Theorem mulgval 12837

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 12478	. . 3 |- ( X e. B -> G e. _V )
3	2	adantl 275	. 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 12836	. . . 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 275	. . 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 108	. . . . . 6 |- ( ( n = N /\ x = X ) -> n = N )
11	10	eqeq1d 2180	. . . . 5 |- ( ( n = N /\ x = X ) -> ( n = 0 <-> N = 0 ) )
12	10	breq2d 4002	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( 0 < n <-> 0 < N ) )
13		simpr 109	. . . . . . . . . . 11 |- ( ( n = N /\ x = X ) -> x = X )
14	13	sneqd 3597	. . . . . . . . . 10 |- ( ( n = N /\ x = X ) -> { x } = { X } )
15	14	xpeq2d 4636	. . . . . . . . 9 |- ( ( n = N /\ x = X ) -> ( NN X. { x } ) = ( NN X. { X } ) )
16	15	seqeq3d 10413	. . . . . . . 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 2222	. . . . . . 7 |- ( ( n = N /\ x = X ) -> seq 1 ( .+ , ( NN X. { x } ) ) = S )
19	18, 10	fveq12d 5506	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = ( S ` N ) )
20	10	negeqd 8118	. . . . . . . 8 |- ( ( n = N /\ x = X ) -> -u n = -u N )
21	18, 20	fveq12d 5506	. . . . . . 7 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) = ( S ` -u N ) )
22	21	fveq2d 5503	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) = ( I ` ( S ` -u N ) ) )
23	12, 19, 22	ifbieq12d 3553	. . . . 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 3551	. . . 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 275	. . 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 525	. . 3 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> N e. ZZ )
27		simplr 526	. . 3 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> X e. B )
28		fn0g 12651	. . . . . . 7 |- 0g Fn _V
29		funfvex 5516	. . . . . . . 8 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
30	29	funfni 5300	. . . . . . 7 |- ( ( 0g Fn _V /\ G e. _V ) -> ( 0g ` G ) e. _V )
31	28, 30	mpan 422	. . . . . 6 |- ( G e. _V -> ( 0g ` G ) e. _V )
32	5, 31	eqeltrid 2258	. . . . 5 |- ( G e. _V -> .0. e. _V )
33	32	ad2antlr 487	. . . 4 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ N = 0 ) -> .0. e. _V )
34		nnuz 9526	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
35		1zzd 9243	. . . . . . . . 9 |- ( ( X e. B /\ G e. _V ) -> 1 e. ZZ )
36		fvconst2g 5714	. . . . . . . . . . . 12 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) = X )
37		simpl 108	. . . . . . . . . . . 12 |- ( ( X e. B /\ u e. NN ) -> X e. B )
38	36, 37	eqeltrd 2248	. . . . . . . . . . 11 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. B )
39	38	elexd 2744	. . . . . . . . . 10 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
40	39	adantlr 475	. . . . . . . . 9 |- ( ( ( X e. B /\ G e. _V ) /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
41		simprl 527	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> u e. _V )
42		plusgslid 12517	. . . . . . . . . . . . 13 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 12447	. . . . . . . . . . . 12 |- ( G e. _V -> ( +g ` G ) e. _V )
44	4, 43	eqeltrid 2258	. . . . . . . . . . 11 |- ( G e. _V -> .+ e. _V )
45	44	ad2antlr 487	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> .+ e. _V )
46		simprr 528	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> v e. _V )
47		ovexg 5891	. . . . . . . . . 10 |- ( ( u e. _V /\ .+ e. _V /\ v e. _V ) -> ( u .+ v ) e. _V )
48	41, 45, 46, 47	syl3anc 1234	. . . . . . . . 9 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> ( u .+ v ) e. _V )
49	34, 35, 40, 48	seqf 10421	. . . . . . . 8 |- ( ( X e. B /\ G e. _V ) -> seq 1 ( .+ , ( NN X. { X } ) ) : NN --> _V )
50	17	feq1i 5342	. . . . . . . 8 |- ( S : NN --> _V <-> seq 1 ( .+ , ( NN X. { X } ) ) : NN --> _V )
51	49, 50	sylibr 133	. . . . . . 7 |- ( ( X e. B /\ G e. _V ) -> S : NN --> _V )
52	51	ad5ant23 520	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> S : NN --> _V )
53		simp-4l 537	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> N e. ZZ )
54		simpr 109	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> 0 < N )
55		elnnz 9226	. . . . . . 7 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
56	53, 54, 55	sylanbrc 415	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> N e. NN )
57	52, 56	ffvelrnd 5636	. . . . 5 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> ( S ` N ) e. _V )
58	1, 6	grpinvfng 12769	. . . . . . . 8 |- ( G e. _V -> I Fn B )
59		basfn 12477	. . . . . . . . . 10 |- Base Fn _V
60		funfvex 5516	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
61	60	funfni 5300	. . . . . . . . . 10 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
62	59, 61	mpan 422	. . . . . . . . 9 |- ( G e. _V -> ( Base ` G ) e. _V )
63	1, 62	eqeltrid 2258	. . . . . . . 8 |- ( G e. _V -> B e. _V )
64		fnex 5722	. . . . . . . 8 |- ( ( I Fn B /\ B e. _V ) -> I e. _V )
65	58, 63, 64	syl2anc 409	. . . . . . 7 |- ( G e. _V -> I e. _V )
66	65	ad3antlr 491	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> I e. _V )
67	51	ad5ant23 520	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> S : NN --> _V )
68		znegcl 9247	. . . . . . . . 9 |- ( N e. ZZ -> -u N e. ZZ )
69	68	ad4antr 492	. . . . . . . 8 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. ZZ )
70		simplr 526	. . . . . . . . . 10 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. N = 0 )
71		simpr 109	. . . . . . . . . 10 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 < N )
72		ztri3or0 9258	. . . . . . . . . . 11 |- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
73	72	ad4antr 492	. . . . . . . . . 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 1346	. . . . . . . . 9 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> N < 0 )
75		zre 9220	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. RR )
76	75	ad4antr 492	. . . . . . . . . 10 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. RR )
77	76	lt0neg1d 8438	. . . . . . . . 9 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( N < 0 <-> 0 < -u N ) )
78	74, 77	mpbid 146	. . . . . . . 8 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> 0 < -u N )
79		elnnz 9226	. . . . . . . 8 |- ( -u N e. NN <-> ( -u N e. ZZ /\ 0 < -u N ) )
80	69, 78, 79	sylanbrc 415	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. NN )
81	67, 80	ffvelrnd 5636	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( S ` -u N ) e. _V )
82		fvexg 5518	. . . . . 6 |- ( ( I e. _V /\ ( S ` -u N ) e. _V ) -> ( I ` ( S ` -u N ) ) e. _V )
83	66, 81, 82	syl2anc 409	. . . . 5 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( I ` ( S ` -u N ) ) e. _V )
84		0zd 9228	. . . . . 6 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> 0 e. ZZ )
85		simplll 529	. . . . . 6 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> N e. ZZ )
86		zdclt 9293	. . . . . 6 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> DECID 0 < N )
87	84, 85, 86	syl2anc 409	. . . . 5 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> DECID 0 < N )
88	57, 83, 87	ifcldadc 3556	. . . 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 9228	. . . . 5 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> 0 e. ZZ )
90		zdceq 9291	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
91	26, 89, 90	syl2anc 409	. . . 4 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> DECID N = 0 )
92	33, 88, 91	ifcldadc 3556	. . 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 5984	. 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 419	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 103  DECID wdc 830   \/ w3o 973   = wceq 1349   e. wcel 2142   _Vcvv 2731   ifcif 3527   {csn 3584   class class class wbr 3990   X. cxp 4610   Fn wfn 5195   -->wf 5196   ` cfv 5200  (class class class)co 5857   e. cmpo 5859   RRcr 7777   0cc0 7778   1c1 7779   < clt 7958   -ucneg 8095   NNcn 8882   ZZcz 9216   seqcseq 10405   Basecbs 12420   +g cplusg 12484   0gc0g 12618   invgcminusg 12731  ._gcmg 12834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-addcom 7878  ax-addass 7880  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-0id 7886  ax-rnegex 7887  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-ltwlin 7891  ax-pre-lttrn 7892  ax-pre-ltadd 7894

This theorem depends on definitions:  df-bi 116  df-dc 831  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-if 3528  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-id 4279  df-iord 4352  df-on 4354  df-ilim 4355  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-recs 6288  df-frec 6374  df-pnf 7960  df-mnf 7961  df-xr 7962  df-ltxr 7963  df-le 7964  df-sub 8096  df-neg 8097  df-inn 8883  df-2 8941  df-n0 9140  df-z 9217  df-uz 9492  df-seqfrec 10406  df-ndx 12423  df-slot 12424  df-base 12426  df-plusg 12497  df-0g 12620  df-minusg 12734  df-mulg 12835

This theorem is referenced by:  mulg0  12839  mulgnn  12840  mulgnegnn  12844