Theorem mulgval 13839

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 13272	. . 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 13838	. . . 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 2241	. . . . 5 |- ( ( n = N /\ x = X ) -> ( n = 0 <-> N = 0 ) )
12	10	breq2d 4121	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( 0 < n <-> 0 < N ) )
13		simpr 110	. . . . . . . . . . 11 |- ( ( n = N /\ x = X ) -> x = X )
14	13	sneqd 3702	. . . . . . . . . 10 |- ( ( n = N /\ x = X ) -> { x } = { X } )
15	14	xpeq2d 4773	. . . . . . . . 9 |- ( ( n = N /\ x = X ) -> ( NN X. { x } ) = ( NN X. { X } ) )
16	15	seqeq3d 10817	. . . . . . . 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 2283	. . . . . . 7 |- ( ( n = N /\ x = X ) -> seq 1 ( .+ , ( NN X. { x } ) ) = S )
19	18, 10	fveq12d 5677	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = ( S ` N ) )
20	10	negeqd 8468	. . . . . . . 8 |- ( ( n = N /\ x = X ) -> -u n = -u N )
21	18, 20	fveq12d 5677	. . . . . . 7 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) = ( S ` -u N ) )
22	21	fveq2d 5674	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) = ( I ` ( S ` -u N ) ) )
23	12, 19, 22	ifbieq12d 3649	. . . . 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 3647	. . . 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 529	. . 3 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> X e. B )
28		fn0g 13588	. . . . . . 7 |- 0g Fn _V
29		funfvex 5687	. . . . . . . 8 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
30	29	funfni 5458	. . . . . . 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 2319	. . . . 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 9890	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
35		1zzd 9604	. . . . . . . . 9 |- ( ( X e. B /\ G e. _V ) -> 1 e. ZZ )
36		fvconst2g 5898	. . . . . . . . . . . 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 2309	. . . . . . . . . . 11 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. B )
39	38	elexd 2827	. . . . . . . . . 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 531	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> u e. _V )
42		plusgslid 13325	. . . . . . . . . . . . 13 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13239	. . . . . . . . . . . 12 |- ( G e. _V -> ( +g ` G ) e. _V )
44	4, 43	eqeltrid 2319	. . . . . . . . . . 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 533	. . . . . . . . . 10 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> v e. _V )
47		ovexg 6084	. . . . . . . . . 10 |- ( ( u e. _V /\ .+ e. _V /\ v e. _V ) -> ( u .+ v ) e. _V )
48	41, 45, 46, 47	syl3anc 1274	. . . . . . . . 9 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> ( u .+ v ) e. _V )
49	34, 35, 40, 48	seqf 10826	. . . . . . . 8 |- ( ( X e. B /\ G e. _V ) -> seq 1 ( .+ , ( NN X. { X } ) ) : NN --> _V )
50	17	feq1i 5501	. . . . . . . 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 543	. . . . . . 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 9587	. . . . . . 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 5813	. . . . 5 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> ( S ` N ) e. _V )
58	1, 6	grpinvfng 13757	. . . . . . . 8 |- ( G e. _V -> I Fn B )
59		basfn 13271	. . . . . . . . . 10 |- Base Fn _V
60		funfvex 5687	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
61	60	funfni 5458	. . . . . . . . . 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 2319	. . . . . . . 8 |- ( G e. _V -> B e. _V )
64		fnex 5906	. . . . . . . 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 9608	. . . . . . . . 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 529	. . . . . . . . . 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 9619	. . . . . . . . . . 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 1387	. . . . . . . . 9 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> N < 0 )
75		zre 9581	. . . . . . . . . . 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 8789	. . . . . . . . 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 9587	. . . . . . . 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 5813	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( S ` -u N ) e. _V )
82		fvexg 5689	. . . . . 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 9589	. . . . . 6 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> 0 e. ZZ )
85		simplll 535	. . . . . 6 |- ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) -> N e. ZZ )
86		zdclt 9655	. . . . . 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 3652	. . . 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 9589	. . . . 5 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> 0 e. ZZ )
90		zdceq 9653	. . . . 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 3652	. . 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 6181	. 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 842   \/ w3o 1004   = wceq 1398   e. wcel 2203   _Vcvv 2813   ifcif 3620   {csn 3689   class class class wbr 4109   X. cxp 4747   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   RRcr 8126   0cc0 8127   1c1 8128   < clt 8308   -ucneg 8445   NNcn 9237   ZZcz 9577   seqcseq 10809   Basecbs 13212   +g cplusg 13290   0gc0g 13469   invgcminusg 13714  ._gcmg 13836

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-seqfrec 10810  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-minusg 13717  df-mulg 13837

This theorem is referenced by:  mulg0  13842  mulgnn  13843  mulgnegnn  13849  subgmulg  13905