Theorem rrextust 30052

Description: The uniformity of an extension of RR is the uniformity generated by its distance. (Contributed by Thierry Arnoux, 2-May-2018.)

Hypotheses

Ref	Expression
rrextust.b	|- B = ( Base ` R )
rrextust.d	|- D = ( ( dist ` R ) |` ( B X. B ) )

Assertion

Ref	Expression
rrextust	|- ( R e. ℝExt -> (UnifSt ` R ) = (metUnif ` D ) )

Proof of Theorem rrextust

Step	Hyp	Ref	Expression
1		rrextust.b	. . . 4 |- B = ( Base ` R )
2		rrextust.d	. . . 4 |- D = ( ( dist ` R ) |` ( B X. B ) )
3		eqid 2622	. . . 4 |- ( ZMod ` R ) = ( ZMod ` R )
4	1, 2, 3	isrrext 30044	. . 3 |- ( R e. ℝExt <-> ( ( R e. NrmRing /\ R e. DivRing ) /\ ( ( ZMod ` R ) e. NrmMod /\ (chr ` R ) = 0 ) /\ ( R e. CUnifSp /\ (UnifSt ` R ) = (metUnif ` D ) ) ) )
5	4	simp3bi 1078	. 2 |- ( R e. ℝExt -> ( R e. CUnifSp /\ (UnifSt ` R ) = (metUnif ` D ) ) )
6	5	simprd 479	1 |- ( R e. ℝExt -> (UnifSt ` R ) = (metUnif ` D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   X. cxp 5112   |` cres 5116   ` cfv 5888   0cc0 9936   Basecbs 15857   distcds 15950   DivRingcdr 18747  metUnifcmetu 19737   ZModczlm 19849  chrcchr 19850  UnifStcuss 22057  CUnifSpccusp 22101  NrmRingcnrg 22384  NrmModcnlm 22385   ℝExt crrext 30038

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896  df-rrext 30043

This theorem is referenced by:  rrhfe  30056  rrhcne  30057  sitgclg  30404