Theorem mulgfvalg 13490

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 13489	. . 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 2206	. . . 4 |- ( w = G -> ZZ = ZZ )
4		fveq2 5578	. . . . 5 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
5		mulgval.b	. . . . 5 |- B = ( Base ` G )
6	4, 5	eqtr4di 2256	. . . 4 |- ( w = G -> ( Base ` w ) = B )
7		fveq2 5578	. . . . . 6 |- ( w = G -> ( 0g ` w ) = ( 0g ` G ) )
8		mulgval.o	. . . . . 6 |- .0. = ( 0g ` G )
9	7, 8	eqtr4di 2256	. . . . 5 |- ( w = G -> ( 0g ` w ) = .0. )
10		seqex 10596	. . . . . . 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 5578	. . . . . . . . . . 11 |- ( w = G -> ( +g ` w ) = ( +g ` G ) )
14		mulgval.p	. . . . . . . . . . 11 |- .+ = ( +g ` G )
15	13, 14	eqtr4di 2256	. . . . . . . . . 10 |- ( w = G -> ( +g ` w ) = .+ )
16	15	seqeq2d 10601	. . . . . . . . 9 |- ( w = G -> seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) = seq 1 ( .+ , ( NN X. { x } ) ) )
17	12, 16	sylan9eqr 2260	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> s = seq 1 ( .+ , ( NN X. { x } ) ) )
18	17	fveq1d 5580	. . . . . . 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 5582	. . . . . . . . 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 2256	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( invg ` w ) = I )
23	17	fveq1d 5580	. . . . . . . 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 5585	. . . . . . 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 3590	. . . . . 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 3140	. . . . 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 3590	. . . 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 6009	. . 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 2783	. . 3 |- ( G e. V -> G e. _V )
30		zex 9383	. . . 4 |- ZZ e. _V
31		basfn 12923	. . . . . 6 |- Base Fn _V
32		funfvex 5595	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
33	32	funfni 5377	. . . . . 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 2292	. . . 4 |- ( G e. V -> B e. _V )
36		mpoexga 6300	. . . 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 5675	. 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 2250	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 1373   e. wcel 2176   _Vcvv 2772   [_csb 3093   ifcif 3571   {csn 3633   class class class wbr 4045   X. cxp 4674   Fn wfn 5267   ` cfv 5272   e. cmpo 5948   0cc0 7927   1c1 7928   < clt 8109   -ucneg 8246   NNcn 9038   ZZcz 9374   seqcseq 10594   Basecbs 12865   +g cplusg 12942   0gc0g 13121   invgcminusg 13366  ._gcmg 13488

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-neg 8248  df-inn 9039  df-z 9375  df-seqfrec 10595  df-ndx 12868  df-slot 12869  df-base 12871  df-mulg 13489

This theorem is referenced by:  mulgval  13491  mulgex  13492  mulgfng  13493  mulgpropdg  13533