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Theorem orngmul 28232
Description: In an ordered ring, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Hypotheses
Ref Expression
orngmul.0  |-  B  =  ( Base `  R
)
orngmul.1  |-  .<_  =  ( le `  R )
orngmul.2  |-  .0.  =  ( 0g `  R )
orngmul.3  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
orngmul  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  .0.  .<_  ( X 
.x.  Y ) )

Proof of Theorem orngmul
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2r 1024 . 2  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  .0.  .<_  X )
2 simp3r 1026 . 2  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  .0.  .<_  Y )
3 simp2l 1023 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  X  e.  B
)
4 simp3l 1025 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  Y  e.  B
)
5 orngmul.0 . . . . . 6  |-  B  =  ( Base `  R
)
6 orngmul.2 . . . . . 6  |-  .0.  =  ( 0g `  R )
7 orngmul.3 . . . . . 6  |-  .x.  =  ( .r `  R )
8 orngmul.1 . . . . . 6  |-  .<_  =  ( le `  R )
95, 6, 7, 8isorng 28228 . . . . 5  |-  ( 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 ) ) ) )
109simp3bi 1014 . . . 4  |-  ( R  e. oRing  ->  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a 
.x.  b ) ) )
11103ad2ant1 1018 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a 
.x.  b ) ) )
12 breq2 4398 . . . . . 6  |-  ( a  =  X  ->  (  .0.  .<_  a  <->  .0.  .<_  X ) )
1312anbi1d 703 . . . . 5  |-  ( a  =  X  ->  (
(  .0.  .<_  a  /\  .0.  .<_  b )  <->  (  .0.  .<_  X  /\  .0.  .<_  b ) ) )
14 oveq1 6284 . . . . . 6  |-  ( a  =  X  ->  (
a  .x.  b )  =  ( X  .x.  b ) )
1514breq2d 4406 . . . . 5  |-  ( a  =  X  ->  (  .0.  .<_  ( a  .x.  b )  <->  .0.  .<_  ( X 
.x.  b ) ) )
1613, 15imbi12d 318 . . . 4  |-  ( a  =  X  ->  (
( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a 
.x.  b ) )  <-> 
( (  .0.  .<_  X  /\  .0.  .<_  b )  ->  .0.  .<_  ( X 
.x.  b ) ) ) )
17 breq2 4398 . . . . . 6  |-  ( b  =  Y  ->  (  .0.  .<_  b  <->  .0.  .<_  Y ) )
1817anbi2d 702 . . . . 5  |-  ( b  =  Y  ->  (
(  .0.  .<_  X  /\  .0.  .<_  b )  <->  (  .0.  .<_  X  /\  .0.  .<_  Y ) ) )
19 oveq2 6285 . . . . . 6  |-  ( b  =  Y  ->  ( X  .x.  b )  =  ( X  .x.  Y
) )
2019breq2d 4406 . . . . 5  |-  ( b  =  Y  ->  (  .0.  .<_  ( X  .x.  b )  <->  .0.  .<_  ( X 
.x.  Y ) ) )
2118, 20imbi12d 318 . . . 4  |-  ( b  =  Y  ->  (
( (  .0.  .<_  X  /\  .0.  .<_  b )  ->  .0.  .<_  ( X 
.x.  b ) )  <-> 
( (  .0.  .<_  X  /\  .0.  .<_  Y )  ->  .0.  .<_  ( X 
.x.  Y ) ) ) )
2216, 21rspc2va 3169 . . 3  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b
) ) )  -> 
( (  .0.  .<_  X  /\  .0.  .<_  Y )  ->  .0.  .<_  ( X 
.x.  Y ) ) )
233, 4, 11, 22syl21anc 1229 . 2  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  ( (  .0. 
.<_  X  /\  .0.  .<_  Y )  ->  .0.  .<_  ( X 
.x.  Y ) ) )
241, 2, 23mp2and 677 1  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  .0.  .<_  ( X 
.x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   .rcmulr 14908   lecple 14914   0gc0g 15052   Ringcrg 17516  oGrpcogrp 28126  oRingcorng 28224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-orng 28226
This theorem is referenced by:  orngsqr  28233  ornglmulle  28234  orngrmulle  28235  orngmullt  28238  suborng  28244
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