Theorem pwsmgp 18618

Description: The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.)

Hypotheses

Ref	Expression
pwsmgp.y	|- Y = ( R ^s I )
pwsmgp.m	|- M = (mulGrp ` R )
pwsmgp.z	|- Z = ( M ^s I )
pwsmgp.n	|- N = (mulGrp ` Y )
pwsmgp.b	|- B = ( Base ` N )
pwsmgp.c	|- C = ( Base ` Z )
pwsmgp.p	|- .+ = ( +g ` N )
pwsmgp.q	|- .+b = ( +g ` Z )

Assertion

Ref	Expression
pwsmgp	|- ( ( R e. V /\ I e. W ) -> ( B = C /\ .+ = .+b ) )

Proof of Theorem pwsmgp

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- ( (Scalar ` R ) X_s ( I X. { R } ) ) = ( (Scalar ` R ) X_s ( I X. { R } ) )
2		eqid 2622	. . . . . 6 |- (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
3		eqid 2622	. . . . . 6 |- ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) = ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) )
4		simpr 477	. . . . . 6 |- ( ( R e. V /\ I e. W ) -> I e. W )
5		fvexd 6203	. . . . . 6 |- ( ( R e. V /\ I e. W ) -> (Scalar ` R ) e. _V )
6		fnconstg 6093	. . . . . . 7 |- ( R e. V -> ( I X. { R } ) Fn I )
7	6	adantr 481	. . . . . 6 |- ( ( R e. V /\ I e. W ) -> ( I X. { R } ) Fn I )
8	1, 2, 3, 4, 5, 7	prdsmgp 18610	. . . . 5 |- ( ( R e. V /\ I e. W ) -> ( ( Base ` (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ) = ( Base ` ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) ) /\ ( +g ` (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ) = ( +g ` ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) ) ) )
9	8	simpld 475	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( Base ` (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ) = ( Base ` ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) ) )
10		pwsmgp.n	. . . . . 6 |- N = (mulGrp ` Y )
11		pwsmgp.y	. . . . . . . 8 |- Y = ( R ^s I )
12		eqid 2622	. . . . . . . 8 |- (Scalar ` R ) = (Scalar ` R )
13	11, 12	pwsval 16146	. . . . . . 7 |- ( ( R e. V /\ I e. W ) -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
14	13	fveq2d 6195	. . . . . 6 |- ( ( R e. V /\ I e. W ) -> (mulGrp ` Y ) = (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
15	10, 14	syl5eq 2668	. . . . 5 |- ( ( R e. V /\ I e. W ) -> N = (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
16	15	fveq2d 6195	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( Base ` N ) = ( Base ` (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ) )
17		pwsmgp.z	. . . . . 6 |- Z = ( M ^s I )
18		pwsmgp.m	. . . . . . . . 9 |- M = (mulGrp ` R )
19		fvex 6201	. . . . . . . . 9 |- (mulGrp ` R ) e. _V
20	18, 19	eqeltri 2697	. . . . . . . 8 |- M e. _V
21		eqid 2622	. . . . . . . . 9 |- ( M ^s I ) = ( M ^s I )
22		eqid 2622	. . . . . . . . 9 |- (Scalar ` M ) = (Scalar ` M )
23	21, 22	pwsval 16146	. . . . . . . 8 |- ( ( M e. _V /\ I e. W ) -> ( M ^s I ) = ( (Scalar ` M ) X_s ( I X. { M } ) ) )
24	20, 4, 23	sylancr 695	. . . . . . 7 |- ( ( R e. V /\ I e. W ) -> ( M ^s I ) = ( (Scalar ` M ) X_s ( I X. { M } ) ) )
25	18, 12	mgpsca 18496	. . . . . . . . . 10 |- (Scalar ` R ) = (Scalar ` M )
26	25	eqcomi 2631	. . . . . . . . 9 |- (Scalar ` M ) = (Scalar ` R )
27	26	a1i 11	. . . . . . . 8 |- ( ( R e. V /\ I e. W ) -> (Scalar ` M ) = (Scalar ` R ) )
28		fnmgp 18491	. . . . . . . . . 10 |- mulGrp Fn _V
29		elex 3212	. . . . . . . . . . 11 |- ( R e. V -> R e. _V )
30	29	adantr 481	. . . . . . . . . 10 |- ( ( R e. V /\ I e. W ) -> R e. _V )
31		fcoconst 6401	. . . . . . . . . 10 |- ( (mulGrp Fn _V /\ R e. _V ) -> (mulGrp o. ( I X. { R } ) ) = ( I X. { (mulGrp ` R ) } ) )
32	28, 30, 31	sylancr 695	. . . . . . . . 9 |- ( ( R e. V /\ I e. W ) -> (mulGrp o. ( I X. { R } ) ) = ( I X. { (mulGrp ` R ) } ) )
33	18	sneqi 4188	. . . . . . . . . 10 |- { M } = { (mulGrp ` R ) }
34	33	xpeq2i 5136	. . . . . . . . 9 |- ( I X. { M } ) = ( I X. { (mulGrp ` R ) } )
35	32, 34	syl6reqr 2675	. . . . . . . 8 |- ( ( R e. V /\ I e. W ) -> ( I X. { M } ) = (mulGrp o. ( I X. { R } ) ) )
36	27, 35	oveq12d 6668	. . . . . . 7 |- ( ( R e. V /\ I e. W ) -> ( (Scalar ` M ) X_s ( I X. { M } ) ) = ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) )
37	24, 36	eqtrd 2656	. . . . . 6 |- ( ( R e. V /\ I e. W ) -> ( M ^s I ) = ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) )
38	17, 37	syl5eq 2668	. . . . 5 |- ( ( R e. V /\ I e. W ) -> Z = ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) )
39	38	fveq2d 6195	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( Base ` Z ) = ( Base ` ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) ) )
40	9, 16, 39	3eqtr4d 2666	. . 3 |- ( ( R e. V /\ I e. W ) -> ( Base ` N ) = ( Base ` Z ) )
41		pwsmgp.b	. . 3 |- B = ( Base ` N )
42		pwsmgp.c	. . 3 |- C = ( Base ` Z )
43	40, 41, 42	3eqtr4g 2681	. 2 |- ( ( R e. V /\ I e. W ) -> B = C )
44	8	simprd 479	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( +g ` (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ) = ( +g ` ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) ) )
45	15	fveq2d 6195	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( +g ` N ) = ( +g ` (mulGrp ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ) )
46	38	fveq2d 6195	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( +g ` Z ) = ( +g ` ( (Scalar ` R ) X_s (mulGrp o. ( I X. { R } ) ) ) ) )
47	44, 45, 46	3eqtr4d 2666	. . 3 |- ( ( R e. V /\ I e. W ) -> ( +g ` N ) = ( +g ` Z ) )
48		pwsmgp.p	. . 3 |- .+ = ( +g ` N )
49		pwsmgp.q	. . 3 |- .+b = ( +g ` Z )
50	47, 48, 49	3eqtr4g 2681	. 2 |- ( ( R e. V /\ I e. W ) -> .+ = .+b )
51	43, 50	jca 554	1 |- ( ( R e. V /\ I e. W ) -> ( B = C /\ .+ = .+b ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   o. ccom 5118   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   X__scprds 16106   ^_s cpws 16107  mulGrpcmgp 18489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108  df-pws 16110  df-mgp 18490

This theorem is referenced by:  pwsco1rhm  18738  pwsco2rhm  18739  pwsdiagrhm  18813  evl1expd  19709