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Theorem orngmul 26271
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 1015 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  .0.  .<_  X )
2 simp3r 1017 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  .0.  .<_  Y )
31, 2jca 532 . 2  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  (  .0.  .<_  X  /\  .0.  .<_  Y ) )
4 simp2l 1014 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  X  e.  B
)
5 simp3l 1016 . . 3  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  Y  e.  B
)
6 orngmul.0 . . . . . 6  |-  B  =  ( Base `  R
)
7 orngmul.2 . . . . . 6  |-  .0.  =  ( 0g `  R )
8 orngmul.3 . . . . . 6  |-  .x.  =  ( .r `  R )
9 orngmul.1 . . . . . 6  |-  .<_  =  ( le `  R )
106, 7, 8, 9isorng 26267 . . . . 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 ) ) ) )
1110simp3bi 1005 . . . 4  |-  ( R  e. oRing  ->  A. a  e.  B  A. b  e.  B  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a 
.x.  b ) ) )
12113ad2ant1 1009 . . 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 ) ) )
13 breq2 4296 . . . . . . 7  |-  ( a  =  X  ->  (  .0.  .<_  a  <->  .0.  .<_  X ) )
14 biidd 237 . . . . . . 7  |-  ( a  =  X  ->  (  .0.  .<_  b  <->  .0.  .<_  b ) )
1513, 14anbi12d 710 . . . . . 6  |-  ( a  =  X  ->  (
(  .0.  .<_  a  /\  .0.  .<_  b )  <->  (  .0.  .<_  X  /\  .0.  .<_  b ) ) )
16 oveq1 6098 . . . . . . 7  |-  ( a  =  X  ->  (
a  .x.  b )  =  ( X  .x.  b ) )
1716breq2d 4304 . . . . . 6  |-  ( a  =  X  ->  (  .0.  .<_  ( a  .x.  b )  <->  .0.  .<_  ( X 
.x.  b ) ) )
1815, 17imbi12d 320 . . . . 5  |-  ( a  =  X  ->  (
( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a 
.x.  b ) )  <-> 
( (  .0.  .<_  X  /\  .0.  .<_  b )  ->  .0.  .<_  ( X 
.x.  b ) ) ) )
19 biidd 237 . . . . . . 7  |-  ( b  =  Y  ->  (  .0.  .<_  X  <->  .0.  .<_  X ) )
20 breq2 4296 . . . . . . 7  |-  ( b  =  Y  ->  (  .0.  .<_  b  <->  .0.  .<_  Y ) )
2119, 20anbi12d 710 . . . . . 6  |-  ( b  =  Y  ->  (
(  .0.  .<_  X  /\  .0.  .<_  b )  <->  (  .0.  .<_  X  /\  .0.  .<_  Y ) ) )
22 oveq2 6099 . . . . . . 7  |-  ( b  =  Y  ->  ( X  .x.  b )  =  ( X  .x.  Y
) )
2322breq2d 4304 . . . . . 6  |-  ( b  =  Y  ->  (  .0.  .<_  ( X  .x.  b )  <->  .0.  .<_  ( X 
.x.  Y ) ) )
2421, 23imbi12d 320 . . . . 5  |-  ( b  =  Y  ->  (
( (  .0.  .<_  X  /\  .0.  .<_  b )  ->  .0.  .<_  ( X 
.x.  b ) )  <-> 
( (  .0.  .<_  X  /\  .0.  .<_  Y )  ->  .0.  .<_  ( X 
.x.  Y ) ) ) )
2518, 24rspc2v 3079 . . . 4  |-  ( ( 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 ) ) ) )
2625imp 429 . . 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 ) ) )
274, 5, 12, 26syl21anc 1217 . 2  |-  ( ( R  e. oRing  /\  ( X  e.  B  /\  .0.  .<_  X )  /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  ( (  .0. 
.<_  X  /\  .0.  .<_  Y )  ->  .0.  .<_  ( X 
.x.  Y ) ) )
283, 27mpd 15 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   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:  orngsqr  26272  ornglmulle  26273  orngrmulle  26274  orngmullt  26277  suborng  26283
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