Theorem mulgval 13984

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 13470	. . 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 13983	. . . 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 2247	. . . . 5 |- ( ( n = N /\ x = X ) -> ( n = 0 <-> N = 0 ) )
12	10	breq2d 4143	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( 0 < n <-> 0 < N ) )
13		simpr 110	. . . . . . . . . . 11 |- ( ( n = N /\ x = X ) -> x = X )
14	13	sneqd 3722	. . . . . . . . . 10 |- ( ( n = N /\ x = X ) -> { x } = { X } )
15	14	xpeq2d 4800	. . . . . . . . 9 |- ( ( n = N /\ x = X ) -> ( NN X. { x } ) = ( NN X. { X } ) )
16	15	seqeq3d 10913	. . . . . . . 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 2289	. . . . . . 7 |- ( ( n = N /\ x = X ) -> seq 1 ( .+ , ( NN X. { x } ) ) = S )
19	18, 10	fveq12d 5705	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = ( S ` N ) )
20	10	negeqd 8528	. . . . . . . 8 |- ( ( n = N /\ x = X ) -> -u n = -u N )
21	18, 20	fveq12d 5705	. . . . . . 7 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) = ( S ` -u N ) )
22	21	fveq2d 5702	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) = ( I ` ( S ` -u N ) ) )
23	12, 19, 22	ifbieq12d 3667	. . . . 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 3665	. . . 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 531	. . 3 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> N e. ZZ )
27		simplr 533	. . 3 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> X e. B )
28		fn0g 13754	. . . . . . 7 |- 0g Fn _V
29		funfvex 5715	. . . . . . . 8 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
30	29	funfni 5486	. . . . . . 7 |- ( ( 0g Fn _V /\ G e. _V ) -> ( 0g ` G ) e. _V )
31	28, 30	mpan 428	. . . . . 6 |- ( G e. _V -> ( 0g ` G ) e. _V )
32	5, 31	eqeltrid 2325	. . . . 5 |- ( G e. _V -> .0. e. _V )
33	32	ad2antlr 493	. . . 4 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ N = 0 ) -> .0. e. _V )
34		nnuz 9974	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
35		1zzd 9682	. . . . . . . . 9 |- ( ( X e. B /\ G e. _V ) -> 1 e. ZZ )
36		fvconst2g 5932	. . . . . . . . . . . 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 2315	. . . . . . . . . . 11 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. B )
39	38	elexd 2835	. . . . . . . . . 10 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
40	39	adantlr 481	. . . . . . . . 9 |- ( ( ( X e. B /\ G e. _V ) /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
41		simprl 535	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> u e. _V )
42		plusgslid 13525	. . . . . . . . . . . . 13 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13437	. . . . . . . . . . . 12 |- ( G e. _V -> ( +g ` G ) e. _V )
44	4, 43	eqeltrid 2325	. . . . . . . . . . 11 |- ( G e. _V -> .+ e. _V )
45	44	ad2antlr 493	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> .+ e. _V )
46		simprr 537	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> v e. _V )
47		ovexg 6122	. . . . . . . . . 10 |- ( ( u e. _V /\ .+ e. _V /\ v e. _V ) -> ( u .+ v ) e. _V )
48	41, 45, 46, 47	syl3anc 1278	. . . . . . . . 9 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> ( u .+ v ) e. _V )
49	34, 35, 40, 48	seqf 10922	. . . . . . . 8 |- ( ( X e. B /\ G e. _V ) -> seq 1 ( .+ , ( NN X. { X } ) ) : NN --> _V )
50	17	feq1i 5529	. . . . . . . 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 526	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> S : NN --> _V )
53		simp-4l 547	. . . . . . 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 9665	. . . . . . 7 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
56	53, 54, 55	sylanbrc 421	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> N e. NN )
57	52, 56	ffvelcdmd 5847	. . . . 5 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> ( S ` N ) e. _V )
58	1, 6	grpinvfng 13908	. . . . . . . 8 |- ( G e. _V -> I Fn B )
59		basfn 13469	. . . . . . . . . 10 |- Base Fn _V
60		funfvex 5715	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
61	60	funfni 5486	. . . . . . . . . 10 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
62	59, 61	mpan 428	. . . . . . . . 9 |- ( G e. _V -> ( Base ` G ) e. _V )
63	1, 62	eqeltrid 2325	. . . . . . . 8 |- ( G e. _V -> B e. _V )
64		fnex 5940	. . . . . . . 8 |- ( ( I Fn B /\ B e. _V ) -> I e. _V )
65	58, 63, 64	syl2anc 415	. . . . . . 7 |- ( G e. _V -> I e. _V )
66	65	ad3antlr 497	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> I e. _V )
67	51	ad5ant23 526	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> S : NN --> _V )
68		znegcl 9686	. . . . . . . . 9 |- ( N e. ZZ -> -u N e. ZZ )
69	68	ad4antr 498	. . . . . . . 8 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. ZZ )
70		simplr 533	. . . . . . . . . 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 9697	. . . . . . . . . . 11 |- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
73	72	ad4antr 498	. . . . . . . . . 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 1391	. . . . . . . . 9 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> N < 0 )
75		zre 9659	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. RR )
76	75	ad4antr 498	. . . . . . . . . 10 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. RR )
77	76	lt0neg1d 8850	. . . . . . . . 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 9665	. . . . . . . 8 |- ( -u N e. NN <-> ( -u N e. ZZ /\ 0 < -u N ) )
80	69, 78, 79	sylanbrc 421	. . . . . . 7 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. NN )
81	67, 80	ffvelcdmd 5847	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( S ` -u N ) e. _V )
82		fvexg 5717	. . . . . 6 |- ( ( I e. _V /\ ( S ` -u N ) e. _V ) -> ( I ` ( S ` -u N ) ) e. _V )
83	66, 81, 82	syl2anc 415	. . . . 5 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( I ` ( S ` -u N ) ) e. _V )
84		0zd 9667	. . . . . 6 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> 0 e. ZZ )
85		simplll 539	. . . . . 6 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> N e. ZZ )
86		zdclt 9733	. . . . . 6 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> DECID 0 < N )
87	84, 85, 86	syl2anc 415	. . . . 5 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> DECID 0 < N )
88	57, 83, 87	ifcldadc 3670	. . . 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 9667	. . . . 5 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> 0 e. ZZ )
90		zdceq 9731	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
91	26, 89, 90	syl2anc 415	. . . 4 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> DECID N = 0 )
92	33, 88, 91	ifcldadc 3670	. . 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 6219	. 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 425	1 |- ( ( N e. ZZ /\ X e. B ) -> ( N .x. X ) = if ( N = 0 , .0. , if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 846   \/ w3o 1008   = wceq 1402   e. wcel 2209   _Vcvv 2821   ifcif 3638   {csn 3709   class class class wbr 4131   X. cxp 4774   Fn wfn 5375   -->wf 5376   ` cfv 5380  (class class class)co 6088   e. cmpo 6090   RRcr 8184   0cc0 8185   1c1 8186   < clt 8366   -ucneg 8505   NNcn 9313   ZZcz 9655   seqcseq 10905   Basecbs 13410   +g cplusg 13490   0gc0g 13669   invgcminusg 13865  ._gcmg 13981

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-minusg 13868  df-mulg 13982

This theorem is used by:  mulg0  13987  mulgnn  13988  mulgnegnn  13994  subgmulg  14050