Theorem mulgval 13533

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 12966	. . 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 13532	. . . 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 2215	. . . . 5 |- ( ( n = N /\ x = X ) -> ( n = 0 <-> N = 0 ) )
12	10	breq2d 4063	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( 0 < n <-> 0 < N ) )
13		simpr 110	. . . . . . . . . . 11 |- ( ( n = N /\ x = X ) -> x = X )
14	13	sneqd 3651	. . . . . . . . . 10 |- ( ( n = N /\ x = X ) -> { x } = { X } )
15	14	xpeq2d 4707	. . . . . . . . 9 |- ( ( n = N /\ x = X ) -> ( NN X. { x } ) = ( NN X. { X } ) )
16	15	seqeq3d 10622	. . . . . . . 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 2257	. . . . . . 7 |- ( ( n = N /\ x = X ) -> seq 1 ( .+ , ( NN X. { x } ) ) = S )
19	18, 10	fveq12d 5596	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = ( S ` N ) )
20	10	negeqd 8287	. . . . . . . 8 |- ( ( n = N /\ x = X ) -> -u n = -u N )
21	18, 20	fveq12d 5596	. . . . . . 7 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) = ( S ` -u N ) )
22	21	fveq2d 5593	. . . . . 6 |- ( ( n = N /\ x = X ) -> ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) = ( I ` ( S ` -u N ) ) )
23	12, 19, 22	ifbieq12d 3602	. . . . 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 3600	. . . 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 13282	. . . . . . 7 |- 0g Fn _V
29		funfvex 5606	. . . . . . . 8 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
30	29	funfni 5385	. . . . . . 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 2293	. . . . 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 9704	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
35		1zzd 9419	. . . . . . . . 9 |- ( ( X e. B /\ G e. _V ) -> 1 e. ZZ )
36		fvconst2g 5811	. . . . . . . . . . . 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 2283	. . . . . . . . . . 11 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. B )
39	38	elexd 2787	. . . . . . . . . 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 13019	. . . . . . . . . . . . 13 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 12934	. . . . . . . . . . . 12 |- ( G e. _V -> ( +g ` G ) e. _V )
44	4, 43	eqeltrid 2293	. . . . . . . . . . 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 5991	. . . . . . . . . 10 |- ( ( u e. _V /\ .+ e. _V /\ v e. _V ) -> ( u .+ v ) e. _V )
48	41, 45, 46, 47	syl3anc 1250	. . . . . . . . 9 |- ( ( ( X e. B /\ G e. _V ) /\ ( u e. _V /\ v e. _V ) ) -> ( u .+ v ) e. _V )
49	34, 35, 40, 48	seqf 10631	. . . . . . . 8 |- ( ( X e. B /\ G e. _V ) -> seq 1 ( .+ , ( NN X. { X } ) ) : NN --> _V )
50	17	feq1i 5428	. . . . . . . 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 9402	. . . . . . 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 5729	. . . . 5 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ 0 < N ) -> ( S ` N ) e. _V )
58	1, 6	grpinvfng 13451	. . . . . . . 8 |- ( G e. _V -> I Fn B )
59		basfn 12965	. . . . . . . . . 10 |- Base Fn _V
60		funfvex 5606	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
61	60	funfni 5385	. . . . . . . . . 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 2293	. . . . . . . 8 |- ( G e. _V -> B e. _V )
64		fnex 5819	. . . . . . . 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 9423	. . . . . . . . 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 9434	. . . . . . . . . . 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 1363	. . . . . . . . 9 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> N < 0 )
75		zre 9396	. . . . . . . . . . 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 8608	. . . . . . . . 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 9402	. . . . . . . 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 5729	. . . . . 6 |- ( ( ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( S ` -u N ) e. _V )
82		fvexg 5608	. . . . . 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 9404	. . . . . 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 9470	. . . . . 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 3605	. . . 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 9404	. . . . 5 |- ( ( ( N e. ZZ /\ X e. B ) /\ G e. _V ) -> 0 e. ZZ )
90		zdceq 9468	. . . . 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 3605	. . 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 6086	. 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 836   \/ w3o 980   = wceq 1373   e. wcel 2177   _Vcvv 2773   ifcif 3575   {csn 3638   class class class wbr 4051   X. cxp 4681   Fn wfn 5275   -->wf 5276   ` cfv 5280  (class class class)co 5957   e. cmpo 5959   RRcr 7944   0cc0 7945   1c1 7946   < clt 8127   -ucneg 8264   NNcn 9056   ZZcz 9392   seqcseq 10614   Basecbs 12907   +g cplusg 12984   0gc0g 13163   invgcminusg 13408  ._gcmg 13530

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-ltadd 8061

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-inn 9057  df-2 9115  df-n0 9316  df-z 9393  df-uz 9669  df-seqfrec 10615  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-0g 13165  df-minusg 13411  df-mulg 13531

This theorem is referenced by:  mulg0  13536  mulgnn  13537  mulgnegnn  13543  subgmulg  13599