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Theorem isrrext 28643
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 3655 . . 3  |-  ( R  e.  (NrmRing  i^i  DivRing )  <->  ( R  e. NrmRing  /\  R  e.  DivRing ) )
21anbi1i 699 . 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 5881 . . . . . . 7  |-  ( r  =  R  ->  ( ZMod `  r )  =  ( ZMod `  R
) )
43eleq1d 2498 . . . . . 6  |-  ( r  =  R  ->  (
( ZMod `  r
)  e. NrmMod  <->  ( ZMod `  R )  e. NrmMod )
)
5 isrrext.z . . . . . . 7  |-  Z  =  ( ZMod `  R
)
65eleq1i 2506 . . . . . 6  |-  ( Z  e. NrmMod 
<->  ( ZMod `  R
)  e. NrmMod )
74, 6syl6bbr 266 . . . . 5  |-  ( r  =  R  ->  (
( ZMod `  r
)  e. NrmMod  <->  Z  e. NrmMod ) )
8 fveq2 5881 . . . . . 6  |-  ( r  =  R  ->  (chr `  r )  =  (chr
`  R ) )
98eqeq1d 2431 . . . . 5  |-  ( r  =  R  ->  (
(chr `  r )  =  0  <->  (chr `  R
)  =  0 ) )
107, 9anbi12d 715 . . . 4  |-  ( r  =  R  ->  (
( ( ZMod `  r )  e. NrmMod  /\  (chr `  r )  =  0 )  <->  ( Z  e. NrmMod  /\  (chr `  R )  =  0 ) ) )
11 eleq1 2501 . . . . 5  |-  ( r  =  R  ->  (
r  e. CUnifSp  <->  R  e. CUnifSp ) )
12 fveq2 5881 . . . . . 6  |-  ( r  =  R  ->  (UnifSt `  r )  =  (UnifSt `  R ) )
13 fveq2 5881 . . . . . . . . 9  |-  ( r  =  R  ->  ( dist `  r )  =  ( dist `  R
) )
14 fveq2 5881 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
15 isrrext.b . . . . . . . . . . 11  |-  B  =  ( Base `  R
)
1614, 15syl6eqr 2488 . . . . . . . . . 10  |-  ( r  =  R  ->  ( Base `  r )  =  B )
1716sqxpeqd 4880 . . . . . . . . 9  |-  ( r  =  R  ->  (
( Base `  r )  X.  ( Base `  r
) )  =  ( B  X.  B ) )
1813, 17reseq12d 5126 . . . . . . . 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 2488 . . . . . . 7  |-  ( r  =  R  ->  (
( dist `  r )  |`  ( ( Base `  r
)  X.  ( Base `  r ) ) )  =  D )
2120fveq2d 5885 . . . . . 6  |-  ( r  =  R  ->  (metUnif `  ( ( dist `  r
)  |`  ( ( Base `  r )  X.  ( Base `  r ) ) ) )  =  (metUnif `  D ) )
2212, 21eqeq12d 2451 . . . . 5  |-  ( r  =  R  ->  (
(UnifSt `  r )  =  (metUnif `  ( ( dist `  r )  |`  ( ( Base `  r
)  X.  ( Base `  r ) ) ) )  <->  (UnifSt `  R )  =  (metUnif `  D )
) )
2311, 22anbi12d 715 . . . 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 715 . . 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 28642 . . 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 3237 . 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 986 . 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 280 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    i^i cin 3441    X. cxp 4852    |` cres 4856   ` cfv 5601   0cc0 9538   Basecbs 15084   distcds 15161   DivRingcdr 17910  metUnifcmetu 18896   ZModczlm 19003  chrcchr 19004  UnifStcuss 21199  CUnifSpccusp 21243  NrmRingcnrg 21525  NrmModcnlm 21526   ℝExt crrext 28637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-xp 4860  df-res 4866  df-iota 5565  df-fv 5609  df-rrext 28642
This theorem is referenced by:  rrextnrg  28644  rrextdrg  28645  rrextnlm  28646  rrextchr  28647  rrextcusp  28648  rrextust  28651  rerrext  28652  cnrrext  28653
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