . . . . . 9 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> N < 0 )
75		zre 9473	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. RR )
76	75	ad4antr 494	. . . . . . . . . 10 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. RR )
77	76	lt0neg1d 8685	. . . . . . . . 9 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( N < 0 <-> 0 < -u N ) )
78	74, 77	mpbid 147	. . . . . . . 8 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> 0 < -u N )
79		elnnz 9479	. . . . . . . 8 |- ( -u N e. NN <-> ( -u N e. ZZ /\ 0 < -u N ) )
80	69, 78, 79	sylanbrc 417	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. NN )
81	67, 80	ffvelcdmd 5779	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( S ` -u N ) e. _V )
82		fvexg 5654	. . . . . 6 |- ( ( I e. _V /\ ( S ` -u N ) e. _V ) -> ( I ` ( S ` -u N ) ) e. _V )
83	66, 81, 82	syl2anc 411	. . . . 5 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( I ` ( S ` -u N ) ) e. _V )
84		0zd 9481	. . . . . 6 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> 0 e. ZZ )
85		simplll 533	. . . . . 6 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> N e. ZZ )
86		zdclt 9547	. . . . . 6 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> DECID 0 < N )
87	84, 85, 86	syl2anc 411	. . . . 5 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> DECID 0 < N )
88	57, 83, 87	ifcldadc 3633	. . . 4 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) e. _V )
89		0zd 9481	. . . . 5 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> 0 e. ZZ )
90		zdceq 9545	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
91	26, 89, 90	syl2anc 411	. . . 4 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> DECID N = 0 )
92	33, 88, 91	ifcldadc 3633	. . 3 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> if ( N = 0 , .0. , if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) ) e. _V )
93	9, 25, 26, 27, 92	ovmpod 6144	. 2 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> ( N .x. X ) = if ( N = 0 , .0. , if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) ) )
94	3, 93	mpdan 421	1 |- ( ( N e. ZZ /\ X e. B ) -> ( N .x. X ) = if ( N = 0 , .0. , if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 839   \/ w3o 1001   = wceq 1395   e. wcel 2200   _Vcvv 2800   ifcif 3603   {csn 3667   class class class wbr 4086   X. cxp 4721   Fn wfn 5319   -->wf 5320   ` cfv 5324  (class class class)co 6013   e. cmpo 6015   RRcr 8021   0cc0 8022   1c1 8023   < clt 8204   -ucneg 8341   NNcn 9133   ZZcz 9469   seqcseq 10699   Basecbs 13072   +g cplusg 13150   0gc0g 13329   invgcminusg 13574  ._gcmg 13696

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

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-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-seqfrec 10700  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-minusg 13577  df-mulg 13697

This theorem is referenced by:  mulg0  13702  mulgnn  13703  mulgnegnn  13709  subgmulg  13765