Theorem mulgfvalg 13060

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 13059	. . 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 2190	. . . 4 |- ( w = G -> ZZ = ZZ )
4		fveq2 5534	. . . . 5 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
5		mulgval.b	. . . . 5 |- B = ( Base ` G )
6	4, 5	eqtr4di 2240	. . . 4 |- ( w = G -> ( Base ` w ) = B )
7		fveq2 5534	. . . . . 6 |- ( w = G -> ( 0g ` w ) = ( 0g ` G ) )
8		mulgval.o	. . . . . 6 |- .0. = ( 0g ` G )
9	7, 8	eqtr4di 2240	. . . . 5 |- ( w = G -> ( 0g ` w ) = .0. )
10		seqex 10477	. . . . . . 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 5534	. . . . . . . . . . 11 |- ( w = G -> ( +g ` w ) = ( +g ` G ) )
14		mulgval.p	. . . . . . . . . . 11 |- .+ = ( +g ` G )
15	13, 14	eqtr4di 2240	. . . . . . . . . 10 |- ( w = G -> ( +g ` w ) = .+ )
16	15	seqeq2d 10482	. . . . . . . . 9 |- ( w = G -> seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) = seq 1 ( .+ , ( NN X. { x } ) ) )
17	12, 16	sylan9eqr 2244	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> s = seq 1 ( .+ , ( NN X. { x } ) ) )
18	17	fveq1d 5536	. . . . . . 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 5538	. . . . . . . . 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 2240	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( invg ` w ) = I )
23	17	fveq1d 5536	. . . . . . . 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 5541	. . . . . . 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 3568	. . . . . 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 3118	. . . . 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 3568	. . . 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 5957	. . 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 2763	. . 3 |- ( G e. V -> G e. _V )
30		zex 9291	. . . 4 |- ZZ e. _V
31		basfn 12569	. . . . . 6 |- Base Fn _V
32		funfvex 5551	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
33	32	funfni 5335	. . . . . 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 2276	. . . 4 |- ( G e. V -> B e. _V )
36		mpoexga 6236	. . . 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 5629	. 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 2234	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 1364   e. wcel 2160   _Vcvv 2752   [_csb 3072   ifcif 3549   {csn 3607   class class class wbr 4018   X. cxp 4642   Fn wfn 5230   ` cfv 5235   e. cmpo 5897   0cc0 7840   1c1 7841   < clt 8021   -ucneg 8158   NNcn 8948   ZZcz 9282   seqcseq 10475   Basecbs 12511   +g cplusg 12586   0gc0g 12758   invgcminusg 12943  ._gcmg 13058

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-frec 6415  df-neg 8160  df-inn 8949  df-z 9283  df-seqfrec 10476  df-ndx 12514  df-slot 12515  df-base 12517  df-mulg 13059

This theorem is referenced by:  mulgval  13061  mulgex  13062  mulgfng  13063  mulgpropdg  13101