Theorem mgpscag 13885

Description: The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)

Hypotheses

Ref	Expression
mgpbas.1	|- M = (mulGrp ` R )
mgpsca.s	|- S = (Scalar ` R )

Assertion

Ref	Expression
mgpscag	|- ( R e. V -> S = (Scalar ` M ) )

Proof of Theorem mgpscag

Step	Hyp	Ref	Expression
1		mgpsca.s	. 2 |- S = (Scalar ` R )
2		mulrslid 13160	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
3	2	slotex 13054	. . . 4 |- ( R e. V -> ( .r ` R ) e. _V )
4		scaslid 13181	. . . . 5 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
5		scandxnplusgndx 13183	. . . . 5 |- (Scalar ` ndx ) =/= ( +g ` ndx )
6		plusgslid 13140	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
7	6	simpri 113	. . . . 5 |- ( +g ` ndx ) e. NN
8	4, 5, 7	setsslnid 13079	. . . 4 |- ( ( R e. V /\ ( .r ` R ) e. _V ) -> (Scalar ` R ) = (Scalar ` ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) ) )
9	3, 8	mpdan 421	. . 3 |- ( R e. V -> (Scalar ` R ) = (Scalar ` ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) ) )
10		mgpbas.1	. . . . 5 |- M = (mulGrp ` R )
11		eqid 2229	. . . . 5 |- ( .r ` R ) = ( .r ` R )
12	10, 11	mgpvalg 13881	. . . 4 |- ( R e. V -> M = ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) )
13	12	fveq2d 5630	. . 3 |- ( R e. V -> (Scalar ` M ) = (Scalar ` ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) ) )
14	9, 13	eqtr4d 2265	. 2 |- ( R e. V -> (Scalar ` R ) = (Scalar ` M ) )
15	1, 14	eqtrid 2274	1 |- ( R e. V -> S = (Scalar ` M ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   <.cop 3669   ` cfv 5317  (class class class)co 6000   NNcn 9106   ndxcnx 13024   sSet csts 13025  Slot cslot 13026   +g cplusg 13105   .rcmulr 13106  Scalarcsca 13108  mulGrpcmgp 13878

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-ndx 13030  df-slot 13031  df-sets 13034  df-plusg 13118  df-mulr 13119  df-sca 13121  df-mgp 13879

This theorem is referenced by: (None)