Theorem mulgfval 17542

Description: 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 )

Assertion

Ref	Expression
mulgfval	|- .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 ) ) ) ) )

Distinct variable groups:   x, n, .0.   n, G, x   n, I, x   B, n, x

Allowed substitution hints:   .+ ( x, n)   .x. ( x, n)

Proof of Theorem mulgfval

Dummy variables w s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulgval.t	. 2 |- .x. = (.g ` G )
2		eqidd 2623	. . . . 5 |- ( w = G -> ZZ = ZZ )
3		fveq2 6191	. . . . . 6 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
4		mulgval.b	. . . . . 6 |- B = ( Base ` G )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( w = G -> ( Base ` w ) = B )
6		fveq2 6191	. . . . . . 7 |- ( w = G -> ( 0g ` w ) = ( 0g ` G ) )
7		mulgval.o	. . . . . . 7 |- .0. = ( 0g ` G )
8	6, 7	syl6eqr 2674	. . . . . 6 |- ( w = G -> ( 0g ` w ) = .0. )
9		seqex 12803	. . . . . . . 8 |- seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) e. _V
10	9	a1i 11	. . . . . . 7 |- ( w = G -> seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) e. _V )
11		id 22	. . . . . . . . . 10 |- ( s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) -> s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) )
12		fveq2 6191	. . . . . . . . . . . 12 |- ( w = G -> ( +g ` w ) = ( +g ` G ) )
13		mulgval.p	. . . . . . . . . . . 12 |- .+ = ( +g ` G )
14	12, 13	syl6eqr 2674	. . . . . . . . . . 11 |- ( w = G -> ( +g ` w ) = .+ )
15	14	seqeq2d 12808	. . . . . . . . . 10 |- ( w = G -> seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) = seq 1 ( .+ , ( NN X. { x } ) ) )
16	11, 15	sylan9eqr 2678	. . . . . . . . 9 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> s = seq 1 ( .+ , ( NN X. { x } ) ) )
17	16	fveq1d 6193	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( s ` n ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) )
18		simpl 473	. . . . . . . . . . 11 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> w = G )
19	18	fveq2d 6195	. . . . . . . . . 10 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( invg ` w ) = ( invg ` G ) )
20		mulgval.i	. . . . . . . . . 10 |- I = ( invg ` G )
21	19, 20	syl6eqr 2674	. . . . . . . . 9 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( invg ` w ) = I )
22	16	fveq1d 6193	. . . . . . . . 9 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( s ` -u n ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) )
23	21, 22	fveq12d 6197	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( ( invg ` w ) ` ( s ` -u n ) ) = ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) )
24	17, 23	ifeq12d 4106	. . . . . . 7 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) = if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) )
25	10, 24	csbied 3560	. . . . . 6 |- ( w = G -> [_ seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) = if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) )
26	8, 25	ifeq12d 4106	. . . . 5 |- ( w = G -> if ( n = 0 , ( 0g ` w ) , [_ seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) ) = if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) )
27	2, 5, 26	mpt2eq123dv 6717	. . . 4 |- ( w = G -> ( n e. ZZ , x e. ( Base ` w ) |-> if ( n = 0 , ( 0g ` w ) , [_ seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) ) ) = ( 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 ) ) ) ) ) )
28		df-mulg 17541	. . . 4 |- .g = ( w e. _V |-> ( n e. ZZ , x e. ( Base ` w ) |-> if ( n = 0 , ( 0g ` w ) , [_ seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) ) ) )
29		zex 11386	. . . . 5 |- ZZ e. _V
30		fvex 6201	. . . . . 6 |- ( Base ` G ) e. _V
31	4, 30	eqeltri 2697	. . . . 5 |- B e. _V
32	29, 31	mpt2ex 7247	. . . 4 |- ( 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 ) ) ) ) ) e. _V
33	27, 28, 32	fvmpt 6282	. . 3 |- ( G e. _V -> (.g ` G ) = ( 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 ) ) ) ) ) )
34		fvprc 6185	. . . 4 |- ( -. G e. _V -> (.g ` G ) = (/) )
35		eqid 2622	. . . . . . 7 |- ( 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 ) ) ) ) ) = ( 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 ) ) ) ) )
36		fvex 6201	. . . . . . . . 9 |- ( 0g ` G ) e. _V
37	7, 36	eqeltri 2697	. . . . . . . 8 |- .0. e. _V
38		fvex 6201	. . . . . . . . 9 |- ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. _V
39		fvex 6201	. . . . . . . . 9 |- ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) e. _V
40	38, 39	ifex 4156	. . . . . . . 8 |- if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) e. _V
41	37, 40	ifex 4156	. . . . . . 7 |- if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) e. _V
42	35, 41	fnmpt2i 7239	. . . . . 6 |- ( 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 ) ) ) ) ) Fn ( ZZ X. B )
43		fvprc 6185	. . . . . . . . . 10 |- ( -. G e. _V -> ( Base ` G ) = (/) )
44	4, 43	syl5eq 2668	. . . . . . . . 9 |- ( -. G e. _V -> B = (/) )
45	44	xpeq2d 5139	. . . . . . . 8 |- ( -. G e. _V -> ( ZZ X. B ) = ( ZZ X. (/) ) )
46		xp0 5552	. . . . . . . 8 |- ( ZZ X. (/) ) = (/)
47	45, 46	syl6eq 2672	. . . . . . 7 |- ( -. G e. _V -> ( ZZ X. B ) = (/) )
48	47	fneq2d 5982	. . . . . 6 |- ( -. G e. _V -> ( ( 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 ) ) ) ) ) Fn ( ZZ X. B ) <-> ( 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 ) ) ) ) ) Fn (/) ) )
49	42, 48	mpbii 223	. . . . 5 |- ( -. G e. _V -> ( 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 ) ) ) ) ) Fn (/) )
50		fn0 6011	. . . . 5 |- ( ( 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 ) ) ) ) ) Fn (/) <-> ( 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 ) ) ) ) ) = (/) )
51	49, 50	sylib 208	. . . 4 |- ( -. G e. _V -> ( 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 ) ) ) ) ) = (/) )
52	34, 51	eqtr4d 2659	. . 3 |- ( -. G e. _V -> (.g ` G ) = ( 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 ) ) ) ) ) )
53	33, 52	pm2.61i 176	. 2 |- (.g ` G ) = ( 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 ) ) ) ) )
54	1, 53	eqtri 2644	1 |- .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 ) ) ) ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   (/)c0 3915   ifcif 4086   {csn 4177   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   ` cfv 5888   |-> cmpt2 6652   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:  mulgval  17543  mulgfn  17544  mulgpropd  17584