Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ornglmulle Structured version   Unicode version

Theorem ornglmulle 26224
Description: In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.)
Hypotheses
Ref Expression
ornglmullt.b  |-  B  =  ( Base `  R
)
ornglmullt.t  |-  .x.  =  ( .r `  R )
ornglmullt.0  |-  .0.  =  ( 0g `  R )
ornglmullt.1  |-  ( ph  ->  R  e. oRing )
ornglmullt.2  |-  ( ph  ->  X  e.  B )
ornglmullt.3  |-  ( ph  ->  Y  e.  B )
ornglmullt.4  |-  ( ph  ->  Z  e.  B )
orngmulle.l  |-  .<_  =  ( le `  R )
orngmulle.5  |-  ( ph  ->  X  .<_  Y )
orngmulle.6  |-  ( ph  ->  .0.  .<_  Z )
Assertion
Ref Expression
ornglmulle  |-  ( ph  ->  ( Z  .x.  X
)  .<_  ( Z  .x.  Y ) )

Proof of Theorem ornglmulle
StepHypRef Expression
1 ornglmullt.1 . . . . 5  |-  ( ph  ->  R  e. oRing )
2 orngogrp 26220 . . . . 5  |-  ( R  e. oRing  ->  R  e. oGrp )
31, 2syl 16 . . . 4  |-  ( ph  ->  R  e. oGrp )
4 isogrp 26116 . . . . 5  |-  ( R  e. oGrp 
<->  ( R  e.  Grp  /\  R  e. oMnd ) )
54simprbi 464 . . . 4  |-  ( R  e. oGrp  ->  R  e. oMnd )
63, 5syl 16 . . 3  |-  ( ph  ->  R  e. oMnd )
7 orngrng 26219 . . . . . 6  |-  ( R  e. oRing  ->  R  e.  Ring )
81, 7syl 16 . . . . 5  |-  ( ph  ->  R  e.  Ring )
9 rnggrp 16638 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
108, 9syl 16 . . . 4  |-  ( ph  ->  R  e.  Grp )
11 ornglmullt.b . . . . 5  |-  B  =  ( Base `  R
)
12 ornglmullt.0 . . . . 5  |-  .0.  =  ( 0g `  R )
1311, 12grpidcl 15557 . . . 4  |-  ( R  e.  Grp  ->  .0.  e.  B )
1410, 13syl 16 . . 3  |-  ( ph  ->  .0.  e.  B )
15 ornglmullt.4 . . . . 5  |-  ( ph  ->  Z  e.  B )
16 ornglmullt.3 . . . . 5  |-  ( ph  ->  Y  e.  B )
17 ornglmullt.t . . . . . 6  |-  .x.  =  ( .r `  R )
1811, 17rngcl 16646 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .x.  Y )  e.  B )
198, 15, 16, 18syl3anc 1218 . . . 4  |-  ( ph  ->  ( Z  .x.  Y
)  e.  B )
20 ornglmullt.2 . . . . 5  |-  ( ph  ->  X  e.  B )
2111, 17rngcl 16646 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  X  e.  B )  ->  ( Z  .x.  X )  e.  B )
228, 15, 20, 21syl3anc 1218 . . . 4  |-  ( ph  ->  ( Z  .x.  X
)  e.  B )
23 eqid 2438 . . . . 5  |-  ( -g `  R )  =  (
-g `  R )
2411, 23grpsubcl 15597 . . . 4  |-  ( ( R  e.  Grp  /\  ( Z  .x.  Y )  e.  B  /\  ( Z  .x.  X )  e.  B )  ->  (
( Z  .x.  Y
) ( -g `  R
) ( Z  .x.  X ) )  e.  B )
2510, 19, 22, 24syl3anc 1218 . . 3  |-  ( ph  ->  ( ( Z  .x.  Y ) ( -g `  R ) ( Z 
.x.  X ) )  e.  B )
26 orngmulle.6 . . . . 5  |-  ( ph  ->  .0.  .<_  Z )
2711, 23grpsubcl 15597 . . . . . 6  |-  ( ( R  e.  Grp  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y ( -g `  R ) X )  e.  B )
2810, 16, 20, 27syl3anc 1218 . . . . 5  |-  ( ph  ->  ( Y ( -g `  R ) X )  e.  B )
2911, 12, 23grpsubid 15601 . . . . . . 7  |-  ( ( R  e.  Grp  /\  X  e.  B )  ->  ( X ( -g `  R ) X )  =  .0.  )
3010, 20, 29syl2anc 661 . . . . . 6  |-  ( ph  ->  ( X ( -g `  R ) X )  =  .0.  )
31 orngmulle.5 . . . . . . 7  |-  ( ph  ->  X  .<_  Y )
32 orngmulle.l . . . . . . . 8  |-  .<_  =  ( le `  R )
3311, 32, 23ogrpsub 26131 . . . . . . 7  |-  ( ( R  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  X  e.  B )  /\  X  .<_  Y )  ->  ( X (
-g `  R ) X )  .<_  ( Y ( -g `  R
) X ) )
343, 20, 16, 20, 31, 33syl131anc 1231 . . . . . 6  |-  ( ph  ->  ( X ( -g `  R ) X ) 
.<_  ( Y ( -g `  R ) X ) )
3530, 34eqbrtrrd 4309 . . . . 5  |-  ( ph  ->  .0.  .<_  ( Y (
-g `  R ) X ) )
3611, 32, 12, 17orngmul 26222 . . . . 5  |-  ( ( R  e. oRing  /\  ( Z  e.  B  /\  .0.  .<_  Z )  /\  ( ( Y (
-g `  R ) X )  e.  B  /\  .0.  .<_  ( Y (
-g `  R ) X ) ) )  ->  .0.  .<_  ( Z 
.x.  ( Y (
-g `  R ) X ) ) )
371, 15, 26, 28, 35, 36syl122anc 1227 . . . 4  |-  ( ph  ->  .0.  .<_  ( Z  .x.  ( Y ( -g `  R
) X ) ) )
3811, 17, 23, 8, 15, 16, 20rngsubdi 16678 . . . 4  |-  ( ph  ->  ( Z  .x.  ( Y ( -g `  R
) X ) )  =  ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) )
3937, 38breqtrd 4311 . . 3  |-  ( ph  ->  .0.  .<_  ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) )
40 eqid 2438 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4111, 32, 40omndadd 26120 . . 3  |-  ( ( R  e. oMnd  /\  (  .0.  e.  B  /\  (
( Z  .x.  Y
) ( -g `  R
) ( Z  .x.  X ) )  e.  B  /\  ( Z 
.x.  X )  e.  B )  /\  .0.  .<_  ( ( Z  .x.  Y ) ( -g `  R ) ( Z 
.x.  X ) ) )  ->  (  .0.  ( +g  `  R ) ( Z  .x.  X
) )  .<_  ( ( ( Z  .x.  Y
) ( -g `  R
) ( Z  .x.  X ) ) ( +g  `  R ) ( Z  .x.  X
) ) )
426, 14, 25, 22, 39, 41syl131anc 1231 . 2  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( Z 
.x.  X ) ) 
.<_  ( ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) ( +g  `  R ) ( Z 
.x.  X ) ) )
4311, 40, 12grplid 15559 . . 3  |-  ( ( R  e.  Grp  /\  ( Z  .x.  X )  e.  B )  -> 
(  .0.  ( +g  `  R ) ( Z 
.x.  X ) )  =  ( Z  .x.  X ) )
4410, 22, 43syl2anc 661 . 2  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( Z 
.x.  X ) )  =  ( Z  .x.  X ) )
4511, 40, 23grpnpcan 15608 . . 3  |-  ( ( R  e.  Grp  /\  ( Z  .x.  Y )  e.  B  /\  ( Z  .x.  X )  e.  B )  ->  (
( ( Z  .x.  Y ) ( -g `  R ) ( Z 
.x.  X ) ) ( +g  `  R
) ( Z  .x.  X ) )  =  ( Z  .x.  Y
) )
4610, 19, 22, 45syl3anc 1218 . 2  |-  ( ph  ->  ( ( ( Z 
.x.  Y ) (
-g `  R )
( Z  .x.  X
) ) ( +g  `  R ) ( Z 
.x.  X ) )  =  ( Z  .x.  Y ) )
4742, 44, 463brtr3d 4316 1  |-  ( ph  ->  ( Z  .x.  X
)  .<_  ( Z  .x.  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   .rcmulr 14231   lecple 14237   0gc0g 14370   Grpcgrp 15402   -gcsg 15405   Ringcrg 16633  oMndcomnd 26111  oGrpcogrp 26112  oRingcorng 26214
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mgp 16580  df-ur 16592  df-rng 16635  df-omnd 26113  df-ogrp 26114  df-orng 26216
This theorem is referenced by:  ornglmullt  26226
  Copyright terms: Public domain W3C validator