Theorem mulgfvalg 13838

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
mulgfvalg	|- ( 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 ) ) ) ) ) )

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

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

Proof of Theorem mulgfvalg

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		df-mulg 13837	. . 3 |- .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 ) ) ) ) ) )
3		eqidd 2233	. . . 4 |- ( w = G -> ZZ = ZZ )
4		fveq2 5670	. . . . 5 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
5		mulgval.b	. . . . 5 |- B = ( Base ` G )
6	4, 5	eqtr4di 2283	. . . 4 |- ( w = G -> ( Base ` w ) = B )
7		fveq2 5670	. . . . . 6 |- ( w = G -> ( 0g ` w ) = ( 0g ` G ) )
8		mulgval.o	. . . . . 6 |- .0. = ( 0g ` G )
9	7, 8	eqtr4di 2283	. . . . 5 |- ( w = G -> ( 0g ` w ) = .0. )
10		seqex 10811	. . . . . . 7 |- seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) e. _V
11	10	a1i 9	. . . . . 6 |- ( w = G -> seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) e. _V )
12		id 19	. . . . . . . . 9 |- ( s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) -> s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) )
13		fveq2 5670	. . . . . . . . . . 11 |- ( w = G -> ( +g ` w ) = ( +g ` G ) )
14		mulgval.p	. . . . . . . . . . 11 |- .+ = ( +g ` G )
15	13, 14	eqtr4di 2283	. . . . . . . . . 10 |- ( w = G -> ( +g ` w ) = .+ )
16	15	seqeq2d 10816	. . . . . . . . 9 |- ( w = G -> seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) = seq 1 ( .+ , ( NN X. { x } ) ) )
17	12, 16	sylan9eqr 2287	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> s = seq 1 ( .+ , ( NN X. { x } ) ) )
18	17	fveq1d 5672	. . . . . . 7 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( s ` n ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) )
19		simpl 109	. . . . . . . . . 10 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> w = G )
20	19	fveq2d 5674	. . . . . . . . 9 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( invg ` w ) = ( invg ` G ) )
21		mulgval.i	. . . . . . . . 9 |- I = ( invg ` G )
22	20, 21	eqtr4di 2283	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( invg ` w ) = I )
23	17	fveq1d 5672	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( s ` -u n ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) )
24	22, 23	fveq12d 5677	. . . . . . 7 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( ( invg ` w ) ` ( s ` -u n ) ) = ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) )
25	18, 24	ifeq12d 3642	. . . . . 6 |- ( ( 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 ) ) ) )
26	11, 25	csbied 3185	. . . . 5 |- ( 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 ) ) ) )
27	9, 26	ifeq12d 3642	. . . 4 |- ( 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 ) ) ) ) )
28	3, 6, 27	mpoeq123dv 6115	. . 3 |- ( 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 ) ) ) ) ) )
29		elex 2825	. . 3 |- ( G e. V -> G e. _V )
30		zex 9586	. . . 4 |- ZZ e. _V
31		basfn 13271	. . . . . 6 |- Base Fn _V
32		funfvex 5687	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
33	32	funfni 5458	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
34	31, 29, 33	sylancr 414	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
35	5, 34	eqeltrid 2319	. . . 4 |- ( G e. V -> B e. _V )
36		mpoexga 6408	. . . 4 |- ( ( ZZ e. _V /\ B 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 ) ) ) ) ) e. _V )
37	30, 35, 36	sylancr 414	. . 3 |- ( 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 ) ) ) ) ) e. _V )
38	2, 28, 29, 37	fvmptd3 5771	. 2 |- ( 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 ) ) ) ) ) )
39	1, 38	eqtrid 2277	1 |- ( 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 ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   _Vcvv 2813   [_csb 3138   ifcif 3620   {csn 3689   class class class wbr 4109   X. cxp 4747   Fn wfn 5347   ` cfv 5352   e. cmpo 6052   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-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-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  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-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  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-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-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-neg 8447  df-inn 9238  df-z 9578  df-seqfrec 10810  df-ndx 13215  df-slot 13216  df-base 13218  df-mulg 13837

This theorem is referenced by:  mulgval  13839  mulgex  13840  mulgfng  13841  mulgpropdg  13881