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Theorem isorng 26267
Description: An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
Hypotheses
Ref Expression
isorng.0  |-  B  =  ( Base `  R
)
isorng.1  |-  .0.  =  ( 0g `  R )
isorng.2  |-  .x.  =  ( .r `  R )
isorng.3  |-  .<_  =  ( le `  R )
Assertion
Ref Expression
isorng  |-  ( R  e. oRing 
<->  ( R  e.  Ring  /\  R  e. oGrp  /\  A. a  e.  B  A. b  e.  B  (
(  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b ) ) ) )
Distinct variable groups:    a, b, B    R, a, b
Allowed substitution hints:    .x. ( a, b)    .<_ ( a, b)    .0. ( a,
b)

Proof of Theorem isorng
Dummy variables  l 
r  t  v  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3539 . . 3  |-  ( R  e.  ( Ring  i^i oGrp )  <-> 
( R  e.  Ring  /\  R  e. oGrp ) )
21anbi1i 695 . 2  |-  ( ( R  e.  ( Ring 
i^i oGrp )  /\  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b
) ) )  <->  ( ( R  e.  Ring  /\  R  e. oGrp )  /\  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b
) ) ) )
3 fvex 5701 . . . . . 6  |-  ( .r
`  r )  e. 
_V
43a1i 11 . . . . 5  |-  ( r  =  R  ->  ( .r `  r )  e. 
_V )
5 simpr 461 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
t  =  ( .r
`  r ) )
6 simpl 457 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
r  =  R )
76fveq2d 5695 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
( .r `  r
)  =  ( .r
`  R ) )
8 isorng.2 . . . . . . . . . . . 12  |-  .x.  =  ( .r `  R )
97, 8syl6eqr 2493 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
( .r `  r
)  =  .x.  )
105, 9eqtrd 2475 . . . . . . . . . 10  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
t  =  .x.  )
1110oveqd 6108 . . . . . . . . 9  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
( a t b )  =  ( a 
.x.  b ) )
1211breq2d 4304 . . . . . . . 8  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
(  .0.  l ( a t b )  <-> 
.0.  l ( a 
.x.  b ) ) )
1312imbi2d 316 . . . . . . 7  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
( ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a t b ) )  <->  ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a  .x.  b ) ) ) )
14132ralbidv 2757 . . . . . 6  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
( A. a  e.  B  A. b  e.  B  ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a t b ) )  <->  A. a  e.  B  A. b  e.  B  ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a  .x.  b ) ) ) )
1514sbcbidv 3245 . . . . 5  |-  ( ( r  =  R  /\  t  =  ( .r `  r ) )  -> 
( [. ( le `  r )  /  l ]. A. a  e.  B  A. b  e.  B  ( (  .0.  l
a  /\  .0.  l
b )  ->  .0.  l ( a t b ) )  <->  [. ( le
`  r )  / 
l ]. A. a  e.  B  A. b  e.  B  ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a  .x.  b ) ) ) )
164, 15sbcied 3223 . . . 4  |-  ( r  =  R  ->  ( [. ( .r `  r
)  /  t ]. [. ( le `  r
)  /  l ]. A. a  e.  B  A. b  e.  B  ( (  .0.  l
a  /\  .0.  l
b )  ->  .0.  l ( a t b ) )  <->  [. ( le
`  r )  / 
l ]. A. a  e.  B  A. b  e.  B  ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a  .x.  b ) ) ) )
17 fvex 5701 . . . . . . 7  |-  ( Base `  r )  e.  _V
1817a1i 11 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  e. 
_V )
19 simpr 461 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  v  =  ( Base `  r ) )  -> 
v  =  ( Base `  r ) )
20 fveq2 5691 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
21 isorng.0 . . . . . . . . . . . . 13  |-  B  =  ( Base `  R
)
2220, 21syl6eqr 2493 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( Base `  r )  =  B )
2322adantr 465 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  v  =  ( Base `  r ) )  -> 
( Base `  r )  =  B )
2419, 23eqtrd 2475 . . . . . . . . . 10  |-  ( ( r  =  R  /\  v  =  ( Base `  r ) )  -> 
v  =  B )
25 raleq 2917 . . . . . . . . . . 11  |-  ( v  =  B  ->  ( A. b  e.  v 
( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  A. b  e.  B  ( (
z l a  /\  z l b )  ->  z l ( a t b ) ) ) )
2625raleqbi1dv 2925 . . . . . . . . . 10  |-  ( v  =  B  ->  ( A. a  e.  v  A. b  e.  v 
( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  A. a  e.  B  A. b  e.  B  ( (
z l a  /\  z l b )  ->  z l ( a t b ) ) ) )
2724, 26syl 16 . . . . . . . . 9  |-  ( ( r  =  R  /\  v  =  ( Base `  r ) )  -> 
( A. a  e.  v  A. b  e.  v  ( ( z l a  /\  z
l b )  -> 
z l ( a t b ) )  <->  A. a  e.  B  A. b  e.  B  ( ( z l a  /\  z l b )  ->  z
l ( a t b ) ) ) )
2827sbcbidv 3245 . . . . . . . 8  |-  ( ( r  =  R  /\  v  =  ( Base `  r ) )  -> 
( [. ( le `  r )  /  l ]. A. a  e.  v 
A. b  e.  v  ( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  [. ( le
`  r )  / 
l ]. A. a  e.  B  A. b  e.  B  ( ( z l a  /\  z
l b )  -> 
z l ( a t b ) ) ) )
2928sbcbidv 3245 . . . . . . 7  |-  ( ( r  =  R  /\  v  =  ( Base `  r ) )  -> 
( [. ( .r `  r )  /  t ]. [. ( le `  r )  /  l ]. A. a  e.  v 
A. b  e.  v  ( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  [. ( .r
`  r )  / 
t ]. [. ( le
`  r )  / 
l ]. A. a  e.  B  A. b  e.  B  ( ( z l a  /\  z
l b )  -> 
z l ( a t b ) ) ) )
3029sbcbidv 3245 . . . . . 6  |-  ( ( r  =  R  /\  v  =  ( Base `  r ) )  -> 
( [. ( 0g `  r )  /  z ]. [. ( .r `  r )  /  t ]. [. ( le `  r )  /  l ]. A. a  e.  v 
A. b  e.  v  ( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  [. ( 0g
`  r )  / 
z ]. [. ( .r
`  r )  / 
t ]. [. ( le
`  r )  / 
l ]. A. a  e.  B  A. b  e.  B  ( ( z l a  /\  z
l b )  -> 
z l ( a t b ) ) ) )
3118, 30sbcied 3223 . . . . 5  |-  ( r  =  R  ->  ( [. ( Base `  r
)  /  v ]. [. ( 0g `  r
)  /  z ]. [. ( .r `  r
)  /  t ]. [. ( le `  r
)  /  l ]. A. a  e.  v  A. b  e.  v 
( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  [. ( 0g
`  r )  / 
z ]. [. ( .r
`  r )  / 
t ]. [. ( le
`  r )  / 
l ]. A. a  e.  B  A. b  e.  B  ( ( z l a  /\  z
l b )  -> 
z l ( a t b ) ) ) )
32 fvex 5701 . . . . . . 7  |-  ( 0g
`  r )  e. 
_V
3332a1i 11 . . . . . 6  |-  ( r  =  R  ->  ( 0g `  r )  e. 
_V )
34 simpr 461 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
z  =  ( 0g
`  r ) )
35 fveq2 5691 . . . . . . . . . . . . . . 15  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
36 isorng.1 . . . . . . . . . . . . . . 15  |-  .0.  =  ( 0g `  R )
3735, 36syl6eqr 2493 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
3837adantr 465 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
( 0g `  r
)  =  .0.  )
3934, 38eqtrd 2475 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
z  =  .0.  )
4039breq1d 4302 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
( z l a  <-> 
.0.  l a ) )
4139breq1d 4302 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
( z l b  <-> 
.0.  l b ) )
4240, 41anbi12d 710 . . . . . . . . . 10  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
( ( z l a  /\  z l b )  <->  (  .0.  l a  /\  .0.  l b ) ) )
4339breq1d 4302 . . . . . . . . . 10  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
( z l ( a t b )  <-> 
.0.  l ( a t b ) ) )
4442, 43imbi12d 320 . . . . . . . . 9  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
( ( ( z l a  /\  z
l b )  -> 
z l ( a t b ) )  <-> 
( (  .0.  l
a  /\  .0.  l
b )  ->  .0.  l ( a t b ) ) ) )
45442ralbidv 2757 . . . . . . . 8  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
( A. a  e.  B  A. b  e.  B  ( ( z l a  /\  z
l b )  -> 
z l ( a t b ) )  <->  A. a  e.  B  A. b  e.  B  ( (  .0.  l
a  /\  .0.  l
b )  ->  .0.  l ( a t b ) ) ) )
4645sbcbidv 3245 . . . . . . 7  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
( [. ( le `  r )  /  l ]. A. a  e.  B  A. b  e.  B  ( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  [. ( le
`  r )  / 
l ]. A. a  e.  B  A. b  e.  B  ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a t b ) ) ) )
4746sbcbidv 3245 . . . . . 6  |-  ( ( r  =  R  /\  z  =  ( 0g `  r ) )  -> 
( [. ( .r `  r )  /  t ]. [. ( le `  r )  /  l ]. A. a  e.  B  A. b  e.  B  ( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  [. ( .r
`  r )  / 
t ]. [. ( le
`  r )  / 
l ]. A. a  e.  B  A. b  e.  B  ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a t b ) ) ) )
4833, 47sbcied 3223 . . . . 5  |-  ( r  =  R  ->  ( [. ( 0g `  r
)  /  z ]. [. ( .r `  r
)  /  t ]. [. ( le `  r
)  /  l ]. A. a  e.  B  A. b  e.  B  ( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  [. ( .r
`  r )  / 
t ]. [. ( le
`  r )  / 
l ]. A. a  e.  B  A. b  e.  B  ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a t b ) ) ) )
4931, 48bitr2d 254 . . . 4  |-  ( r  =  R  ->  ( [. ( .r `  r
)  /  t ]. [. ( le `  r
)  /  l ]. A. a  e.  B  A. b  e.  B  ( (  .0.  l
a  /\  .0.  l
b )  ->  .0.  l ( a t b ) )  <->  [. ( Base `  r )  /  v ]. [. ( 0g `  r )  /  z ]. [. ( .r `  r )  /  t ]. [. ( le `  r )  /  l ]. A. a  e.  v 
A. b  e.  v  ( ( z l a  /\  z l b )  ->  z
l ( a t b ) ) ) )
50 fvex 5701 . . . . . 6  |-  ( le
`  r )  e. 
_V
5150a1i 11 . . . . 5  |-  ( r  =  R  ->  ( le `  r )  e. 
_V )
52 simpr 461 . . . . . . . . . 10  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
l  =  ( le
`  r ) )
53 simpl 457 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
r  =  R )
5453fveq2d 5695 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
( le `  r
)  =  ( le
`  R ) )
55 isorng.3 . . . . . . . . . . 11  |-  .<_  =  ( le `  R )
5654, 55syl6eqr 2493 . . . . . . . . . 10  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
( le `  r
)  =  .<_  )
5752, 56eqtrd 2475 . . . . . . . . 9  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
l  =  .<_  )
5857breqd 4303 . . . . . . . 8  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
(  .0.  l a  <-> 
.0.  .<_  a ) )
5957breqd 4303 . . . . . . . 8  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
(  .0.  l b  <-> 
.0.  .<_  b ) )
6058, 59anbi12d 710 . . . . . . 7  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
( (  .0.  l
a  /\  .0.  l
b )  <->  (  .0.  .<_  a  /\  .0.  .<_  b ) ) )
6157breqd 4303 . . . . . . 7  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
(  .0.  l ( a  .x.  b )  <-> 
.0.  .<_  ( a  .x.  b ) ) )
6260, 61imbi12d 320 . . . . . 6  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
( ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a  .x.  b ) )  <->  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b
) ) ) )
63622ralbidv 2757 . . . . 5  |-  ( ( r  =  R  /\  l  =  ( le `  r ) )  -> 
( A. a  e.  B  A. b  e.  B  ( (  .0.  l a  /\  .0.  l b )  ->  .0.  l ( a  .x.  b ) )  <->  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b
) ) ) )
6451, 63sbcied 3223 . . . 4  |-  ( r  =  R  ->  ( [. ( le `  r
)  /  l ]. A. a  e.  B  A. b  e.  B  ( (  .0.  l
a  /\  .0.  l
b )  ->  .0.  l ( a  .x.  b ) )  <->  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b
) ) ) )
6516, 49, 643bitr3d 283 . . 3  |-  ( r  =  R  ->  ( [. ( Base `  r
)  /  v ]. [. ( 0g `  r
)  /  z ]. [. ( .r `  r
)  /  t ]. [. ( le `  r
)  /  l ]. A. a  e.  v  A. b  e.  v 
( ( z l a  /\  z l b )  ->  z
l ( a t b ) )  <->  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b
) ) ) )
66 df-orng 26265 . . 3  |- oRing  =  {
r  e.  ( Ring 
i^i oGrp )  |  [. ( Base `  r )  / 
v ]. [. ( 0g
`  r )  / 
z ]. [. ( .r
`  r )  / 
t ]. [. ( le
`  r )  / 
l ]. A. a  e.  v  A. b  e.  v  ( ( z l a  /\  z
l b )  -> 
z l ( a t b ) ) }
6765, 66elrab2 3119 . 2  |-  ( R  e. oRing 
<->  ( R  e.  (
Ring  i^i oGrp )  /\  A. a  e.  B  A. b  e.  B  (
(  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b ) ) ) )
68 df-3an 967 . 2  |-  ( ( R  e.  Ring  /\  R  e. oGrp  /\  A. a  e.  B  A. b  e.  B  ( (  .0. 
.<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a 
.x.  b ) ) )  <->  ( ( R  e.  Ring  /\  R  e. oGrp
)  /\  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b
) ) ) )
692, 67, 683bitr4i 277 1  |-  ( R  e. oRing 
<->  ( R  e.  Ring  /\  R  e. oGrp  /\  A. a  e.  B  A. b  e.  B  (
(  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972   [.wsbc 3186    i^i cin 3327   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   .rcmulr 14239   lecple 14245   0gc0g 14378   Ringcrg 16645  oGrpcogrp 26161  oRingcorng 26263
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  ax-nul 4421
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-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-orng 26265
This theorem is referenced by:  orngrng  26268  orngogrp  26269  orngmul  26271  suborng  26283  reofld  26308
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