Theorem mgpplusg 14205

Description: Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)

Hypotheses

Ref	Expression
mgpval.1	|- M = (mulGrp ` R )
mgpval.2	|- .x. = ( .r ` R )

Assertion

Ref	Expression
mgpplusg	|- .x. = ( +g ` M )

Proof of Theorem mgpplusg

Dummy variables x j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-slot 13339	. . . . . . . . 9 |- Slot ( .r ` ndx ) = ( x e. _V |-> ( x ` ( .r ` ndx ) ) )
2	1	funmpt2 5414	. . . . . . . 8 |- Fun Slot ( .r ` ndx )
3		mulridx 13468	. . . . . . . . 9 |- .r = Slot ( .r ` ndx )
4	3	funeqi 5396	. . . . . . . 8 |- ( Fun .r <-> Fun Slot ( .r ` ndx ) )
5	2, 4	mpbir 146	. . . . . . 7 |- Fun .r
6		funrel 5392	. . . . . . 7 |- ( Fun .r -> Rel .r )
7	5, 6	ax-mp 5	. . . . . 6 |- Rel .r
8		relelfvdm 5725	. . . . . 6 |- ( ( Rel .r /\ x e. ( .r ` R ) ) -> R e. dom .r )
9	7, 8	mpan 428	. . . . 5 |- ( x e. ( .r ` R ) -> R e. dom .r )
10	9	elexd 2835	. . . 4 |- ( x e. ( .r ` R ) -> R e. _V )
11		mgpval.2	. . . 4 |- .x. = ( .r ` R )
12	10, 11	eleq2s 2333	. . 3 |- ( x e. .x. -> R e. _V )
13		plusgslid 13449	. . . . 5 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
14	13	slotm 13398	. . . 4 |- ( x e. ( +g ` M ) -> E. j j e. M )
15		fnmgp 14202	. . . . . . . . 9 |- mulGrp Fn _V
16		fnrel 5477	. . . . . . . . 9 |- (mulGrp Fn _V -> Rel mulGrp )
17	15, 16	ax-mp 5	. . . . . . . 8 |- Rel mulGrp
18		relelfvdm 5725	. . . . . . . 8 |- ( ( Rel mulGrp /\ j e. (mulGrp ` R ) ) -> R e. dom mulGrp )
19	17, 18	mpan 428	. . . . . . 7 |- ( j e. (mulGrp ` R ) -> R e. dom mulGrp )
20	19	elexd 2835	. . . . . 6 |- ( j e. (mulGrp ` R ) -> R e. _V )
21		mgpval.1	. . . . . 6 |- M = (mulGrp ` R )
22	20, 21	eleq2s 2333	. . . . 5 |- ( j e. M -> R e. _V )
23	22	exlimiv 1651	. . . 4 |- ( E. j j e. M -> R e. _V )
24	14, 23	syl 14	. . 3 |- ( x e. ( +g ` M ) -> R e. _V )
25	21, 11	mgpplusgg 14204	. . . 4 |- ( R e. _V -> .x. = ( +g ` M ) )
26	25	eleq2d 2308	. . 3 |- ( R e. _V -> ( x e. .x. <-> x e. ( +g ` M ) ) )
27	12, 24, 26	pm5.21nii 716	. 2 |- ( x e. .x. <-> x e. ( +g ` M ) )
28	27	eqriv 2235	1 |- .x. = ( +g ` M )

Syntax hints:   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   dom cdm 4772   Rel wrel 4777   Fun wfun 5369   Fn wfn 5370   ` cfv 5375   ndxcnx 13332  Slot cslot 13334   +g cplusg 13414   .rcmulr 13415  mulGrpcmgp 14200

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-sets 13342  df-plusg 13427  df-mulr 13428  df-mgp 14201

This theorem is referenced by:  assamulgscmlem2  15025