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Theorem isrrext 26434
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 3544 . . 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 5696 . . . . . . 7  |-  ( r  =  R  ->  ( ZMod `  r )  =  ( ZMod `  R
) )
43eleq1d 2509 . . . . . 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 263 . . . . 5  |-  ( r  =  R  ->  (
( ZMod `  r
)  e. NrmMod  <->  Z  e. NrmMod ) )
8 fveq2 5696 . . . . . 6  |-  ( r  =  R  ->  (chr `  r )  =  (chr
`  R ) )
98eqeq1d 2451 . . . . 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 2503 . . . . 5  |-  ( r  =  R  ->  (
r  e. CUnifSp  <->  R  e. CUnifSp ) )
12 fveq2 5696 . . . . . 6  |-  ( r  =  R  ->  (UnifSt `  r )  =  (UnifSt `  R ) )
13 fveq2 5696 . . . . . . . . 9  |-  ( r  =  R  ->  ( dist `  r )  =  ( dist `  R
) )
14 fveq2 5696 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
15 isrrext.b . . . . . . . . . . 11  |-  B  =  ( Base `  R
)
1614, 15syl6eqr 2493 . . . . . . . . . 10  |-  ( r  =  R  ->  ( Base `  r )  =  B )
1716, 16xpeq12d 4870 . . . . . . . . 9  |-  ( r  =  R  ->  (
( Base `  r )  X.  ( Base `  r
) )  =  ( B  X.  B ) )
1813, 17reseq12d 5116 . . . . . . . 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 2493 . . . . . . 7  |-  ( r  =  R  ->  (
( dist `  r )  |`  ( ( Base `  r
)  X.  ( Base `  r ) ) )  =  D )
2120fveq2d 5700 . . . . . 6  |-  ( r  =  R  ->  (metUnif `  ( ( dist `  r
)  |`  ( ( Base `  r )  X.  ( Base `  r ) ) ) )  =  (metUnif `  D ) )
2212, 21eqeq12d 2457 . . . . 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 26433 . . 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 3124 . 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 969 . 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 965    = wceq 1369    e. wcel 1756    i^i cin 3332    X. cxp 4843    |` cres 4847   ` cfv 5423   0cc0 9287   Basecbs 14179   distcds 14252   DivRingcdr 16837  metUnifcmetu 17813   ZModczlm 17937  chrcchr 17938  UnifStcuss 19833  CUnifSpccusp 19877  NrmRingcnrg 20177  NrmModcnlm 20178   ℝExt crrext 26428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-res 4857  df-iota 5386  df-fv 5431  df-rrext 26433
This theorem is referenced by:  rrextnrg  26435  rrextdrg  26436  rrextnlm  26437  rrextchr  26438  rrextcusp  26439  rrextust  26442  rerrext  26443  cnrrext  26444
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