Theorem mulgval 17543

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 n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( n = N /\ x = X ) -> n = N )
2	1	eqeq1d 2624	. . 3 |- ( ( n = N /\ x = X ) -> ( n = 0 <-> N = 0 ) )
3	1	breq2d 4665	. . . 4 |- ( ( n = N /\ x = X ) -> ( 0 < n <-> 0 < N ) )
4		simpr 477	. . . . . . . . 9 |- ( ( n = N /\ x = X ) -> x = X )
5	4	sneqd 4189	. . . . . . . 8 |- ( ( n = N /\ x = X ) -> { x } = { X } )
6	5	xpeq2d 5139	. . . . . . 7 |- ( ( n = N /\ x = X ) -> ( NN X. { x } ) = ( NN X. { X } ) )
7	6	seqeq3d 12809	. . . . . 6 |- ( ( n = N /\ x = X ) -> seq 1 ( .+ , ( NN X. { x } ) ) = seq 1 ( .+ , ( NN X. { X } ) ) )
8		mulgval.s	. . . . . 6 |- S = seq 1 ( .+ , ( NN X. { X } ) )
9	7, 8	syl6eqr 2674	. . . . 5 |- ( ( n = N /\ x = X ) -> seq 1 ( .+ , ( NN X. { x } ) ) = S )
10	9, 1	fveq12d 6197	. . . 4 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = ( S ` N ) )
11	1	negeqd 10275	. . . . . 6 |- ( ( n = N /\ x = X ) -> -u n = -u N )
12	9, 11	fveq12d 6197	. . . . 5 |- ( ( n = N /\ x = X ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) = ( S ` -u N ) )
13	12	fveq2d 6195	. . . 4 |- ( ( n = N /\ x = X ) -> ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) = ( I ` ( S ` -u N ) ) )
14	3, 10, 13	ifbieq12d 4113	. . 3 |- ( ( 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 ) ) ) )
15	2, 14	ifbieq2d 4111	. 2 |- ( ( 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 ) ) ) ) )
16		mulgval.b	. . 3 |- B = ( Base ` G )
17		mulgval.p	. . 3 |- .+ = ( +g ` G )
18		mulgval.o	. . 3 |- .0. = ( 0g ` G )
19		mulgval.i	. . 3 |- I = ( invg ` G )
20		mulgval.t	. . 3 |- .x. = (.g ` G )
21	16, 17, 18, 19, 20	mulgfval 17542	. 2 |- .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 ) ) ) ) )
22		fvex 6201	. . . 4 |- ( 0g ` G ) e. _V
23	18, 22	eqeltri 2697	. . 3 |- .0. e. _V
24		fvex 6201	. . . 4 |- ( S ` N ) e. _V
25		fvex 6201	. . . 4 |- ( I ` ( S ` -u N ) ) e. _V
26	24, 25	ifex 4156	. . 3 |- if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) e. _V
27	23, 26	ifex 4156	. 2 |- if ( N = 0 , .0. , if ( 0 < N , ( S ` N ) , ( I ` ( S ` -u N ) ) ) ) e. _V
28	15, 21, 27	ovmpt2a 6791	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:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   {csn 4177   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   < clt 10074   -ucneg 10267   NNcn 11020   ZZcz 11377   seqcseq 12801   Basecbs 15857   +g cplusg 15941   0gc0g 16100   invgcminusg 17423  ._gcmg 17540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-neg 10269  df-z 11378  df-seq 12802  df-mulg 17541

This theorem is referenced by:  mulg0  17546  mulgnn  17547  mulgnegnn  17551  subgmulg  17608