Theorem mulgfvalg 13532

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 13531	. . 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 2207	. . . 4 |- ( w = G -> ZZ = ZZ )
4		fveq2 5589	. . . . 5 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
5		mulgval.b	. . . . 5 |- B = ( Base ` G )
6	4, 5	eqtr4di 2257	. . . 4 |- ( w = G -> ( Base ` w ) = B )
7		fveq2 5589	. . . . . 6 |- ( w = G -> ( 0g ` w ) = ( 0g ` G ) )
8		mulgval.o	. . . . . 6 |- .0. = ( 0g ` G )
9	7, 8	eqtr4di 2257	. . . . 5 |- ( w = G -> ( 0g ` w ) = .0. )
10		seqex 10616	. . . . . . 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 5589	. . . . . . . . . . 11 |- ( w = G -> ( +g ` w ) = ( +g ` G ) )
14		mulgval.p	. . . . . . . . . . 11 |- .+ = ( +g ` G )
15	13, 14	eqtr4di 2257	. . . . . . . . . 10 |- ( w = G -> ( +g ` w ) = .+ )
16	15	seqeq2d 10621	. . . . . . . . 9 |- ( w = G -> seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) = seq 1 ( .+ , ( NN X. { x } ) ) )
17	12, 16	sylan9eqr 2261	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> s = seq 1 ( .+ , ( NN X. { x } ) ) )
18	17	fveq1d 5591	. . . . . . 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 5593	. . . . . . . . 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 2257	. . . . . . . 8 |- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( invg ` w ) = I )
23	17	fveq1d 5591	. . . . . . . 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 5596	. . . . . . 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 3595	. . . . . 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 3144	. . . . 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 3595	. . . 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 6020	. . 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 2785	. . 3 |- ( G e. V -> G e. _V )
30		zex 9401	. . . 4 |- ZZ e. _V
31		basfn 12965	. . . . . 6 |- Base Fn _V
32		funfvex 5606	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
33	32	funfni 5385	. . . . . 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 2293	. . . 4 |- ( G e. V -> B e. _V )
36		mpoexga 6311	. . . 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 5686	. 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 2251	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 2177   _Vcvv 2773   [_csb 3097   ifcif 3575   {csn 3638   class class class wbr 4051   X. cxp 4681   Fn wfn 5275   ` cfv 5280   e. cmpo 5959   0cc0 7945   1c1 7946   < clt 8127   -ucneg 8264   NNcn 9056   ZZcz 9392   seqcseq 10614   Basecbs 12907   +g cplusg 12984   0gc0g 13163   invgcminusg 13408  ._gcmg 13530

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-neg 8266  df-inn 9057  df-z 9393  df-seqfrec 10615  df-ndx 12910  df-slot 12911  df-base 12913  df-mulg 13531

This theorem is referenced by:  mulgval  13533  mulgex  13534  mulgfng  13535  mulgpropdg  13575