Theorem rrxds 23181

Description: The distance over generalized Euclidean spaces. Compare with df-rrn 33625. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)

Hypotheses

Ref	Expression
rrxval.r	|- H = (ℝ^ ` I )
rrxbase.b	|- B = ( Base ` H )

Assertion

Ref	Expression
rrxds	|- ( I e. V -> ( f e. B , g e. B |-> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( ( f ` x ) - ( g ` x ) ) ^ 2 ) ) ) ) ) = ( dist ` H ) )

Distinct variable groups:   f, g, x, B   f, I, g, x   f, V, g, x

Allowed substitution hints:   H( x, f, g)

Proof of Theorem rrxds

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxval.r	. . . 4 |- H = (ℝ^ ` I )
2	1	rrxval 23175	. . 3 |- ( I e. V -> H = (toCHil ` (RRfld freeLMod I ) ) )
3	2	fveq2d 6195	. 2 |- ( I e. V -> ( dist ` H ) = ( dist ` (toCHil ` (RRfld freeLMod I ) ) ) )
4		recrng 19967	. . . . 5 |- RRfld e. *Ring
5		srngring 18852	. . . . 5 |- (RRfld e. *Ring -> RRfld e. Ring )
6	4, 5	ax-mp 5	. . . 4 |- RRfld e. Ring
7		eqid 2622	. . . . 5 |- (RRfld freeLMod I ) = (RRfld freeLMod I )
8	7	frlmlmod 20093	. . . 4 |- ( (RRfld e. Ring /\ I e. V ) -> (RRfld freeLMod I ) e. LMod )
9	6, 8	mpan 706	. . 3 |- ( I e. V -> (RRfld freeLMod I ) e. LMod )
10		lmodgrp 18870	. . 3 |- ( (RRfld freeLMod I ) e. LMod -> (RRfld freeLMod I ) e. Grp )
11		eqid 2622	. . . 4 |- (toCHil ` (RRfld freeLMod I ) ) = (toCHil ` (RRfld freeLMod I ) )
12		eqid 2622	. . . 4 |- ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) = ( norm ` (toCHil ` (RRfld freeLMod I ) ) )
13		eqid 2622	. . . 4 |- ( -g ` (RRfld freeLMod I ) ) = ( -g ` (RRfld freeLMod I ) )
14	11, 12, 13	tchds 23030	. . 3 |- ( (RRfld freeLMod I ) e. Grp -> ( ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) o. ( -g ` (RRfld freeLMod I ) ) ) = ( dist ` (toCHil ` (RRfld freeLMod I ) ) ) )
15	9, 10, 14	3syl 18	. 2 |- ( I e. V -> ( ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) o. ( -g ` (RRfld freeLMod I ) ) ) = ( dist ` (toCHil ` (RRfld freeLMod I ) ) ) )
16		eqid 2622	. . . . . . . 8 |- ( Base ` (RRfld freeLMod I ) ) = ( Base ` (RRfld freeLMod I ) )
17	16, 13	grpsubf 17494	. . . . . . 7 |- ( (RRfld freeLMod I ) e. Grp -> ( -g ` (RRfld freeLMod I ) ) : ( ( Base ` (RRfld freeLMod I ) ) X. ( Base ` (RRfld freeLMod I ) ) ) --> ( Base ` (RRfld freeLMod I ) ) )
18	9, 10, 17	3syl 18	. . . . . 6 |- ( I e. V -> ( -g ` (RRfld freeLMod I ) ) : ( ( Base ` (RRfld freeLMod I ) ) X. ( Base ` (RRfld freeLMod I ) ) ) --> ( Base ` (RRfld freeLMod I ) ) )
19		rrxbase.b	. . . . . . . . . 10 |- B = ( Base ` H )
20	1, 19	rrxbase 23176	. . . . . . . . 9 |- ( I e. V -> B = { h e. ( RR ^m I ) | h finSupp 0 } )
21		rebase 19952	. . . . . . . . . . 11 |- RR = ( Base ` RRfld )
22		re0g 19958	. . . . . . . . . . 11 |- 0 = ( 0g ` RRfld )
23		eqid 2622	. . . . . . . . . . 11 |- { h e. ( RR ^m I ) | h finSupp 0 } = { h e. ( RR ^m I ) | h finSupp 0 }
24	7, 21, 22, 23	frlmbas 20099	. . . . . . . . . 10 |- ( (RRfld e. Ring /\ I e. V ) -> { h e. ( RR ^m I ) | h finSupp 0 } = ( Base ` (RRfld freeLMod I ) ) )
25	6, 24	mpan 706	. . . . . . . . 9 |- ( I e. V -> { h e. ( RR ^m I ) | h finSupp 0 } = ( Base ` (RRfld freeLMod I ) ) )
26	20, 25	eqtrd 2656	. . . . . . . 8 |- ( I e. V -> B = ( Base ` (RRfld freeLMod I ) ) )
27	26	sqxpeqd 5141	. . . . . . 7 |- ( I e. V -> ( B X. B ) = ( ( Base ` (RRfld freeLMod I ) ) X. ( Base ` (RRfld freeLMod I ) ) ) )
28	27, 26	feq23d 6040	. . . . . 6 |- ( I e. V -> ( ( -g ` (RRfld freeLMod I ) ) : ( B X. B ) --> B <-> ( -g ` (RRfld freeLMod I ) ) : ( ( Base ` (RRfld freeLMod I ) ) X. ( Base ` (RRfld freeLMod I ) ) ) --> ( Base ` (RRfld freeLMod I ) ) ) )
29	18, 28	mpbird 247	. . . . 5 |- ( I e. V -> ( -g ` (RRfld freeLMod I ) ) : ( B X. B ) --> B )
30	29	fovrnda 6805	. . . 4 |- ( ( I e. V /\ ( f e. B /\ g e. B ) ) -> ( f ( -g ` (RRfld freeLMod I ) ) g ) e. B )
31		ffn 6045	. . . . . 6 |- ( ( -g ` (RRfld freeLMod I ) ) : ( B X. B ) --> B -> ( -g ` (RRfld freeLMod I ) ) Fn ( B X. B ) )
32	29, 31	syl 17	. . . . 5 |- ( I e. V -> ( -g ` (RRfld freeLMod I ) ) Fn ( B X. B ) )
33		fnov 6768	. . . . 5 |- ( ( -g ` (RRfld freeLMod I ) ) Fn ( B X. B ) <-> ( -g ` (RRfld freeLMod I ) ) = ( f e. B , g e. B |-> ( f ( -g ` (RRfld freeLMod I ) ) g ) ) )
34	32, 33	sylib 208	. . . 4 |- ( I e. V -> ( -g ` (RRfld freeLMod I ) ) = ( f e. B , g e. B |-> ( f ( -g ` (RRfld freeLMod I ) ) g ) ) )
35	1, 19	rrxnm 23179	. . . . 5 |- ( I e. V -> ( h e. B |-> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( h ` x ) ^ 2 ) ) ) ) ) = ( norm ` H ) )
36	2	fveq2d 6195	. . . . 5 |- ( I e. V -> ( norm ` H ) = ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) )
37	35, 36	eqtr2d 2657	. . . 4 |- ( I e. V -> ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) = ( h e. B |-> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( h ` x ) ^ 2 ) ) ) ) ) )
38		fveq1 6190	. . . . . . . 8 |- ( h = ( f ( -g ` (RRfld freeLMod I ) ) g ) -> ( h ` x ) = ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) )
39	38	oveq1d 6665	. . . . . . 7 |- ( h = ( f ( -g ` (RRfld freeLMod I ) ) g ) -> ( ( h ` x ) ^ 2 ) = ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) )
40	39	mpteq2dv 4745	. . . . . 6 |- ( h = ( f ( -g ` (RRfld freeLMod I ) ) g ) -> ( x e. I |-> ( ( h ` x ) ^ 2 ) ) = ( x e. I |-> ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) ) )
41	40	oveq2d 6666	. . . . 5 |- ( h = ( f ( -g ` (RRfld freeLMod I ) ) g ) -> (RRfld gsumg ( x e. I |-> ( ( h ` x ) ^ 2 ) ) ) = (RRfld gsumg ( x e. I |-> ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) ) ) )
42	41	fveq2d 6195	. . . 4 |- ( h = ( f ( -g ` (RRfld freeLMod I ) ) g ) -> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( h ` x ) ^ 2 ) ) ) ) = ( sqr ` (RRfld gsumg ( x e. I |-> ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) ) ) ) )
43	30, 34, 37, 42	fmpt2co 7260	. . 3 |- ( I e. V -> ( ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) o. ( -g ` (RRfld freeLMod I ) ) ) = ( f e. B , g e. B |-> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) ) ) ) ) )
44		simp1 1061	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ f e. B /\ g e. B ) -> I e. V )
45		simprl 794	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ ( f e. B /\ g e. B ) ) -> f e. B )
46	26	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ ( f e. B /\ g e. B ) ) -> B = ( Base ` (RRfld freeLMod I ) ) )
47	45, 46	eleqtrd 2703	. . . . . . . . . . . . . . 15 |- ( ( I e. V /\ ( f e. B /\ g e. B ) ) -> f e. ( Base ` (RRfld freeLMod I ) ) )
48	47	3impb 1260	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ f e. B /\ g e. B ) -> f e. ( Base ` (RRfld freeLMod I ) ) )
49	7, 21, 16	frlmbasmap 20103	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ f e. ( Base ` (RRfld freeLMod I ) ) ) -> f e. ( RR ^m I ) )
50	44, 48, 49	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( I e. V /\ f e. B /\ g e. B ) -> f e. ( RR ^m I ) )
51		elmapi 7879	. . . . . . . . . . . . 13 |- ( f e. ( RR ^m I ) -> f : I --> RR )
52	50, 51	syl 17	. . . . . . . . . . . 12 |- ( ( I e. V /\ f e. B /\ g e. B ) -> f : I --> RR )
53		ffn 6045	. . . . . . . . . . . 12 |- ( f : I --> RR -> f Fn I )
54	52, 53	syl 17	. . . . . . . . . . 11 |- ( ( I e. V /\ f e. B /\ g e. B ) -> f Fn I )
55		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( I e. V /\ ( f e. B /\ g e. B ) ) -> g e. B )
56	55, 46	eleqtrd 2703	. . . . . . . . . . . . . . 15 |- ( ( I e. V /\ ( f e. B /\ g e. B ) ) -> g e. ( Base ` (RRfld freeLMod I ) ) )
57	56	3impb 1260	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ f e. B /\ g e. B ) -> g e. ( Base ` (RRfld freeLMod I ) ) )
58	7, 21, 16	frlmbasmap 20103	. . . . . . . . . . . . . 14 |- ( ( I e. V /\ g e. ( Base ` (RRfld freeLMod I ) ) ) -> g e. ( RR ^m I ) )
59	44, 57, 58	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( I e. V /\ f e. B /\ g e. B ) -> g e. ( RR ^m I ) )
60		elmapi 7879	. . . . . . . . . . . . 13 |- ( g e. ( RR ^m I ) -> g : I --> RR )
61	59, 60	syl 17	. . . . . . . . . . . 12 |- ( ( I e. V /\ f e. B /\ g e. B ) -> g : I --> RR )
62		ffn 6045	. . . . . . . . . . . 12 |- ( g : I --> RR -> g Fn I )
63	61, 62	syl 17	. . . . . . . . . . 11 |- ( ( I e. V /\ f e. B /\ g e. B ) -> g Fn I )
64		inidm 3822	. . . . . . . . . . 11 |- ( I i^i I ) = I
65		eqidd 2623	. . . . . . . . . . 11 |- ( ( ( I e. V /\ f e. B /\ g e. B ) /\ x e. I ) -> ( f ` x ) = ( f ` x ) )
66		eqidd 2623	. . . . . . . . . . 11 |- ( ( ( I e. V /\ f e. B /\ g e. B ) /\ x e. I ) -> ( g ` x ) = ( g ` x ) )
67	54, 63, 44, 44, 64, 65, 66	offval 6904	. . . . . . . . . 10 |- ( ( I e. V /\ f e. B /\ g e. B ) -> ( f oF ( -g ` RRfld ) g ) = ( x e. I |-> ( ( f ` x ) ( -g ` RRfld ) ( g ` x ) ) ) )
68	6	a1i 11	. . . . . . . . . . . 12 |- ( ( I e. V /\ ( f e. B /\ g e. B ) ) -> RRfld e. Ring )
69		simpl 473	. . . . . . . . . . . 12 |- ( ( I e. V /\ ( f e. B /\ g e. B ) ) -> I e. V )
70		eqid 2622	. . . . . . . . . . . 12 |- ( -g ` RRfld ) = ( -g ` RRfld )
71	7, 16, 68, 69, 47, 56, 70, 13	frlmsubgval 20108	. . . . . . . . . . 11 |- ( ( I e. V /\ ( f e. B /\ g e. B ) ) -> ( f ( -g ` (RRfld freeLMod I ) ) g ) = ( f oF ( -g ` RRfld ) g ) )
72	71	3impb 1260	. . . . . . . . . 10 |- ( ( I e. V /\ f e. B /\ g e. B ) -> ( f ( -g ` (RRfld freeLMod I ) ) g ) = ( f oF ( -g ` RRfld ) g ) )
73	52	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ f e. B /\ g e. B ) /\ x e. I ) -> ( f ` x ) e. RR )
74	61	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ( I e. V /\ f e. B /\ g e. B ) /\ x e. I ) -> ( g ` x ) e. RR )
75	70	resubgval 19955	. . . . . . . . . . . 12 |- ( ( ( f ` x ) e. RR /\ ( g ` x ) e. RR ) -> ( ( f ` x ) - ( g ` x ) ) = ( ( f ` x ) ( -g ` RRfld ) ( g ` x ) ) )
76	73, 74, 75	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( I e. V /\ f e. B /\ g e. B ) /\ x e. I ) -> ( ( f ` x ) - ( g ` x ) ) = ( ( f ` x ) ( -g ` RRfld ) ( g ` x ) ) )
77	76	mpteq2dva 4744	. . . . . . . . . 10 |- ( ( I e. V /\ f e. B /\ g e. B ) -> ( x e. I |-> ( ( f ` x ) - ( g ` x ) ) ) = ( x e. I |-> ( ( f ` x ) ( -g ` RRfld ) ( g ` x ) ) ) )
78	67, 72, 77	3eqtr4d 2666	. . . . . . . . 9 |- ( ( I e. V /\ f e. B /\ g e. B ) -> ( f ( -g ` (RRfld freeLMod I ) ) g ) = ( x e. I |-> ( ( f ` x ) - ( g ` x ) ) ) )
79	73, 74	resubcld 10458	. . . . . . . . 9 |- ( ( ( I e. V /\ f e. B /\ g e. B ) /\ x e. I ) -> ( ( f ` x ) - ( g ` x ) ) e. RR )
80	78, 79	fvmpt2d 6293	. . . . . . . 8 |- ( ( ( I e. V /\ f e. B /\ g e. B ) /\ x e. I ) -> ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) = ( ( f ` x ) - ( g ` x ) ) )
81	80	oveq1d 6665	. . . . . . 7 |- ( ( ( I e. V /\ f e. B /\ g e. B ) /\ x e. I ) -> ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) = ( ( ( f ` x ) - ( g ` x ) ) ^ 2 ) )
82	81	mpteq2dva 4744	. . . . . 6 |- ( ( I e. V /\ f e. B /\ g e. B ) -> ( x e. I |-> ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) ) = ( x e. I |-> ( ( ( f ` x ) - ( g ` x ) ) ^ 2 ) ) )
83	82	oveq2d 6666	. . . . 5 |- ( ( I e. V /\ f e. B /\ g e. B ) -> (RRfld gsumg ( x e. I |-> ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) ) ) = (RRfld gsumg ( x e. I |-> ( ( ( f ` x ) - ( g ` x ) ) ^ 2 ) ) ) )
84	83	fveq2d 6195	. . . 4 |- ( ( I e. V /\ f e. B /\ g e. B ) -> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) ) ) ) = ( sqr ` (RRfld gsumg ( x e. I |-> ( ( ( f ` x ) - ( g ` x ) ) ^ 2 ) ) ) ) )
85	84	mpt2eq3dva 6719	. . 3 |- ( I e. V -> ( f e. B , g e. B |-> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( ( f ( -g ` (RRfld freeLMod I ) ) g ) ` x ) ^ 2 ) ) ) ) ) = ( f e. B , g e. B |-> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( ( f ` x ) - ( g ` x ) ) ^ 2 ) ) ) ) ) )
86	43, 85	eqtrd 2656	. 2 |- ( I e. V -> ( ( norm ` (toCHil ` (RRfld freeLMod I ) ) ) o. ( -g ` (RRfld freeLMod I ) ) ) = ( f e. B , g e. B |-> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( ( f ` x ) - ( g ` x ) ) ^ 2 ) ) ) ) ) )
87	3, 15, 86	3eqtr2rd 2663	1 |- ( I e. V -> ( f e. B , g e. B |-> ( sqr ` (RRfld gsumg ( x e. I |-> ( ( ( f ` x ) - ( g ` x ) ) ^ 2 ) ) ) ) ) = ( dist ` H ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   ^m cmap 7857   finSupp cfsupp 8275   RRcr 9935   0cc0 9936   - cmin 10266   2c2 11070   ^cexp 12860   sqrcsqrt 13973   Basecbs 15857   distcds 15950   gsum_g cgsu 16101   Grpcgrp 17422   -gcsg 17424   Ringcrg 18547   *Ringcsr 18844   LModclmod 18863  RRfldcrefld 19950   freeLMod cfrlm 20090   normcnm 22381  toCHilctch 22967  ℝ^crrx 23171

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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

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-rmo 2920  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  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-fsupp 8276  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-div 10685  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-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-field 18750  df-subrg 18778  df-staf 18845  df-srng 18846  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-cnfld 19747  df-refld 19951  df-dsmm 20076  df-frlm 20091  df-nm 22387  df-tng 22389  df-tch 22969  df-rrx 23173

This theorem is referenced by:  rrxmval  23188  rrxmfval  23189  rrxtopn  40501  rrxdsfi  40505