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Theorem isrrext 27645
Description: Express the property " R is an extension of  RR". (Contributed by Thierry Arnoux, 2-May-2018.)
Hypotheses
Ref Expression
isrrext.b  |-  B  =  ( Base `  R
)
isrrext.v  |-  D  =  ( ( dist `  R
)  |`  ( B  X.  B ) )
isrrext.z  |-  Z  =  ( ZMod `  R
)
Assertion
Ref Expression
isrrext  |-  ( R  e. ℝExt 
<->  ( ( R  e. NrmRing  /\  R  e.  DivRing )  /\  ( Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  ( R  e. CUnifSp  /\  (UnifSt `  R
)  =  (metUnif `  D
) ) ) )

Proof of Theorem isrrext
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 elin 3687 . . 3  |-  ( R  e.  (NrmRing  i^i  DivRing )  <->  ( R  e. NrmRing  /\  R  e.  DivRing ) )
21anbi1i 695 . 2  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  ( ( Z  e. NrmMod  /\  (chr `  R
)  =  0 )  /\  ( R  e. CUnifSp  /\  (UnifSt `  R )  =  (metUnif `  D )
) ) )  <->  ( ( R  e. NrmRing  /\  R  e.  DivRing )  /\  ( ( Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  ( R  e. CUnifSp  /\  (UnifSt `  R
)  =  (metUnif `  D
) ) ) ) )
3 fveq2 5866 . . . . . . 7  |-  ( r  =  R  ->  ( ZMod `  r )  =  ( ZMod `  R
) )
43eleq1d 2536 . . . . . 6  |-  ( r  =  R  ->  (
( ZMod `  r
)  e. NrmMod  <->  ( ZMod `  R )  e. NrmMod )
)
5 isrrext.z . . . . . . 7  |-  Z  =  ( ZMod `  R
)
65eleq1i 2544 . . . . . 6  |-  ( Z  e. NrmMod 
<->  ( ZMod `  R
)  e. NrmMod )
74, 6syl6bbr 263 . . . . 5  |-  ( r  =  R  ->  (
( ZMod `  r
)  e. NrmMod  <->  Z  e. NrmMod ) )
8 fveq2 5866 . . . . . 6  |-  ( r  =  R  ->  (chr `  r )  =  (chr
`  R ) )
98eqeq1d 2469 . . . . 5  |-  ( r  =  R  ->  (
(chr `  r )  =  0  <->  (chr `  R
)  =  0 ) )
107, 9anbi12d 710 . . . 4  |-  ( r  =  R  ->  (
( ( ZMod `  r )  e. NrmMod  /\  (chr `  r )  =  0 )  <->  ( Z  e. NrmMod  /\  (chr `  R )  =  0 ) ) )
11 eleq1 2539 . . . . 5  |-  ( r  =  R  ->  (
r  e. CUnifSp  <->  R  e. CUnifSp ) )
12 fveq2 5866 . . . . . 6  |-  ( r  =  R  ->  (UnifSt `  r )  =  (UnifSt `  R ) )
13 fveq2 5866 . . . . . . . . 9  |-  ( r  =  R  ->  ( dist `  r )  =  ( dist `  R
) )
14 fveq2 5866 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
15 isrrext.b . . . . . . . . . . 11  |-  B  =  ( Base `  R
)
1614, 15syl6eqr 2526 . . . . . . . . . 10  |-  ( r  =  R  ->  ( Base `  r )  =  B )
1716, 16xpeq12d 5024 . . . . . . . . 9  |-  ( r  =  R  ->  (
( Base `  r )  X.  ( Base `  r
) )  =  ( B  X.  B ) )
1813, 17reseq12d 5274 . . . . . . . 8  |-  ( r  =  R  ->  (
( dist `  r )  |`  ( ( Base `  r
)  X.  ( Base `  r ) ) )  =  ( ( dist `  R )  |`  ( B  X.  B ) ) )
19 isrrext.v . . . . . . . 8  |-  D  =  ( ( dist `  R
)  |`  ( B  X.  B ) )
2018, 19syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  (
( dist `  r )  |`  ( ( Base `  r
)  X.  ( Base `  r ) ) )  =  D )
2120fveq2d 5870 . . . . . 6  |-  ( r  =  R  ->  (metUnif `  ( ( dist `  r
)  |`  ( ( Base `  r )  X.  ( Base `  r ) ) ) )  =  (metUnif `  D ) )
2212, 21eqeq12d 2489 . . . . 5  |-  ( r  =  R  ->  (
(UnifSt `  r )  =  (metUnif `  ( ( dist `  r )  |`  ( ( Base `  r
)  X.  ( Base `  r ) ) ) )  <->  (UnifSt `  R )  =  (metUnif `  D )
) )
2311, 22anbi12d 710 . . . 4  |-  ( r  =  R  ->  (
( r  e. CUnifSp  /\  (UnifSt `  r )  =  (metUnif `  ( ( dist `  r
)  |`  ( ( Base `  r )  X.  ( Base `  r ) ) ) ) )  <->  ( R  e. CUnifSp  /\  (UnifSt `  R
)  =  (metUnif `  D
) ) ) )
2410, 23anbi12d 710 . . 3  |-  ( r  =  R  ->  (
( ( ( ZMod
`  r )  e. NrmMod  /\  (chr `  r )  =  0 )  /\  ( r  e. CUnifSp  /\  (UnifSt `  r )  =  (metUnif `  ( ( dist `  r
)  |`  ( ( Base `  r )  X.  ( Base `  r ) ) ) ) ) )  <-> 
( ( Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  ( R  e. CUnifSp  /\  (UnifSt `  R )  =  (metUnif `  D ) ) ) ) )
25 df-rrext 27644 . . 3  |- ℝExt  =  {
r  e.  (NrmRing  i^i  DivRing )  |  ( (
( ZMod `  r
)  e. NrmMod  /\  (chr `  r )  =  0 )  /\  ( r  e. CUnifSp  /\  (UnifSt `  r
)  =  (metUnif `  (
( dist `  r )  |`  ( ( Base `  r
)  X.  ( Base `  r ) ) ) ) ) ) }
2624, 25elrab2 3263 . 2  |-  ( R  e. ℝExt 
<->  ( R  e.  (NrmRing  i^i 
DivRing )  /\  ( ( Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  ( R  e. CUnifSp  /\  (UnifSt `  R
)  =  (metUnif `  D
) ) ) ) )
27 3anass 977 . 2  |-  ( ( ( R  e. NrmRing  /\  R  e.  DivRing )  /\  ( Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  ( R  e. CUnifSp  /\  (UnifSt `  R
)  =  (metUnif `  D
) ) )  <->  ( ( R  e. NrmRing  /\  R  e.  DivRing )  /\  ( ( Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  ( R  e. CUnifSp  /\  (UnifSt `  R
)  =  (metUnif `  D
) ) ) ) )
282, 26, 273bitr4i 277 1  |-  ( R  e. ℝExt 
<->  ( ( R  e. NrmRing  /\  R  e.  DivRing )  /\  ( Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  ( R  e. CUnifSp  /\  (UnifSt `  R
)  =  (metUnif `  D
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    X. cxp 4997    |` cres 5001   ` cfv 5588   0cc0 9492   Basecbs 14490   distcds 14564   DivRingcdr 17196  metUnifcmetu 18209   ZModczlm 18333  chrcchr 18334  UnifStcuss 20519  CUnifSpccusp 20563  NrmRingcnrg 20863  NrmModcnlm 20864   ℝExt crrext 27639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-res 5011  df-iota 5551  df-fv 5596  df-rrext 27644
This theorem is referenced by:  rrextnrg  27646  rrextdrg  27647  rrextnlm  27648  rrextchr  27649  rrextcusp  27650  rrextust  27653  rerrext  27654  cnrrext  27655
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